Welcome to the documentation for the Axelrod Python library¶
Here is quick overview of the current capabilities of the library:
Over 230 strategies including many from the literature and exciting original contributions
 Classic strategies like TiTForTat, WSLS, and variants
 ZeroDeterminant and other MemoryOne strategies
 Many generic strategies that can be used to define an array of popular strategies, including finite state machines, strategies that hunt for patterns in other strategies, and strategies that combine the effects of many others
 Strategy transformers that augment the abilities of any strategy
HeadtoHead matches
Round Robin tournaments with a variety of options, including:
 noisy environments
 spatial tournaments
 probabilistically chosen match lengths
Population dynamics
 The Moran process
 An ecological model
Multiprocessor support (not currently supported on Windows), caching for deterministic interactions, automatically generate figures and statistics
Every strategy is categorized on a number of dimensions, including:
 Deterministic or Stochastic
 How many rounds of history used
 Whether the strategy makes use of the game matrix, the length of the match, etc.
Furthermore the library is extensively tested with 100% coverage, ensuring validity and reproducibility of results!
Quick start¶
Count the number of available players:
>>> import axelrod as axl
>>> len(axl.strategies)
238
Create matches between two players:
>>> import axelrod as axl
>>> players = (axl.Alternator(), axl.TitForTat())
>>> match = axl.Match(players, 5)
>>> interactions = match.play()
>>> interactions
[(C, C), (D, C), (C, D), (D, C), (C, D)]
Build full tournaments between groups of players:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator(), axl.TitForTat())
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
>>> results.ranked_names
['Alternator', 'Tit For Tat', 'Cooperator']
Study the evolutionary process using a Moran process:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator(), axl.TitForTat())
>>> mp = axl.MoranProcess(players)
>>> populations = mp.play()
>>> populations
[Counter({'Alternator': 1, 'Cooperator': 1, 'Tit For Tat': 1}),
Counter({'Alternator': 1, 'Cooperator': 1, 'Tit For Tat': 1}),
Counter({'Cooperator': 1, 'Tit For Tat': 2}),
Counter({'Cooperator': 1, 'Tit For Tat': 2}),
Counter({'Tit For Tat': 3})]
As well as this, the library has a growing collection of strategies. The Strategy index gives a description of them.
For further details there is a library of Tutorials available and a Community page with information about how to get support and/or make contributions.
Table of Contents¶
Tutorials¶
This section contains a variety of tutorials related to the Axelrod library.
Contents:
New to Game Theory and/or Python¶
This section contains a variety of tutorials that should help get you started with the Axelrod library.
Contents:
Installation¶
The library requires Python 3.5.
The simplest way to install the package is to obtain it from the PyPi repository:
$ pip install axelrod
You can also build it from source if you would like to:
$ git clone https://github.com/AxelrodPython/Axelrod.git
$ cd Axelrod
$ python setup.py install
Creating Matches¶
You can create your own match between two players using the Match
class.
This is often useful when designing new strategies in order to study how they
perform against specific opponents.
For example, to create a 5 turn match between Cooperator
and
Alternator
:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator())
>>> match = axl.Match(players, 5)
>>> match.play()
[(C, C), (C, D), (C, C), (C, D), (C, C)]
By default, a match will not be noisy, but you can introduce noise if you wish. Noise is the probability with which any action dictated by a strategy will be swapped:
>>> match = axl.Match(players=players, turns=5, noise=0.2)
>>> match.play()
[(D, C), (C, D), (C, C), (C, D), (D, D)]
The result of the match is held as an attribute within the Match
class.
Each time play()
is called, it will overwrite the content of that
attribute:
>>> match.result
[(D, C), (C, D), (C, C), (C, D), (D, D)]
>>> match.play()
[(C, C), (C, C), (C, D), (C, C), (C, C)]
>>> match.result
[(C, C), (C, C), (C, D), (C, C), (C, C)]
The result of the match can also be viewed as sparklines where cooperation is shown as a solid block and defection as a space. Sparklines are a very concise way to view the result and can be useful for spotting patterns:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator())
>>> match = axl.Match(players, 25)
>>> match.play()
[(C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C)]
>>> print(match.sparklines())
█████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █
The █ character for cooperation and a space for defection are default values but you can use any characters you like:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator())
>>> match = axl.Match(players, 25)
>>> match.play()
[(C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C)]
>>> print(match.sparklines(c_symbol='', d_symbol=''))


A Match class can also score the individual turns of a match. Just call match.scores() after play:
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Alternator())
>>> match = axl.Match(players, 25)
>>> match.play()
[(C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C), (C, D), (C, C)]
>>> match.scores()
[(3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3), (0, 5), (3, 3)]
There are various further methods:
>>> match.final_score()
(39, 99)
>>> match.final_score_per_turn()
(1.56, 3.96)
>>> match.winner()
Alternator
>>> match.cooperation() # The count of cooperations
(25, 13)
>>> match.normalised_cooperation() # The count of cooperations per turn
(1.0, 0.52)
Creating and running a simple tournament¶
The following lines of code creates a list players playing simple strategies:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> players
[Cooperator, Defector, Tit For Tat, Grudger]
We can now create a tournament, play it, save the results and view the rank of each player:
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
>>> results.ranked_names
['Defector', 'Tit For Tat', 'Grudger', 'Cooperator']
We can also plot these results:
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
Note that in this case none of our strategies are stochastic so the boxplot shows that there is no variation. Take a look at the Visualising results section to see plots showing a stochastic effect.
Summarising tournament results¶
As shown in Creating and running a simple tournament let us create a tournament:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> tournament = axl.Tournament(players, turns=10, repetitions=3)
>>> results = tournament.play()
The results set can return a list of named tuples, ordered by strategy rank that summarises the results of the tournament:
>>> summary = results.summarise()
>>> import pprint
>>> pprint.pprint(summary)
[Player(Rank=0, Name='Defector', Median_score=2.6..., Cooperation_rating=0.0, Wins=3.0, Initial_C_rate=0.0, CC_rate=...),
Player(Rank=1, Name='Tit For Tat', Median_score=2.3..., Cooperation_rating=0..., Wins=0.0, Initial_C_rate=1.0, CC_rate=...),
Player(Rank=2, Name='Grudger', Median_score=2.3..., Cooperation_rating=0..., Wins=0.0, Initial_C_rate=1.0, CC_rate=...),
Player(Rank=3, Name='Cooperator', Median_score=2.0..., Cooperation_rating=1.0, Wins=0.0, Initial_C_rate=1.0, CC_rate=...)]
It is also possible to write this data directly to a csv file using the write_summary method:
>>> results.write_summary('summary.csv')
>>> import csv
>>> with open('summary.csv', 'r') as outfile:
... csvreader = csv.reader(outfile)
... for row in csvreader:
... print(row)
['Rank', 'Name', 'Median_score', 'Cooperation_rating', 'Wins', 'Initial_C_rate', 'CC_rate', 'CD_rate', 'DC_rate', 'DD_rate', 'CC_to_C_rate', 'CD_to_C_rate', 'DC_to_C_rate', 'DD_to_C_rate']
['0', 'Defector', '2.6...', '0.0', '3.0', '0.0', '0.0', '0.0', '0.4...', '0.6...', '0', '0', '0', '0']
['1', 'Tit For Tat', '2.3...', '0.7', '0.0', '1.0', '0.66...', '0.03...', '0.0', '0.3...', '1.0', '0', '0', '0']
['2', 'Grudger', '2.3...', '0.7', '0.0', '1.0', '0.66...', '0.03...', '0.0', '0.3...', '1.0', '0', '0', '0']
['3', 'Cooperator', '2.0...', '1.0', '0.0', '1.0', '0.66...', '0.33...', '0.0', '0.0', '1.0', '1.0', '0', '0']
The result set class computes a large number of detailed outcomes read about those in Access tournament results.
Visualising results¶
This tutorial will show you briefly how to visualise some basic results
Visualising the results of the tournament¶
As shown in Creating and running a simple tournament, let us create a tournament, but this time we will include a player that acts randomly:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> players.append(axl.Random())
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
We can view these results (which helps visualise the stochastic effects):
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
Visualising the distributions of wins¶
We can view the distributions of wins for each strategy:
>>> p = plot.winplot()
>>> p.show()
Visualising the payoff matrix¶
We can also easily view the payoff matrix described in Access tournament results, this becomes particularly useful when viewing the outputs of tournaments with a large number of strategies:
>>> p = plot.payoff()
>>> p.show()
Saving all plots¶
The axelrod.Plot
class has a method: save_all_plots
that will
save all the above plots to file.
Passing various objects to plot¶
The library give access to underlying matplotlib axes objects of each plot, thus the user can easily modify various aspects of a plot:
>>> import matplotlib.pyplot as plt
>>> _, ax = plt.subplots()
>>> title = ax.set_title('Payoff')
>>> xlabel = ax.set_xlabel('Strategies')
>>> p = plot.boxplot(ax=ax)
>>> p.show()
Moran Process¶
The strategies in the library can be pitted against one another in the Moran process, a population process simulating natural selection.
The process works as follows. Given an initial population of players, the population is iterated in rounds consisting of:
 matches played between each pair of players, with the cumulative total scores recorded
 a player is chosen to reproduce proportional to the player’s score in the round
 a player is chosen at random to be replaced
The process proceeds in rounds until the population consists of a single player type. That type is declared the winner. To run an instance of the process with the library, proceed as follows:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> mp = axl.MoranProcess(players, seed=1)
>>> populations = mp.play()
>>> mp.winning_strategy_name
'Tit For Tat'
You can access some attributes of the process, such as the number of rounds:
>>> len(mp)
15
The sequence of populations:
>>> import pprint
>>> pprint.pprint(populations)
[Counter({'Defector': 1, 'Tit For Tat': 1, 'Grudger': 1, 'Cooperator': 1}),
Counter({'Defector': 1, 'Tit For Tat': 1, 'Grudger': 1, 'Cooperator': 1}),
Counter({'Cooperator': 2, 'Defector': 1, 'Tit For Tat': 1}),
Counter({'Defector': 2, 'Cooperator': 2}),
Counter({'Cooperator': 3, 'Defector': 1}),
Counter({'Cooperator': 3, 'Defector': 1}),
Counter({'Defector': 2, 'Cooperator': 2}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 3, 'Cooperator': 1}),
Counter({'Defector': 4})]
The scores in each round:
>>> for row in mp.score_history:
... print([round(element, 1) for element in row])
[6.0, 7.0, 7.0, 7.0]
[7.0, 3.1, 7.0, 7.0]
[7.0, 3.1, 7.0, 7.0]
[7.0, 3.1, 7.0, 7.0]
[7.0, 3.1, 7.0, 7.0]
[3.0, 3.0, 5.0, 5.0]
[3.0, 3.0, 5.0, 5.0]
[3.1, 7.0, 7.0, 7.0]
[3.1, 7.0, 7.0, 7.0]
[9.0, 9.0, 9.0, 9.0]
[9.0, 9.0, 9.0, 9.0]
[9.0, 9.0, 9.0, 9.0]
[9.0, 9.0, 9.0, 9.0]
[9.0, 9.0, 9.0, 9.0]
We can plot the results of a Moran process with mp.populations_plot(). Let’s use a larger population to get a bit more data:
>>> import random
>>> import matplotlib.pyplot as plt
>>> players = [axl.Defector(), axl.Defector(), axl.Defector(),
... axl.Cooperator(), axl.Cooperator(), axl.Cooperator(),
... axl.TitForTat(), axl.TitForTat(), axl.TitForTat(),
... axl.Random()]
>>> mp = axl.MoranProcess(players=players, turns=200, seed=2)
>>> populations = mp.play()
>>> mp.winning_strategy_name
'Tit For Tat'
>>> ax = mp.populations_plot()
>>> plt.show()
Moran Process with Mutation¶
The MoranProcess
class also accepts an argument for a mutation rate.
Nonzero mutation changes the Markov process so that it no longer has absorbing
states, and will iterate forever. To prevent this, iterate with a loop (or
function like takewhile
from itertools
):
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> mp = axl.MoranProcess(players, mutation_rate=0.1, seed=10)
>>> for _ in mp:
... if len(mp.population_distribution()) == 1:
... break
>>> mp.population_distribution()
Counter({'Defector': 4})
It is possible to pass a fitness function that scales the utility values. A common one used in the literature, [Ohtsuki2006], is \(f(s) = 1  w + ws\) where \(w\) denotes the intensity of selection:
>>> players = (axl.Cooperator(), axl.Defector(), axl.Defector(), axl.Defector())
>>> w = 0.95
>>> fitness_transformation = lambda score: 1  w + w * score
>>> mp = axl.MoranProcess(players, turns=10, fitness_transformation=fitness_transformation, seed=3)
>>> populations = mp.play()
>>> mp.winning_strategy_name
'Defector'
Other types of implemented Moran processes:
Human Interaction¶
It is possible to play interactively using the Human strategy:
>>> import axelrod as axl
>>> me = axl.Human(name='me')
>>> players = [axl.TitForTat(), me]
>>> match = axl.Match(players, turns=3)
>>> match.play()
You will be prompted for the action to play at each turn:
Starting new match
Turn 1 action [C or D] for me: C
Turn 1: me played C, opponent played C
Turn 2 action [C or D] for me: D
Turn 2: me played D, opponent played C
Turn 3 action [C or D] for me: C
[(C, C), (C, D), (D, C)]
after this, the match
object can be manipulated as described in
Creating Matches
Running Axelrod’s First Tournament¶
This tutorial will bring together topics from the previous tutorials to reproduce Axelrod’s original tournament from [Axelrod1980].
Selecting our players¶
We will use the players from Axelrod’s first tournament which are contained in the axelrod.axelrod_first_strategies list:
>>> import axelrod as axl
>>> first_tournament_participants_ordered_by_reported_rank = [s() for s in axl.axelrod_first_strategies]
>>> number_of_strategies = len(first_tournament_participants_ordered_by_reported_rank)
>>> for player in first_tournament_participants_ordered_by_reported_rank:
... print(player)
Tit For Tat
First by Tideman and Chieruzzi: (D, D)
First by Nydegger
First by Grofman
First by Shubik
First by Stein and Rapoport: 0.05: (D, D)
Grudger
First by Davis: 10
First by Graaskamp: 0.05
First by Downing
First by Feld: 1.0, 0.5, 200
First by Joss: 0.9
First by Tullock
First by Anonymous
Random: 0.5
Creating the tournament¶
Now we create and run the tournament, we will set a seed to ensure reproducibility and 5 repetitions to smooth the random effects. We use 5 repetitions as this is what was done in [Axelrod1980]:
>>> tournament = axl.Tournament(
... players=first_tournament_participants_ordered_by_reported_rank,
... turns=200,
... repetitions=5,
... seed=1,
... )
>>> results = tournament.play()
Viewing the ranks of the participants¶
The results object contains the ranked names:
>>> for name in results.ranked_names:
... print(name)
First by Stein and Rapoport: 0.05: (D, D)
First by Grofman
First by Shubik
Tit For Tat
First by Nydegger
First by Tideman and Chieruzzi: (D, D)
Grudger
First by Davis: 10
First by Graaskamp: 0.05
First by Downing
First by Feld: 1.0, 0.5, 200
First by Tullock
First by Joss: 0.9
First by Anonymous
Random: 0.5
We see that TitForTat does not win. In fact TitForTat typically does not win this tournament, possibly because our implementations differ from the original strategies as their code is not available.
We can plot the reported rank (from [Axelrod1980]) versus the reproduced one:
>>> import matplotlib.pyplot as plt
>>> plt.figure(figsize=(15, 6))
>>> plt.plot((0, 15), (0, 15), color="grey", linestyle="")
>>> for original_rank, strategy in enumerate(first_tournament_participants_ordered_by_reported_rank):
... rank = results.ranked_names.index(str(strategy))
... if rank == original_rank:
... symbol = "+"
... plt.plot((rank, rank), (rank, 0), color="grey")
... else:
... symbol = "o"
... plt.scatter([rank], [original_rank], marker=symbol, color="black", s=50)
>>> plt.xticks(
... range(number_of_strategies),
... results.ranked_names,
... rotation=90
... )
>>> plt.ylabel("Reported rank")
>>> plt.xlabel("Reproduced rank");
>>> plt.show()
Visualising the scores¶
We see that the first 6 strategies do not match the ranks of the original paper, we can take a look the variation in the scores:
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
The first 6 strategies have similar scores which could indicate that the original work by Axelrod was not run with sufficient repetitions. Another explanation is that all the strategies are implemented from the descriptions given in [Axelrod1980] and there is no source code to base this on. This leads to some strategies being ambiguous. These are all clearly explained in the strategy docstrings. For example:
>>> print(axl.FirstByAnonymous.__doc__)
Submitted to Axelrod's first tournament by a graduate student whose name was
withheld.
The description written in [Axelrod1980]_ is:
> "This rule has a probability of cooperating, P, which is initially 30% and
> is updated every 10 moves. P is adjusted if the other player seems random,
> very cooperative, or very uncooperative. P is also adjusted after move 130
> if the rule has a lower score than the other player. Unfortunately, the
> complex process of adjustment frequently left the probability of cooperation
> in the 30% to 70% range, and therefore the rule appeared random to many
> other players."
Given the lack of detail this strategy is implemented based on the final
sentence of the description which is to have a cooperation probability that
is uniformly random in the 30 to 70% range.
Names:
 (Name withheld): [Axelrod1980]_
Other outcomes¶
If we run the tournament with other seeds, the results are different. For example, with 1408 Tit For Tat wins:
>>> tournament = axl.Tournament(
... players=first_tournament_participants_ordered_by_reported_rank,
... turns=200,
... repetitions=5,
... seed=1408,
... )
>>> results = tournament.play()
>>> for name in results.ranked_names:
... print(name)
Tit For Tat
First by Stein and Rapoport: 0.05: (D, D)
First by Grofman
First by Shubik
First by Tideman and Chieruzzi: (D, D)
First by Nydegger
Grudger
First by Davis: 10
First by Graaskamp: 0.05
First by Downing
First by Feld: 1.0, 0.5, 200
First by Tullock
First by Joss: 0.9
First by Anonymous
Random: 0.5
With 136 the strategy submitted by Grofman wins:
>>> tournament = axl.Tournament(
... players=first_tournament_participants_ordered_by_reported_rank,
... turns=200,
... repetitions=5,
... seed=136
... )
>>> results = tournament.play()
>>> for name in results.ranked_names:
... print(name)
First by Grofman
First by Stein and Rapoport: 0.05: (D, D)
Tit For Tat
First by Shubik
First by Tideman and Chieruzzi: (D, D)
First by Nydegger
Grudger
First by Davis: 10
First by Downing
First by Graaskamp: 0.05
First by Feld: 1.0, 0.5, 200
First by Joss: 0.9
First by Tullock
Random: 0.5
First by Anonymous
How to¶
This section contains short descriptions of how to accomplish specific tasks with the Axelrod library.
Include noise¶
A common variation on iterated prisoner’s dilemma tournaments is to add stochasticity in the choice of actions, simply called noise. This noise is introduced by flipping plays between C and D with some probability that is applied to all plays after they are delivered by the player [Bendor1993].
The presence of this persistent background noise causes some strategies to
behave substantially differently. For example, TitForTat
can fall into
defection loops with itself when there is noise. While TitForTat
would
usually cooperate well with itself:
C C C C C ...
C C C C C ...
Noise can cause a C to flip to a D (or vice versa), disrupting the cooperative chain:
C C C D C D C D D D ...
C C C C D C D D D D ...
To create a noisy tournament you simply need to add the noise argument:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> noise = 0.1
>>> tournament = axl.Tournament(players, noise=noise)
>>> results = tournament.play()
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
Here is how the distribution of wins now looks:
>>> p = plot.winplot()
>>> p.show()
Include probabilistic endings¶
It is possible to create a tournament where the length of each Match is not constant for all encounters: after each turn the Match ends with a given probability, [Axelrod1980b]:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> tournament = axl.Tournament(players, prob_end=0.5)
We can view the results in a similar way as described in Access tournament results:
>>> results = tournament.play()
>>> m = results.payoff_matrix
>>> for row in m:
... print([round(ele, 1) for ele in row]) # Rounding output
[3.0, 0.0, 3.0, 3.0]
[5.0, 1.0, 3.7, 3.6]
[3.0, 0.3, 3.0, 3.0]
[3.0, 0.4, 3.0, 3.0]
We see that Cooperator
always scores 0 against Defector
but
other scores seem variable as they are effected by the length of each match.
We can (as before) obtain the ranks for our players:
>>> results.ranked_names
['Defector', 'Tit For Tat', 'Grudger', 'Cooperator']
We can plot the results:
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
We can also view the length of the matches played by each player. The plot shows
that the length of each match (for each player) is not the same. The median
length is 4 which is the expected value with the probability of a match ending
being 0.5
.
>>> p = plot.lengthplot()
>>> p.show()
Create spatial tournaments¶
A spatial tournament is defined on a graph where the nodes correspond to players and edges define whether or not a given player pair will have a match.
The initial work on spatial tournaments was done by Nowak and May in a 1992 paper: [Nowak1992].
Additionally, Szabó and Fáth in their 2007 paper [Szabo2007] consider a variety of graphs, such as lattices, small world, scalefree graphs and evolving networks.
Let’s create a tournament where Cooperator
and Defector
do not
play each other and neither do TitForTat
and Grudger
:
Note that the edges have to be given as a list of tuples of player indices:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> edges = [(0, 2), (0, 3), (1, 2), (1, 3)]
To create a spatial tournament you pass the edges
to the
Tournament
class:
>>> spatial_tournament = axl.Tournament(players, edges=edges)
>>> results = spatial_tournament.play()
We can plot the results:
>>> plot = axl.Plot(results)
>>> p = plot.boxplot()
>>> p.show()
We can, like any other tournament, obtain the ranks for our players:
>>> results.ranked_names
['Cooperator', 'Tit For Tat', 'Grudger', 'Defector']
Let’s run a small tournament of 2 turns
and 2 repetitions
and obtain the interactions:
>>> spatial_tournament = axl.Tournament(players ,turns=2, repetitions=2, edges=edges)
>>> results = spatial_tournament.play()
>>> results.payoffs
[[[], [], [3.0, 3.0], [3.0, 3.0]], [[], [], [3.0, 3.0], [3.0, 3.0]], [[3.0, 3.0], [0.5, 0.5], [], []], [[3.0, 3.0], [0.5, 0.5], [], []]]
As anticipated not all players interact with each other.
It is also possible to create a probabilistic ending spatial tournament:
>>> prob_end_spatial_tournament = axl.Tournament(players, edges=edges, prob_end=.1, repetitions=1, seed=10)
>>> prob_end_results = prob_end_spatial_tournament.play()
We see that the match lengths are no longer all equal:
>>> prob_end_results.match_lengths
[[[0, 0, 20.0, 1.0], [0, 0, 46.0, 13.0], [20.0, 46.0, 0, 0], [1.0, 13.0, 0, 0]]]
Create Moran Processes On Graphs¶
The library also provides a graphbased Moran process [Shakarian2013] with
MoranProcess
. To use this feature you must supply at least one
Axelrod.graph.Graph
object, which can be initialized with just a list of
edges:
edges = [(source_1, target1), (source2, target2), ...]
The nodes can be any hashable object (integers, strings, etc.). For example:
>>> import axelrod as axl
>>> from axelrod.graph import Graph
>>> edges = [(0, 1), (1, 2), (2, 3), (3, 1)]
>>> graph = Graph(edges)
Graphs are undirected by default but you can pass directed=True
to
create a directed graph. Various intermediates such as the list of neighbors
are cached for efficiency by the graph object.
A Moran process can be invoked with one or two graphs. The first graph, the interaction graph, dictates how players are matched up in the scoring phase. Each player plays a match with each neighbor. The second graph dictates how players replace another during reproduction. When an individual is selected to reproduce, it replaces one of its neighbors in the reproduction graph. If only one graph is supplied to the process, the two graphs are assumed to be the same.
To create a graphbased Moran process, use a graph as follows:
>>> from axelrod.graph import Graph
>>> edges = [(0, 1), (1, 2), (2, 3), (3, 1)]
>>> graph = Graph(edges)
>>> players = [axl.Cooperator(), axl.Cooperator(), axl.Cooperator(), axl.Defector()]
>>> mp = axl.MoranProcess(players, interaction_graph=graph, seed=40)
>>> results = mp.play()
>>> mp.population_distribution()
Counter({'Defector': 4})
You can supply the reproduction_graph
as a keyword argument. The
standard Moran process is equivalent to using a complete graph with no loops
for the interaction_graph
and with loops for the
reproduction_graph
.
Create Approximate Moran Process¶
Due to the high computational cost of a single Moran process, an approximate
Moran process is implemented that can make use of cached outcomes of games. The
following code snippet will generate a Moran process in which the outcomes of
the matches played by a Random: 0.5
are sampled from one possible
outcome against each opponent (Defector
and Random: 0.5
). First
the cache is built by passing counter objects of outcomes:
>>> import axelrod as axl
>>> from collections import Counter
>>> cached_outcomes = {}
>>> cached_outcomes[("Random: 0.5", "Defector")] = axl.Pdf(Counter([(1, 1)]))
>>> cached_outcomes[("Random: 0.5", "Random: 0.5")] = axl.Pdf(Counter([(3, 3)]))
>>> cached_outcomes[("Defector", "Defector")] = axl.Pdf(Counter([(1, 1)]))
Now let us create an Approximate Moran Process:
>>> players = [axl.Defector(), axl.Random(), axl.Random()]
>>> amp = axl.ApproximateMoranProcess(players, cached_outcomes, seed=5)
>>> results = amp.play()
>>> amp.population_distribution()
Counter({'Random: 0.5': 3})
Note that by nature of being an approximation, it’s possible that the results of an
ApproximateMoranProcess may not always match the results of a standard MoranProcess,
even for the same random seed. We see that, for this random seed, the Random: 0.5
won this Moran process. This is not what happens in a standard Moran process where the
Random: 0.5
player will not win:
>>> mp = axl.MoranProcess(players, seed=5)
>>> results = mp.play()
>>> mp.population_distribution()
Counter({'Defector': 3})
Calculate Morality Metrics¶
Tyler SingerClark’s June 2014 paper, “Morality Metrics On Iterated Prisoner’s Dilemma Players” [SingerClark2014]), describes several interesting metrics which may be used to analyse IPD tournaments all of which are available within the ResultSet class. (Tyler’s paper is available here: http://www.scottaaronson.com/morality.pdf).
Each metric depends upon the cooperation rate of the players, defined by Tyler SingerClark as:
where C(b) is the total number of turns where a player chose to cooperate and TT is the total number of turns played.
A matrix of cooperation rates is available within a tournament’s ResultSet:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
>>> [[round(float(ele), 3) for ele in row] for row in results.normalised_cooperation]
[[1.0, 1.0, 1.0, 1.0], [0.0, 0.0, 0.0, 0.0], [1.0, 0.005, 1.0, 1.0], [1.0, 0.005, 1.0, 1.0]]
There is also a ‘good partner’ matrix showing how often a player cooperated at least as much as its opponent:
>>> results.good_partner_matrix
[[0, 10, 10, 10], [0, 0, 0, 0], [10, 10, 0, 10], [10, 10, 10, 0]]
Each of the metrics described in Tyler’s paper is available as follows (here they are rounded to 2 digits):
>>> [round(ele, 2) for ele in results.cooperating_rating]
[1.0, 0.0, 0.67..., 0.67...]
>>> [round(ele, 2) for ele in results.good_partner_rating]
[1.0, 0.0, 1.0, 1.0]
>>> [round(ele, 2) for ele in results.eigenjesus_rating]
[0.58, 0.0, 0.58, 0.58]
>>> [round(ele, 2) for ele in results.eigenmoses_rating]
[0.37, 0.37, 0.6, 0.6]
Run Axelrod’s Ecological Variant¶
In Axelrod’s original work an ecological approach based on the payoff matrix of the tournament was used to study the evolutionary stability of each strategy. Whilst this bears some comparison to the Moran Process, the latter is much more widely used in the literature.
To study the evolutionary stability of each strategy it is possible to create an ecosystem based on the payoff matrix of a tournament:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger(),
... axl.Random()]
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
>>> eco = axl.Ecosystem(results)
>>> eco.reproduce(100) # Evolve the population over 100 time steps
Here is how we obtain a nice stackplot of the system evolving over time:
>>> plot = axl.Plot(results)
>>> p = plot.stackplot(eco)
>>> p.show()
Fingerprint¶
Ashlock Fingerprints¶
In [Ashlock2008], [Ashlock2009] a methodology for obtaining visual
representation of a strategy’s behaviour is described. The basic method is to
play the strategy against a probe strategy with varying noise parameters.
These noise parameters are implemented through the JossAnnTransformer
.
The JossAnn of a strategy is a new strategy which has a probability x
of cooperating, a probability y
of defecting, and otherwise uses the
response appropriate to the original strategy. We can then plot the expected
score of the strategy against x
and y
and obtain a heat plot
over the unit square. When x + y >= 1
the JossAnn
is created
with parameters (1y, 1x)
and plays against the Dual of the probe
instead. A full definition and explanation is given in
[Ashlock2008], [Ashlock2009].
Here is how to create a fingerprint of WinStayLoseShift
using
TitForTat
as a probe:
>>> import axelrod as axl
>>> strategy = axl.WinStayLoseShift
>>> probe = axl.TitForTat
>>> af = axl.AshlockFingerprint(strategy, probe)
>>> data = af.fingerprint(turns=10, repetitions=2, step=0.2, seed=1)
>>> data
{...
>>> data[(0, 0)]
3.0
The fingerprint
method returns a dictionary mapping coordinates of the
form (x, y)
to the mean score for the corresponding interactions.
We can then plot the above to get:
>>> p = af.plot()
>>> p.show()
In reality we would need much more detail to make this plot useful.
Running the above with the following parameters:
>>> af.fingerprint(turns=50, repetitions=2, step=0.01)
We get the plot:
We are also able to specify a matplotlib colour map, interpolation and can remove the colorbar and axis labels:
>>> p = af.plot(cmap='PuOr', interpolation='bicubic', colorbar=False, labels=False)
>>> p.show()
Note that it is also possible to pass a player instance to be fingerprinted and/or as a probe. This allows for the fingerprinting of parametrized strategies:
>>> player = axl.Random(p=.1)
>>> probe = axl.GTFT(p=.9)
>>> af = axl.AshlockFingerprint(player, probe)
>>> data = af.fingerprint(turns=10, repetitions=2, step=0.2, seed=2)
>>> data
{...
>>> data[(0, 0)]
3.75
Transitive Fingerprint¶
Another implemented fingerprint is the transitive fingerprint. The transitive fingerprint represents the cooperation rate of a strategy against a set of opponents over a number of turns.
By default the set of opponents consists of 50
Random players that
cooperate with increasing probability. This is how to obtain the transitive
fingerprint for TitForTat
:
>>> player = axl.TitForTat()
>>> tf = axl.TransitiveFingerprint(player)
>>> data = tf.fingerprint(turns=40, seed=3)
The data produced is a numpy
array showing the cooperation rate against
a given opponent (row) in a given turn (column):
>>> data.shape
(50, 40)
It is also possible to visualise the fingerprint:
>>> p = tf.plot()
>>> p.show()
It is also possible to fingerprint against a given set of opponents:
>>> opponents = [s() for s in axl.demo_strategies]
>>> tf = axl.TransitiveFingerprint(player, opponents=opponents)
>>> data = tf.fingerprint(turns=5, repetitions=10, seed=4)
The name of the opponents can be displayed in the plot:
>>> p = tf.plot(display_names=True)
>>> p.show()
Evolve Players¶
Several strategies in the library derive from EvolvablePlayer
which specifies methods
allowing evolutionary or particle swarm algorithms to be used with these strategies. The
Axelrod Dojo library [Axelrod1980]
contains implementations of both algorithms for use with the Axelrod library. Examples include
FSMPlayers, ANN (neural networks), and LookerUp and Gambler (lookup tables).
New EvolvablePlayer
subclasses can be added to the library. Any strategy that can
define mutation
and crossover
methods can be used with the evolutionary algorithm
and the atomic mutation version of the Moran process. To use the particle swarm algorithms, methods
to serialize the strategy to and from a vector of floats must be defined.
Moran Process: Atomic Mutation for Evolvable Players¶
Additionally, the Moran process implementation supports a second style of mutation suitable for
evolving new strategies utilizing the EvolvablePlayer
class via its mutate
method.
This is in contrast to the transitional mutation that selects one of the other player types rather than (possibly)
generating a new player variant. To use this mutation style set mutation_method=atomic in the initialisation
of the Moran process:
>>> import axelrod as axl
>>> C = axl.Action.C
>>> players = [axl.EvolvableFSMPlayer(num_states=2, initial_state=1, initial_action=C) for _ in range(5)]
>>> mp = axl.MoranProcess(players, turns=10, mutation_method="atomic", seed=1)
>>> population = mp.play()
Note that this may cause the Moran process to fail to converge, if the mutation rates are very high or the population size very large. See Moran Process for more information.
Reproducible Seeding¶
EvolvablePlayers
are inherently stochastic. For reproducibility of results, they can be seeded. When
using the Moran process, a process level seed is sufficient. Child seeds will be created and propagated
in a reproducible way. If initialized without a seed, an EvolvablePlayer
will be given a
random seed in a nonreproducible way.
Access collections of strategies¶
All of the strategies are accessible from the main name space of the library. For example:
>>> import axelrod as axl
>>> axl.TitForTat()
Tit For Tat
>>> axl.Cooperator()
Cooperator
The main strategies which obey the rules of Axelrod’s original tournament can be found in a list: axelrod.strategies:
>>> axl.strategies
[...
This makes creating a full tournament very straightforward:
>>> players = [s() for s in axl.strategies]
>>> tournament = axl.Tournament(players)
There are a list of various other strategies in the library to make it easier to create a variety of tournaments:
>>> axl.demo_strategies # 5 simple strategies useful for demonstration.
[...
>>> axl.basic_strategies # A set of basic strategies.
[...
>>> axl.long_run_time_strategies # These have a high computational cost
[...
Furthermore there are some strategies that ‘cheat’ (for example by modifying
their opponents source code). These can be found in
axelrod.cheating_strategies
:
>>> axl.cheating_strategies
[...
All of the strategies in the library are contained in:
axelrod.all_strategies
:
>>> axl.all_strategies
[...
All strategies are also classified, you can read more about that in Classification of strategies.
Classification of strategies¶
Due to the large number of strategies, every class and instance of the class has
a classifier
attribute which classifies that strategy according to
various dimensions.
Here is the classifier
for the Cooperator
strategy:
>>> import axelrod as axl
>>> expected_dictionary = {
... 'manipulates_state': False,
... 'long_run_time': False,
... 'stochastic': False,
... 'manipulates_source': False,
... 'inspects_source': False,
... 'memory_depth': 0
... } # Order of this dictionary might be different on your machine
>>> axl.Cooperator.classifier == expected_dictionary
True
Note that instances of the class also have this classifier:
>>> s = axl.Cooperator()
>>> s.classifier == expected_dictionary
True
The instance starts with a copy of the class’s classifier dictionary, but is allowed to change this classifier dictionary at any point, and many strategies do so upon initialization.
In addition to the classifier dictionary, each classifier is defined with some logic that maps classifier definitions to values. To learn the classification of a strategy, we first look in the strategy’s classifier dictionary, then if the key is not present, then we refer to this logic. This logic must be defined for a class, and not specific instances.
To lookup the classifier of a strategy, using the classifier dict, or the
strategy’s logic as default, we use Classifiers[<classifier>](
<strategy>)
:
>>> from axelrod import Classifiers
>>> Classifiers['memory_depth'](axl.TitForTat())
1
>>> Classifiers['stochastic'](axl.Random())
True
We can use this classification to generate sets of strategies according to filters which we define in a ‘filterset’ dictionary and then pass to the ‘filtered_strategies’ function. For example, to identify all the stochastic strategies:
>>> filterset = {
... 'stochastic': True
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
87
Or, to find out how many strategies only use 1 turn worth of memory to make a decision:
>>> filterset = {
... 'memory_depth': 1
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
32
Multiple filters can be specified within the filterset dictionary. To specify a range of memory_depth values, we can use the ‘min_memory_depth’ and ‘max_memory_depth’ filters:
>>> filterset = {
... 'min_memory_depth': 1,
... 'max_memory_depth': 4
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
55
We can also identify strategies that make use of particular properties of the tournament. For example, here is the number of strategies that make use of the length of each match of the tournament:
>>> filterset = {
... 'makes_use_of': ['length']
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
22
Note that in the filterset dictionary, the value for the ‘makes_use_of’ key must be a list. Here is how we might identify the number of strategies that use both the length of the tournament and the game being played:
>>> filterset = {
... 'makes_use_of': ['length', 'game']
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
15
Some strategies have been classified as having a particularly long run time:
>>> filterset = {
... 'long_run_time': True
... }
>>> strategies = axl.filtered_strategies(filterset)
>>> len(strategies)
18
Strategies that manipulate_source
, manipulate_state
and/or inspect_source
return False
for the
Classifier.obey_axelrod
function:
>>> s = axl.Darwin()
>>> axl.Classifiers.obey_axelrod(s)
False
>>> s = axl.TitForTat()
>>> axl.Classifiers.obey_axelrod(s)
True
Strategy Transformers¶
What is a Strategy Transformer?¶
A strategy transformer is a function that modifies an existing strategy. For
example, FlipTransformer
takes a strategy and flips the actions from
C to D and D to C:
>>> import axelrod as axl
>>> from axelrod.strategy_transformers import *
>>> FlippedCooperator = FlipTransformer()(axl.Cooperator)
>>> player = FlippedCooperator()
>>> opponent = axl.Cooperator()
>>> player.strategy(opponent)
D
>>> opponent.strategy(player)
C
Our player was switched from a Cooperator
to a Defector
when
we applied the transformer. The transformer also changed the name of the
class and player:
>>> player.name
'Flipped Cooperator'
>>> FlippedCooperator.name
'Flipped Cooperator'
This behavior can be suppressed by setting the name_prefix
argument:
>>> FlippedCooperator = FlipTransformer(name_prefix=None)(axl.Cooperator)
>>> player = FlippedCooperator()
>>> player.name
'Cooperator'
Note carefully that the transformer returns a class, not an instance of a class. This means that you need to use the Transformed class as you would normally to create a new instance:
>>> from axelrod.strategy_transformers import NoisyTransformer
>>> player = NoisyTransformer(0.5)(axl.Cooperator)()
rather than NoisyTransformer(0.5)(axl.Cooperator())
or just NoisyTransformer(0.5)(axl.Cooperator)
.
Included Transformers¶
The library includes the following transformers:
ApologyTransformer
: Apologizes after a round of(D, C)
:>>> ApologizingDefector = ApologyTransformer([D], [C])(axl.Defector) >>> player = ApologizingDefector() You can pass any two sequences in. In this example the player would apologize after two consequtive rounds of `(D, C)`:: >>> ApologizingDefector = ApologyTransformer([D, D], [C, C])(axl.Defector) >>> player = ApologizingDefector()
DeadlockBreakingTransformer
: Attempts to break(D, C) > (C, D)
deadlocks by cooperating:>>> DeadlockBreakingTFT = DeadlockBreakingTransformer()(axl.TitForTat) >>> player = DeadlockBreakingTFT()
DualTransformer
: The Dual of a strategy will return the exact opposite set of moves to the original strategy when both are faced with the same history. [Ashlock2008]:>>> DualWSLS = DualTransformer()(axl.WinStayLoseShift) >>> player = DualWSLS()
FlipTransformer
: Flips all actions:>>> FlippedCooperator = FlipTransformer()(axl.Cooperator) >>> player = FlippedCooperator()
FinalTransformer(seq=None)
: Ends the tournament with the moves in the sequenceseq
, if the tournament_length is known. For example, to obtain a cooperator that defects on the last two rounds:>>> FinallyDefectingCooperator = FinalTransformer([D, D])(axl.Cooperator) >>> player = FinallyDefectingCooperator()
ForgiverTransformer(p)
: Flips defections with probabilityp
:>>> ForgivinDefector = ForgiverTransformer(0.1)(axl.Defector) >>> player = ForgivinDefector()
GrudgeTransformer(N)
: Defections unconditionally after more than N defections:>>> GrudgingCooperator = GrudgeTransformer(2)(axl.Cooperator) >>> player = GrudgingCooperator()
InitialTransformer(seq=None)
: First plays the moves in the sequenceseq
, then plays as usual. For example, to obtain a defector that cooperates on the first two rounds:>>> InitiallyCooperatingDefector = InitialTransformer([C, C])(axl.Defector) >>> player = InitiallyCooperatingDefector()
JossAnnTransformer(probability)
: Whereprobability = (x, y)
, the JossAnn of a strategy is a new strategy which has a probabilityx
of choosing the move C, a probabilityy
of choosing the move D, and otherwise uses the response appropriate to the original strategy. [Ashlock2008]:>>> JossAnnTFT = JossAnnTransformer((0.2, 0.3))(axl.TitForTat) >>> player = JossAnnTFT()
MixedTransformer
: Randomly plays a mutation to another strategy (or set of strategies. Here is the syntax to do this with a set of strategies:>>> strategies = [axl.Grudger, axl.TitForTat] >>> probability = [.2, .3] # .5 chance of mutated to one of above >>> player = MixedTransformer(probability, strategies)(axl.Cooperator)
Here is the syntax when passing a single strategy:
>>> strategy = axl.Grudger >>> probability = .2 >>> player = MixedTransformer(probability, strategy)(axl.Cooperator)
NiceTransformer()
: Prevents a strategy from defecting if the opponent has not yet defected:>>> NiceDefector = NiceTransformer()(axl.Defector) >>> player = NiceDefector()
NoisyTransformer(noise)
: Flips actions with probabilitynoise
:>>> NoisyCooperator = NoisyTransformer(0.5)(axl.Cooperator) >>> player = NoisyCooperator()
RetaliationTransformer(N)
: Retaliation N times after a defection:>>> TwoTitsForTat = RetaliationTransformer(2)(axl.Cooperator) >>> player = TwoTitsForTat()
RetaliateUntilApologyTransformer()
: adds TitForTatstyle retaliation:>>> TFT = RetaliateUntilApologyTransformer()(axl.Cooperator) >>> player = TFT()
TrackHistoryTransformer
: Tracks History internally in thePlayer
instance in a variable_recorded_history
. This allows a player to e.g. detect noise.:>>> player = TrackHistoryTransformer()(axl.Random)()
Composing Transformers¶
Transformers can be composed to form new composers, in two ways. You can simply chain together multiple transformers:
>>> cls1 = FinalTransformer([D,D])(InitialTransformer([D,D])(axl.Cooperator))
>>> p1 = cls1()
This defines a strategy that cooperates except on the first two and last two
rounds. Alternatively, you can make a new class using
compose_transformers
:
>>> cls1 = compose_transformers(FinalTransformer([D, D]), InitialTransformer([D, D]))
>>> p1 = cls1(axl.Cooperator)()
>>> p2 = cls1(axl.Defector)()
Usage as Class Decorators¶
Transformers can also be used to decorate existing strategies. For example,
the strategy BackStabber
defects on the last two rounds. We can encode this
behavior with a transformer as a class decorator:
@FinalTransformer([D, D]) # End with two defections
class BackStabber(Player):
"""
Forgives the first 3 defections but on the fourth
will defect forever. Defects on the last 2 rounds unconditionally.
"""
name = 'BackStabber'
classifier = {
'memory_depth': float('inf'),
'stochastic': False,
'inspects_source': False,
'manipulates_source': False,
'manipulates_state': False
}
def strategy(self, opponent):
if not opponent.history:
return C
if opponent.defections > 3:
return D
return C
Writing New Transformers¶
To make a new transformer, you need to define a strategy wrapping function with the following signature:
def strategy_wrapper(player, opponent, proposed_action, *args, **kwargs):
"""
Strategy wrapper functions should be of the following form.
Parameters

player: Player object or subclass (self)
opponent: Player object or subclass
proposed_action: an axelrod.Action, C or D
The proposed action by the wrapped strategy
proposed_action = Player.strategy(...)
args, kwargs:
Any additional arguments that you need.
Returns

action: an axelrod.Action, C or D
"""
# This example just passes through the proposed_action
return proposed_action
The proposed action will be the outcome of:
self.strategy(player)
in the underlying class (the one that is transformed). The strategy_wrapper still has full access to the player and the opponent objects and can have arguments.
To make a transformer from the strategy_wrapper
function, use
StrategyTransformerFactory
, which has signature:
def StrategyTransformerFactory(strategy_wrapper, name_prefix=""):
"""Modify an existing strategy dynamically by wrapping the strategy
method with the argument `strategy_wrapper`.
Parameters

strategy_wrapper: function
A function of the form `strategy_wrapper(player, opponent, proposed_action, *args, **kwargs)`
Can also use a class that implements
def __call__(self, player, opponent, action)
name_prefix: string, "Transformed "
A string to prepend to the strategy and class name
"""
So we use StrategyTransformerFactory
with strategy_wrapper
:
TransformedClass = StrategyTransformerFactory(generic_strategy_wrapper)
Cooperator2 = TransformedClass(*args, **kwargs)(axl.Cooperator)
If your wrapper requires no arguments, you can simply proceed as follows:
>>> TransformedClass = StrategyTransformerFactory(generic_strategy_wrapper)()
>>> Cooperator2 = TransformedClass(axl.Cooperator)
For more examples, see axelrod/strategy_transformers.py
.
Access tournament results¶
This tutorial will show you how to access the various results of a tournament:
 Wins: the number of matches won by each player
 Match lengths: the number of turns of each match played by each player (relevant for tournaments with probabilistic ending).
 Scores: the total scores of each player.
 Normalised scores: the scores normalised by matches played and turns.
 Ranking: ranking of players based on median score.
 Ranked names: names of players in ranked order.
 Payoffs: average payoff per turn of each player.
 Payoff matrix: the payoff matrix showing the payoffs of each row player against each column player.
 Payoff standard deviation: the standard deviation of the payoffs matrix.
 Score differences: the score difference between each player.
 Payoff difference means: the mean score differences.
 Cooperation counts: the number of times each player cooperated.
 Normalised cooperation: cooperation count per turn.
 Normalised cooperation: cooperation count per turn.
 State distribution: the count of each type of state of a match
 Normalised state distribution: the normalised count of each type of state of a match
 State to action distribution: the count of each type of state to action pair of a match
 Normalised state distribution: the normalised count of each type of state to action pair of a match
 Initial cooperation count: the count of initial cooperation by each player.
 Initial cooperation rate: the rate of initial cooperation by each player.
 Cooperation rating: cooperation rating of each player
 Vengeful cooperation: a morality metric from the literature (see Calculate Morality Metrics).
 Good partner matrix: a morality metric from [SingerClark2014].
 Good partner rating: a morality metric from [SingerClark2014].
 Eigenmoses rating: a morality metric from [SingerClark2014].
 Eigenjesus rating: a morality metric from [SingerClark2014].
As shown in Creating and running a simple tournament let us create a tournament:
>>> import axelrod as axl
>>> players = [axl.Cooperator(), axl.Defector(),
... axl.TitForTat(), axl.Grudger()]
>>> tournament = axl.Tournament(players, turns=10, repetitions=3)
>>> results = tournament.play()
Wins¶
This gives the number of wins obtained by each player:
>>> results.wins
[[0, 0, 0], [3, 3, 3], [0, 0, 0], [0, 0, 0]]
The Defector
is the only player to win any matches (all other matches
are ties).
Match lengths¶
This gives the length of the matches played by each player:
>>> import pprint # Nicer formatting of output
>>> pprint.pprint(results.match_lengths)
[[[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0]],
[[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0]],
[[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0],
[10.0, 10.0, 10.0, 10.0]]]
Every player plays 10 turns against every other player (including themselves) for every repetition of the tournament.
Scores¶
This gives all the total tournament scores (per player and per repetition):
>>> results.scores
[[60, 60, 60], [78, 78, 78], [69, 69, 69], [69, 69, 69]]
Normalised scores¶
This gives the scores, averaged per opponent and turns:
>>> results.normalised_scores
[[2.0, 2.0, 2.0], [2.6, 2.6, 2.6], [2.3, 2.3, 2.3], [2.3, 2.3, 2.3]]
We see that Cooperator got on average a score of 2 per turn per opponent:
>>> results.normalised_scores[0]
[2.0, 2.0, 2.0]
Ranking¶
This gives the ranked index of each player:
>>> results.ranking
[1, 2, 3, 0]
The first player has index 1 (Defector
) and the last has index 0
(Cooperator
).
Ranked names¶
This gives the player names in ranked order:
>>> results.ranked_names
['Defector', 'Tit For Tat', 'Grudger', 'Cooperator']
Payoffs¶
This gives for each player, against each opponent every payoff received for each repetition:
>>> pprint.pprint(results.payoffs)
[[[3.0, 3.0, 3.0], [0.0, 0.0, 0.0], [3.0, 3.0, 3.0], [3.0, 3.0, 3.0]],
[[5.0, 5.0, 5.0], [1.0, 1.0, 1.0], [1.4, 1.4, 1.4], [1.4, 1.4, 1.4]],
[[3.0, 3.0, 3.0], [0.9, 0.9, 0.9], [3.0, 3.0, 3.0], [3.0, 3.0, 3.0]],
[[3.0, 3.0, 3.0], [0.9, 0.9, 0.9], [3.0, 3.0, 3.0], [3.0, 3.0, 3.0]]]
Payoff matrix¶
This gives the mean payoff of each player against every opponent:
>>> pprint.pprint(results.payoff_matrix)
[[3.0, 0.0, 3.0, 3.0],
[5.0, 1.0, 1.4, 1.4],
[3.0, 0.9, 3.0, 3.0],
[3.0, 0.9, 3.0, 3.0]]
We see that the Cooperator
gets a mean score of 3 against all players
except the Defector
:
>>> results.payoff_matrix[0]
[3.0, 0.0, 3.0, 3.0]
Payoff standard deviation¶
This gives the standard deviation of the payoff of each player against every opponent:
>>> pprint.pprint(results.payoff_stddevs)
[[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 2.2, 2.2],
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0]]
We see that there is no variation for the payoff for Cooperator
:
>>> results.payoff_stddevs[0]
[0.0, 0.0, 0.0, 0.0]
Score differences¶
This gives the score difference for each player against each opponent for every repetition:
>>> pprint.pprint(results.score_diffs)
[[[0.0, 0.0, 0.0], [5.0, 5.0, 5.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]],
[[5.0, 5.0, 5.0], [0.0, 0.0, 0.0], [0.5, 0.5, 0.5], [0.5, 0.5, 0.5]],
[[0.0, 0.0, 0.0], [0.5, 0.5, 0.5], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]],
[[0.0, 0.0, 0.0], [0.5, 0.5, 0.5], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]]]
We see that Cooperator
has no difference in score with all players
except against the Defector
:
>>> results.score_diffs[0][1]
[5.0, 5.0, 5.0]
Payoff difference means¶
This gives the mean payoff differences over each repetition:
>>> pprint.pprint(results.payoff_diffs_means)
[[0.0, 5.0, 0.0, 0.0],
[5.0, 0.0, 0.49999999999999983, 0.49999999999999983],
[0.0, 0.49999999999999983, 0.0, 0.0],
[0.0, 0.49999999999999983, 0.0, 0.0]]
Here is the mean payoff difference for the Cooperator
strategy, shows
that it has no difference with all players except against the
Defector
:
>>> results.payoff_diffs_means[0]
[0.0, 5.0, 0.0, 0.0]
Cooperation counts¶
This gives a total count of cooperation for each player against each opponent:
>>> results.cooperation
[[30, 30, 30, 30], [0, 0, 0, 0], [30, 3, 30, 30], [30, 3, 30, 30]]
Normalised cooperation¶
This gives the average rate of cooperation against each opponent:
>>> pprint.pprint(results.normalised_cooperation)
[[1.0, 1.0, 1.0, 1.0],
[0.0, 0.0, 0.0, 0.0],
[1.0, 0.1, 1.0, 1.0],
[1.0, 0.1, 1.0, 1.0]]
We see that Cooperator
for all the rounds (as expected):
>>> results.normalised_cooperation[0]
[1.0, 1.0, 1.0, 1.0]
State distribution counts¶
This gives a total state count against each opponent. A state corresponds to 1
turn of a match and can be one of (C, C), (C, D), (D, C),
(D, D)
where the first element is the action of the player in question and
the second the action of the opponent:
>>> pprint.pprint(results.state_distribution)
[[Counter(),
Counter({(C, D): 30}),
Counter({(C, C): 30}),
Counter({(C, C): 30})],
[Counter({(D, C): 30}),
Counter(),
Counter({(D, D): 27, (D, C): 3}),
Counter({(D, D): 27, (D, C): 3})],
[Counter({(C, C): 30}),
Counter({(D, D): 27, (C, D): 3}),
Counter(),
Counter({(C, C): 30})],
[Counter({(C, C): 30}),
Counter({(D, D): 27, (C, D): 3}),
Counter({(C, C): 30}),
Counter()]]
Normalised state distribution¶
This gives the average rate state distribution against each opponent.
A state corresponds to 1
turn of a match and can be one of (C, C), (C, D), (D, C),
(D, D)
where the first element is the action of the player in question and
the second the action of the opponent:
>>> pprint.pprint(results.normalised_state_distribution)
[[Counter(),
Counter({(C, D): 1.0}),
Counter({(C, C): 1.0}),
Counter({(C, C): 1.0})],
[Counter({(D, C): 1.0}),
Counter(),
Counter({(D, D): 0.9..., (D, C): 0.1...}),
Counter({(D, D): 0.9..., (D, C): 0.1...})],
[Counter({(C, C): 1.0}),
Counter({(D, D): 0.9..., (C, D): 0.1...}),
Counter(),
Counter({(C, C): 1.0})],
[Counter({(C, C): 1.0}),
Counter({(D, D): 0.9..., (C, D): 0.1...}),
Counter({(C, C): 1.0}),
Counter()]]
State to action distribution counts¶
This gives a total state action pair count against each opponent. A state
corresponds to 1 turn of a match and can be one of (C, C), (C,
D), (D, C), (D, D)
where the first element is the action of the
player in question and the second the action of the opponent:
>>> pprint.pprint(results.state_to_action_distribution)
[[Counter(),
Counter({((C, D), C): 27}),
Counter({((C, C), C): 27}),
Counter({((C, C), C): 27})],
[Counter({((D, C), D): 27}),
Counter(),
Counter({((D, D), D): 24, ((D, C), D): 3}),
Counter({((D, D), D): 24, ((D, C), D): 3})],
[Counter({((C, C), C): 27}),
Counter({((D, D), D): 24, ((C, D), D): 3}),
Counter(),
Counter({((C, C), C): 27})],
[Counter({((C, C), C): 27}),
Counter({((D, D), D): 24, ((C, D), D): 3}),
Counter({((C, C), C): 27}),
Counter()]]
Normalised state to action distribution¶
This gives the average rate state to action pair distribution against each
opponent. A state corresponds to 1 turn of a match and can be one of
(C, C), (C, D), (D, C), (D, D)
where the first element
is the action of the player in question and the second the action of the
opponent:
>>> pprint.pprint(results.normalised_state_to_action_distribution)
[[Counter(),
Counter({((C, D), C): 1.0}),
Counter({((C, C), C): 1.0}),
Counter({((C, C), C): 1.0})],
[Counter({((D, C), D): 1.0}),
Counter(),
Counter({((D, C), D): 1.0, ((D, D), D): 1.0}),
Counter({((D, C), D): 1.0, ((D, D), D): 1.0})],
[Counter({((C, C), C): 1.0}),
Counter({((C, D), D): 1.0, ((D, D), D): 1.0}),
Counter(),
Counter({((C, C), C): 1.0})],
[Counter({((C, C), C): 1.0}),
Counter({((C, D), D): 1.0, ((D, D), D): 1.0}),
Counter({((C, C), C): 1.0}),
Counter()]]
Initial cooperation counts¶
This gives the count of cooperations made by each player during the first turn of every match:
>>> results.initial_cooperation_count
[9, 0, 9, 9]
Each player plays an opponent a total of 9 times (3 opponents and 3
repetitions). Apart from the Defector
, they all cooperate on the first
turn.
Initial cooperation rates¶
This gives the rate of which a strategy cooperates during the first turn:
>>> results.initial_cooperation_rate
[1.0, 0.0, 1.0, 1.0]
Morality Metrics¶
The following morality metrics are available, they are calculated as a function of the cooperation rating:
>>> results.cooperating_rating
[1.0, 0.0, 0.7, 0.7]
>>> pprint.pprint(results.vengeful_cooperation)
[[1.0, 1.0, 1.0, 1.0],
[1.0, 1.0, 1.0, 1.0],
[1.0, 0.8, 1.0, 1.0],
[1.0, 0.78 1.0, 1.0]]
>>> pprint.pprint(results.good_partner_matrix)
[[0, 3, 3, 3], [0, 0, 0, 0], [3, 3, 0, 3], [3, 3, 3, 0]]
>>> pprint.pprint(results.good_partner_rating)
[1.0, 0.0, 1.0, 1.0]
>>> results.eigenmoses_rating
[0.37..., 0.37..., 0.59..., 0.59...]
>>> results.eigenjesus_rating
[0.57..., 0.0, 0.57..., 0.57...]
For more information about these see Calculate Morality Metrics.
Read and write interactions from/to file¶
When dealing with large tournaments it might be desirable to separate the
analysis from the actual running of the tournaments. This can be done by passing
a filename
argument to the play
method of a tournament:
>>> import axelrod as axl
>>> players = [s() for s in axl.basic_strategies]
>>> tournament = axl.Tournament(players, turns=4, repetitions=2)
>>> results = tournament.play(filename="basic_tournament.csv")
This will create a file basic_tournament.csv with data that looks something like:
Interaction index,Player index,Opponent index,Repetition,Player name,Opponent name,Actions,Score,Score difference,Turns,Score per turn,Score difference per turn,Win,Initial cooperation,Cooperation count,CC count,CD count,DC count,DD count,CC to C count,CC to D count,CD to C count,CD to D count,DC to C count,DC to D count,DD to C count,DD to D count,Good partner
0,0,0,0,Alternator,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
0,0,0,0,Alternator,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
1,0,0,1,Alternator,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
1,0,0,1,Alternator,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
2,0,1,0,Alternator,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
2,1,0,0,Anti Tit For Tat,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
3,0,1,1,Alternator,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
3,1,0,1,Anti Tit For Tat,Alternator,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
4,0,2,0,Alternator,Bully,CDCD,5,5,4,1.25,1.25,0,True,2,1,1,0,2,0,1,0,1,0,0,1,0,1
4,2,0,0,Bully,Alternator,DDCD,10,5,4,2.5,1.25,1,False,1,1,0,1,2,0,1,0,0,0,1,1,0,0
5,0,2,1,Alternator,Bully,CDCD,5,5,4,1.25,1.25,0,True,2,1,1,0,2,0,1,0,1,0,0,1,0,1
5,2,0,1,Bully,Alternator,DDCD,10,5,4,2.5,1.25,1,False,1,1,0,1,2,0,1,0,0,0,1,1,0,0
6,0,3,0,Alternator,Cooperator,CDCD,16,10,4,4.0,2.5,1,True,2,2,0,2,0,0,2,0,0,1,0,0,0,0
6,3,0,0,Cooperator,Alternator,CCCC,6,10,4,1.5,2.5,0,True,4,2,2,0,0,2,0,1,0,0,0,0,0,1
7,0,3,1,Alternator,Cooperator,CDCD,16,10,4,4.0,2.5,1,True,2,2,0,2,0,0,2,0,0,1,0,0,0,0
7,3,0,1,Cooperator,Alternator,CCCC,6,10,4,1.5,2.5,0,True,4,2,2,0,0,2,0,1,0,0,0,0,0,1
8,0,4,0,Alternator,Cycler DC,CDCD,10,0,4,2.5,0.0,0,True,2,0,2,2,0,0,0,0,2,1,0,0,0,1
8,4,0,0,Cycler DC,Alternator,DCDC,10,0,4,2.5,0.0,0,False,2,0,2,2,0,0,0,0,1,2,0,0,0,1
9,0,4,1,Alternator,Cycler DC,CDCD,10,0,4,2.5,0.0,0,True,2,0,2,2,0,0,0,0,2,1,0,0,0,1
9,4,0,1,Cycler DC,Alternator,DCDC,10,0,4,2.5,0.0,0,False,2,0,2,2,0,0,0,0,1,2,0,0,0,1
10,0,5,0,Alternator,Defector,CDCD,2,10,4,0.5,2.5,0,True,2,0,2,0,2,0,0,0,2,0,0,1,0,1
10,5,0,0,Defector,Alternator,DDDD,12,10,4,3.0,2.5,1,False,0,0,0,2,2,0,0,0,0,0,2,0,1,0
11,0,5,1,Alternator,Defector,CDCD,2,10,4,0.5,2.5,0,True,2,0,2,0,2,0,0,0,2,0,0,1,0,1
11,5,0,1,Defector,Alternator,DDDD,12,10,4,3.0,2.5,1,False,0,0,0,2,2,0,0,0,0,0,2,0,1,0
12,0,6,0,Alternator,Grudger,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
12,6,0,0,Grudger,Alternator,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,1,0,0,1
13,0,6,1,Alternator,Grudger,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
13,6,0,1,Grudger,Alternator,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,1,0,0,1
14,0,7,0,Alternator,Suspicious Tit For Tat,CDCD,10,0,4,2.5,0.0,0,True,2,0,2,2,0,0,0,0,2,1,0,0,0,1
14,7,0,0,Suspicious Tit For Tat,Alternator,DCDC,10,0,4,2.5,0.0,0,False,2,0,2,2,0,0,0,0,1,2,0,0,0,1
15,0,7,1,Alternator,Suspicious Tit For Tat,CDCD,10,0,4,2.5,0.0,0,True,2,0,2,2,0,0,0,0,2,1,0,0,0,1
15,7,0,1,Suspicious Tit For Tat,Alternator,DCDC,10,0,4,2.5,0.0,0,False,2,0,2,2,0,0,0,0,1,2,0,0,0,1
16,0,8,0,Alternator,Tit For Tat,CDCD,13,5,4,3.25,1.25,1,True,2,1,1,2,0,0,1,0,1,1,0,0,0,0
16,8,0,0,Tit For Tat,Alternator,CCDC,8,5,4,2.0,1.25,0,True,3,1,2,1,0,1,0,0,1,1,0,0,0,1
17,0,8,1,Alternator,Tit For Tat,CDCD,13,5,4,3.25,1.25,1,True,2,1,1,2,0,0,1,0,1,1,0,0,0,0
17,8,0,1,Tit For Tat,Alternator,CCDC,8,5,4,2.0,1.25,0,True,3,1,2,1,0,1,0,0,1,1,0,0,0,1
18,0,9,0,Alternator,WinShift LoseStay: D,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
18,9,0,0,WinShift LoseStay: D,Alternator,DCCD,9,0,4,2.25,0.0,0,False,2,1,1,1,1,0,1,1,0,1,0,0,0,1
19,0,9,1,Alternator,WinShift LoseStay: D,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
19,9,0,1,WinShift LoseStay: D,Alternator,DCCD,9,0,4,2.25,0.0,0,False,2,1,1,1,1,0,1,1,0,1,0,0,0,1
20,0,10,0,Alternator,WinStay LoseShift: C,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
20,10,0,0,WinStay LoseShift: C,Alternator,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,1,0,0,1
21,0,10,1,Alternator,WinStay LoseShift: C,CDCD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,1,1,0,0,0,1
21,10,0,1,WinStay LoseShift: C,Alternator,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,1,0,0,1
22,1,1,0,Anti Tit For Tat,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
22,1,1,0,Anti Tit For Tat,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
23,1,1,1,Anti Tit For Tat,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
23,1,1,1,Anti Tit For Tat,Anti Tit For Tat,CDCD,8,0,4,2.0,0.0,0,True,2,2,0,0,2,0,2,0,0,0,0,1,0,1
24,1,2,0,Anti Tit For Tat,Bully,CCCC,0,20,4,0.0,5.0,0,True,4,0,4,0,0,0,0,3,0,0,0,0,0,1
24,2,1,0,Bully,Anti Tit For Tat,DDDD,20,20,4,5.0,5.0,1,False,0,0,0,4,0,0,0,0,0,0,3,0,0,0
25,1,2,1,Anti Tit For Tat,Bully,CCCC,0,20,4,0.0,5.0,0,True,4,0,4,0,0,0,0,3,0,0,0,0,0,1
25,2,1,1,Bully,Anti Tit For Tat,DDDD,20,20,4,5.0,5.0,1,False,0,0,0,4,0,0,0,0,0,0,3,0,0,0
26,1,3,0,Anti Tit For Tat,Cooperator,CDDD,18,15,4,4.5,3.75,1,True,1,1,0,3,0,0,1,0,0,0,2,0,0,0
26,3,1,0,Cooperator,Anti Tit For Tat,CCCC,3,15,4,0.75,3.75,0,True,4,1,3,0,0,1,0,2,0,0,0,0,0,1
27,1,3,1,Anti Tit For Tat,Cooperator,CDDD,18,15,4,4.5,3.75,1,True,1,1,0,3,0,0,1,0,0,0,2,0,0,0
27,3,1,1,Cooperator,Anti Tit For Tat,CCCC,3,15,4,0.75,3.75,0,True,4,1,3,0,0,1,0,2,0,0,0,0,0,1
28,1,4,0,Anti Tit For Tat,Cycler DC,CCDC,7,5,4,1.75,1.25,0,True,3,2,1,0,1,0,1,1,0,0,0,1,0,1
28,4,1,0,Cycler DC,Anti Tit For Tat,DCDC,12,5,4,3.0,1.25,1,False,2,2,0,1,1,0,1,0,0,1,0,1,0,0
29,1,4,1,Anti Tit For Tat,Cycler DC,CCDC,7,5,4,1.75,1.25,0,True,3,2,1,0,1,0,1,1,0,0,0,1,0,1
29,4,1,1,Cycler DC,Anti Tit For Tat,DCDC,12,5,4,3.0,1.25,1,False,2,2,0,1,1,0,1,0,0,1,0,1,0,0
30,1,5,0,Anti Tit For Tat,Defector,CCCC,0,20,4,0.0,5.0,0,True,4,0,4,0,0,0,0,3,0,0,0,0,0,1
30,5,1,0,Defector,Anti Tit For Tat,DDDD,20,20,4,5.0,5.0,1,False,0,0,0,4,0,0,0,0,0,0,3,0,0,0
31,1,5,1,Anti Tit For Tat,Defector,CCCC,0,20,4,0.0,5.0,0,True,4,0,4,0,0,0,0,3,0,0,0,0,0,1
31,5,1,1,Defector,Anti Tit For Tat,DDDD,20,20,4,5.0,5.0,1,False,0,0,0,4,0,0,0,0,0,0,3,0,0,0
32,1,6,0,Anti Tit For Tat,Grudger,CDDC,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,0,0,1,1,0,1
32,6,1,0,Grudger,Anti Tit For Tat,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,0,0,1,1
33,1,6,1,Anti Tit For Tat,Grudger,CDDC,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,0,0,0,1,1,0,1
33,6,1,1,Grudger,Anti Tit For Tat,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,1,0,0,1,0,0,0,1,1
34,1,7,0,Anti Tit For Tat,Suspicious Tit For Tat,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,1,0,0,1,0,0,1
34,7,1,0,Suspicious Tit For Tat,Anti Tit For Tat,DCCD,9,0,4,2.25,0.0,0,False,2,1,1,1,1,1,0,0,1,1,0,0,0,1
35,1,7,1,Anti Tit For Tat,Suspicious Tit For Tat,CCDD,9,0,4,2.25,0.0,0,True,2,1,1,1,1,0,1,1,0,0,1,0,0,1
35,7,1,1,Suspicious Tit For Tat,Anti Tit For Tat,DCCD,9,0,4,2.25,0.0,0,False,2,1,1,1,1,1,0,0,1,1,0,0,0,1
...
Note that depending on the order in which the matches have been played, the rows could also be in a different order.
It is possible to read in this data file to obtain interactions:
>>> interactions = axl.interaction_utils.read_interactions_from_file("basic_tournament.csv")
This gives a dictionary mapping pairs of player indices to interaction histories:
>>> interactions[(0, 1)]
[[(C, C), (D, D), (C, C), (D, D)], [(C, C), (D, D), (C, C), (D, D)]]
This should allow for easy manipulation of data outside of the capabilities within the library.
Note that you can supply build_results=False as a keyword argument to tournament.play() to prevent keeping or loading interactions in memory, since the total memory footprint can be large for various combinations of parameters. The memory usage scales as \(O(\text{players}^2 \times \text{turns} \times \text{repetitions})\).
Use Parallel processing¶
When dealing with large tournaments on a multi core machine it is possible to
run the tournament in parallel although this is not currently supported on
Windows. Using processes=0
will simply use all available cores:
>>> import axelrod as axl
>>> players = [s() for s in axl.basic_strategies]
>>> tournament = axl.Tournament(players, turns=4, repetitions=2)
>>> results = tournament.play(processes=0)
Use a cache¶
Whilst for stochastic strategies, every repetition of a Match will give a
different result, for deterministic strategies, when there is no noise there is
no need to re run the match. The library has a DeterministicCache
class
that allows us to quickly replay matches.
Caching a Match¶
To illustrate this, let us time the play of a match without a cache:
>>> import axelrod as axl
>>> import timeit
>>> def run_match():
... p1, p2 = axl.GoByMajority(), axl.Alternator()
... match = axl.Match((p1, p2), turns=200)
... return match.play()
>>> time_with_no_cache = timeit.timeit(run_match, number=500)
>>> time_with_no_cache
2.2295279502868652
Here is how to create a new empty cache:
>>> cache = axl.DeterministicCache()
>>> len(cache)
0
Let us rerun the above match but using the cache:
>>> p1, p2 = axl.GoByMajority(), axl.Alternator()
>>> match = axl.Match((p1, p2), turns=200, deterministic_cache=cache)
>>> match.play()
[(C, C), ..., (C, D)]
We can take a look at the cache:
>>> cache
{('Soft Go By Majority', 'Alternator'): [(C, C), ..., (C, D)]}
>>> len(cache)
1
>>> len(cache[(axl.GoByMajority(), axl.Alternator())])
200
This maps a triplet of 2 player names and the match length to the resulting interactions. We can rerun the code and compare the timing:
>>> def run_match_with_cache():
... p1, p2 = axl.GoByMajority(), axl.Alternator()
... match = axl.Match((p1, p2), turns=200, deterministic_cache=cache)
... return match.play()
>>> time_with_cache = timeit.timeit(run_match_with_cache, number=500)
>>> time_with_cache
0.04215192794799805
>>> time_with_cache < time_with_no_cache
True
We can write the cache to file:
>>> cache.save("cache.txt")
True
Caching a Tournament¶
Tournaments will automatically create caches as needed on a match by match basis.
Caching a Moran Process¶
A prebuilt cache can also be used in a Moran process (by default a new cache is used):
>>> cache = axl.DeterministicCache("cache.txt")
>>> players = [axl.GoByMajority(), axl.Alternator(),
... axl.Cooperator(), axl.Grudger()]
>>> mp = axl.MoranProcess(players, deterministic_cache=cache)
>>> populations = mp.play()
>>> mp.winning_strategy_name
Defector
We see that the cache has been augmented, although note that this particular number will depend on the stochastic behaviour of the Moran process:
>>> len(cache)
18
Use different stage games¶
As described in Play Contexts and Generic Prisoner’s Dilemma the default game used for the Prisoner’s Dilemma is given by:
>>> import axelrod as axl
>>> pd = axl.game.Game()
>>> pd
Axelrod game: (R,P,S,T) = (3, 1, 0, 5)
>>> pd.RPST()
(3, 1, 0, 5)
These Game
objects are used to score matches,
tournaments and Moran processes:
>>> pd.score((axl.Action.C, axl.Action.C))
(3, 3)
>>> pd.score((axl.Action.C, axl.Action.D))
(0, 5)
>>> pd.score((axl.Action.D, axl.Action.C))
(5, 0)
>>> pd.score((axl.Action.D, axl.Action.D))
(1, 1)
It is possible to run a matches, tournaments and Moran processes with a different game. For example here is the game of chicken:
>>> chicken = axl.game.Game(r=0, s=1, t=1, p=10)
>>> chicken
Axelrod game: (R,P,S,T) = (0, 10, 1, 1)
>>> chicken.RPST()
(0, 10, 1, 1)
Here is a simple tournament run with this game:
>>> players = [axl.Cooperator(), axl.Defector(), axl.TitForTat()]
>>> tournament = axl.Tournament(players, game=chicken)
>>> results = tournament.play()
>>> results.ranked_names
['Cooperator', 'Defector', 'Tit For Tat']
The default Prisoner’s dilemma has different results:
>>> tournament = axl.Tournament(players)
>>> results = tournament.play()
>>> results.ranked_names
['Defector', 'Tit For Tat', 'Cooperator']
Set a random seed¶
The library has a variety of strategies whose behaviour is stochastic. To ensure reproducible results a random seed should be set. The library abstracts away the propagation of seeds in matches and tournaments, so you typically only need to supply a seed to those objects.
Matches¶
For a match, set a seed by passing a parameter to Match
>>> import axelrod as axl
>>> players = (axl.Random(), axl.MetaMixer()) # Two stochastic strategies
>>> match = axl.Match(players, turns=3, seed=101)
>>> results = match.play()
We obtain the same results if it is played with the same seed:
>>> match2 = axl.Match(players, turns=3, seed=101)
>>> result2 = match2.play()
>>> results == result2
True
For noisy matches, a seed also needs to be set for reproducibility, even if the players are deterministic.
>>> import axelrod as axl
>>> players = (axl.Cooperator(), axl.Defector()) # Two deterministic strategies
>>> match = axl.Match(players, turns=200, seed=101, noise=0.25)
>>> results = match.play()
>>> match2 = axl.Match(players, turns=200, seed=101, noise=0.25)
>>> results2 = match2.play()
>>> results == results2
True
Tournaments¶
For tournaments, an initial seed is used to generate subsequent seeds for each match in a a manner that will yield identical results. Note that if the tournament is run with multiple processes, the order of the matches may be computed differently, but the seeds used for each match will be the same.
To seed a tournament we also pass a seed to the tournament at creation time:
>>> import axelrod as axl
>>> seed = 201
>>> players = (axl.Random(), axl.Cooperator(), axl.MetaMixer())
>>> tournament = axl.Tournament(players, turns=5, repetitions=5, seed=seed)
>>> results = tournament.play(processes=1)
>>> tournament2 = axl.Tournament(players, turns=5, repetitions=5, seed=seed)
>>> results2 = tournament2.play(processes=1)
>>> results.ranked_names == results2.ranked_names
True
For parallel processing, the ordering of match results may differ, but the actual results, and the final rankings, will be the same.
>>> import axelrod as axl
>>> players = (axl.Random(), axl.Cooperator(), axl.MetaMixer())
>>> tournament = axl.Tournament(players, turns=5, repetitions=5, seed=201)
>>> results = tournament.play(processes=2)
>>> tournament2 = axl.Tournament(players, turns=5, repetitions=5, seed=201)
>>> results2 = tournament2.play(processes=2)
>>> results.ranked_names == results2.ranked_names
True
Moran Process¶
Similarly, a Moran process is essentially another type of tournament. The library’s implementation
will propagate child seeds to each match to ensure reproducibility. See also the documentation on
EvolvablePlayers
.
Fingerprints¶
Since fingerprint generation depends on tournaments, fingerprints can also be given a seed for reproducibility.
Set Player information¶
It is possible to determine what information players know about their matches.
By default all known information is given. For example let us create a match
with 5 turns between FirstBySteinAndRapoport
and Alternator
. The
latter of these two always defects on the last 2 turns:
>>> import axelrod as axl
>>> players = (axl.Alternator(), axl.FirstBySteinAndRapoport())
>>> axl.Match(players, turns=5).play()
[(C, C), (D, C), (C, C), (D, D), (C, D)]
We can play the same match but let us tell the players that the match lasts 6 turns:
>>> axl.Match(players, turns=5, match_attributes={"length": 6}).play()
[(C, C), (D, C), (C, C), (D, C), (C, D)]
We can also pass this information to a tournament. Let us create a tournament with 5 turns but ensure the players believe the match length is infinite (unknown):
>>> tournament = axl.Tournament(players, turns=5,
... match_attributes={"length": float('inf')})
The match_attributes
dictionary can also be used to pass game
and noise
.
Check Player equality¶
It is possible to test for player equality using ==
:
>>> import axelrod as axl
>>> p1, p2, p3 = axl.Alternator(), axl.Alternator(), axl.TitForTat()
>>> p1 == p2
True
>>> p1 == p3
False
Note that this checks all the attributes of an instance:
>>> p1.name = "John Nash"
>>> p1 == p2
False
This however does not check if the players will behave in the same way. For example here are two equivalent players:
>>> p1 = axl.Alternator()
>>> p2 = axl.Cycler("CD")
>>> p1 == p2
False
To check if player strategies are equivalent you can use Fingerprint.
Cite the library¶
We would be delighted if anyone wanted to use and/or reference this library for their own research.
If you do please let us know and reference the library: as described in the CITATION.rst file on the library repository.
Contribute¶
This section contains a variety of tutorials that should help you contribute to the library.
Contents:
Guidelines¶
All contributions to this repository are welcome via pull request on the github repository.
The project follows the following guidelines:
Use the base Python library unless completely necessary. A few external libraries (such as numpy) have been included in requirements.txt – feel free to use these as needed.
Try as best as possible to follow PEP8 which includes using descriptive variable names.
Code Format: Use the Black formatter to format all code and the isort utility to sort import statements. You can run black on all code with:
$ python m black l 80 .
Commits: Please try to use commit messages that give a meaningful history for anyone using git’s log features. Try to use messages that complete sentence, “This commit will…” There is some excellent guidance on the subject from Chris Beams
Testing: the project uses the unittest library and has a nice testing suite that makes some things easy to write tests for. Please try to increase the test coverage on pull requests.
Merging pullrequests: We require two of the (currently three) coreteam maintainers to merge. Opening a PR for early feedback or to check test coverage is OK, just indicate that the PR is not ready to merge (and update when it is).
By submitting a pull request, you are agreeing that your work may be distributed under the terms of the project’s licence and you will become one of the project’s joint copyright holders.
Setting up the environment¶
Installing all dependencies¶
All dependencies can be installed by running:
$ pip install r requirements.txt
It is recommended to do this using a virtual environment tool of your choice.
For example, when using the virtual environment library venv
:
$ python m venv axelrod_development
$ source axelrod_development/bin/activate
$ pip install r requirements.txt
The git workflow¶
There are two important branches in this repository:
dev
: The most up to date branch with no failing tests. This is the default branch on github.release
: The latest release.
When working on a new contribution branch from the latest dev
branch and
open a Pull Request on github from your branch to the dev
branch.
The procedure for a new release (this is carried out by one of core maintainers):
 Create a Pull Request from
dev
torelease
which should include an update toaxelrod/version.py
andCHANGES.md
 Create a git tag.
 Push to github.
 Create a release on github.
 Push to PyPi:
python setup.py sdist bdist_wheel upload
Contributing a strategy¶
This section contains a variety of tutorials that should help you contribute a new strategy to the library.
Contents:
Instructions¶
Here is the file structure for the Axelrod repository:
.
├── axelrod
│ └── __init__.py
│ └── ecosystem.py
│ └── game.py
│ └── player.py
│ └── plot.py
│ └── result_set.py
│ └── round_robin.py
│ └── tournament.py
│ └── /strategies/
│ └── __init__.py
│ └── _strategies.py
│ └── cooperator.py
│ └── defector.py
│ └── grudger.py
│ └── titfortat.py
│ └── gobymajority.py
│ └── ...
│ └── /tests/
│ └── integration
│ └── strategies
│ └── unit
│ └── test_*.py
└── README.md
To contribute a strategy you need to follow as many of the following steps as possible:
 Fork the github repository.
 Add a
<strategy>.py
file to the strategies directory or add a strategy to a pre existing<strategy>.py
file.  Update the
./axelrod/strategies/_strategies.py
file.  If you created a new
<strategy>.py
file add it to.docs/reference/all_strategies.rst
.  Write some unit tests in the
./axelrod/tests/strategies/
directory.  This one is also optional: ping us a message and we’ll add you to the Contributors team. This would add an AxelrodPython organisation badge to your profile.
 Send us a pull request.
If you would like a hand with any of the above please do get in touch: we’re always delighted to have new strategies.
Writing the new strategy¶
If you’re not sure if you have a strategy that has already been implemented, you can search the Strategy index to see if they are implemented. If you are still unsure please get in touch: via the gitter room or open an issue.
Several strategies are special cases of other strategies. For example, both
Cooperator
and Defector
are special cases of Random
,
Random(1)
and Random(0)
respectively. While we could eliminate
Cooperator
in its current
form, these strategies are intentionally left as is as simple examples for new
users and contributors. Nevertheless, please feel free to update the docstrings
of strategies like Random
to point out such cases.
There are a couple of things that need to be created in a strategy.py file. Let
us take a look at the TitForTat
class (located in the
axelrod/strategies/titfortat.py
file):
class TitForTat(Player):
"""
A player starts by cooperating and then mimics previous move by
opponent.
Note that the code for this strategy is written in a fairly verbose
way. This is done so that it can serve as an example strategy for
those who might be new to Python.
Names
 Rapoport's strategy: [Axelrod1980]_
 TitForTat: [Axelrod1980]_
"""
# These are various properties for the strategy
name = 'Tit For Tat'
classifier = {
'memory_depth': 1, # FourVector = (1.,0.,1.,0.)
'stochastic': False,
'inspects_source': False,
'manipulates_source': False,
'manipulates_state': False
}
def strategy(self, opponent):
"""This is the actual strategy"""
# First move
if len(self.history) == 0:
return C
# React to the opponent's last move
if opponent.history[1] == D:
return D
return C
The first thing that is needed is a docstring that explains what the strategy does:
"""A player starts by cooperating and then mimics previous move by opponent."""
Secondly, any alternate names should be included and if possible references provided (this helps when trying to identify if a strategy has already been implemented or not):
 Rapoport's strategy: [Axelrod1980]_
 TitForTat: [Axelrod1980]_
These references can be found in the Bibliography. If a required references is not there please feel free to add it or just get in touch and we’d be happy to help.
After that simply add in the string that will appear as the name of the strategy:
name = 'Tit For Tat'
Note that this is mainly used in plots by matplotlib
so you can use
LaTeX if you want to. For example there is strategy with \(\pi\) as a
name:
name = '$\pi$'
Following that you can add in the classifier
dictionary:
classifier = {
'memory_depth': 1, # FourVector = (1.,0.,1.,0.)
'stochastic': False,
'inspects_source': False,
'manipulates_source': False,
'manipulates_state': False
}
This helps classify the strategy as described in Classification of strategies.
After that the only thing required is to write the strategy
method
which takes an opponent as an argument. In the case of
TitForTat
the
strategy checks if it has any history (if len(self.history) == 0
). If
it does not (ie this is the first play of the match) then it returns C
.
If not, the strategy simply repeats the opponent’s last move (return
opponent.history[1]
):
def strategy(opponent):
"""This is the actual strategy"""
# First move
if len(self.history) == 0:
return C
# Repeat the opponent's last move
return opponent.history[1]
The variables C
and D
represent the cooperate and defect actions
respectively.
Some strategies make specific use of the variables of a match to create their
own attributes. In principle these attributes could change throughout a match
or tournament if the match properties (like the game matrix) change, so we
require that this logic live in the receive_match_attributes
method for
correct dynamic updating. Here is how this is done for Stalker
:
def receive_match_attributes(self)
R, P, S, T = self.match_attributes["game"].RPST()
self.very_good_score = R
self.very_bad_score = P
self.wish_score = (R + P) / 2
There are various examples of helpful functions and properties that make writing strategies easier. Do not hesitate to get in touch with the AxelrodPython team for guidance.
Writing docstrings¶
The project takes pride in its documentation for the strategies and its corresponding bibliography. The docstring is a string which describes a method, module or class. The docstrings help the user in understanding the working of the strategy and the source of the strategy. The docstring must be written in the following way, i.e.:
"""This is a docstring.
It can be written over multiple lines.
"""
The Sections of the docstring are:
Working of the strategy
A brief summary on how the strategy works, E.g.:
class TitForTat(Player): """ A player starts by cooperating and then mimics the previous action of the opponent. """
Bibliography/Source of the strategy
A section to mention the source of the strategy or the paper from which the strategy was taken. The section must start with the Names section. For E.g.:
class TitForTat(Player): """ A player starts by cooperating and then mimics the previous action of the opponent. Names:  Rapoport's strategy: [Axelrod1980]_  TitForTat: [Axelrod1980]_ """
Here, the info written under the Names section tells about the source of the TitforTat strategy.
[Axelrod1980]_
corresponds to the bibliographic item indocs/reference/bibliography.rst
. If you are using a source that is not in the bibliography please add it.
Adding the new strategy¶
To get the strategy to be recognised by the library we need to add it to the
files that initialise when someone types import axelrod
. This is done
in the axelrod/strategies/_strategies.py
file.
To classify the new strategy, run rebuild_classifier_table:
python rebuild_classifier_table.py
This will update axelrod/strategies/_strategies.py
. Check that the
recorded classifications for the strategies are what you expected.
If you have added your strategy to a file that already existed (perhaps you
added a new variant of titfortat
to the titfortat.py
file),
simply add your strategy to the list of strategies already imported from
<file_name>.py
:
from <file_name> import <listofstrategies>
If you have added your strategy to a new file then simply add a line similar to above with your new strategy.
Once you have done that, you need to add the class itself to the
all_strategies
list (in axelrod/strategies/_strategies.py
).
You will also need to increment the doctest in
axelrod/docs/index.rst
.
Finally, if you have created a new module (a new <strategy.py>
file)
please add it to the docs/references/all_strategies.rst
file so that it
will automatically be documented.
Classifying the new strategy¶
Every strategy class has a classifier dictionary that gives some classification of the strategy according to certain dimensions. Some of the classifiers have formulas that try to compute the value for different strategies. Where these exist, they’re overridden by the values defined in this dictionary. When creating a new strategy, you should try to fill out all of the dictionary.
Let us take a look at the dimensions available by looking at TitForTat
:
>>> import axelrod
>>> classifier = axelrod.TitForTat.classifier
>>> for key in sorted(classifier.keys()):
... print(key)
inspects_source
long_run_time
manipulates_source
manipulates_state
memory_depth
stochastic
You can read more about this in the Classification of strategies section but here are some tips about filling this part in correctly.
Note that when an instance of a class is created it gets it’s own copy of the
default classifier dictionary from the class. This might sometimes be modified by
the initialisation depending on input parameters. A good example of this is the
Joss
strategy:
>>> joss = axelrod.FirstByJoss()
>>> boring_joss = axelrod.FirstByJoss(p=1)
>>> axelrod.Classifiers["stochastic"](joss)
True
>>> axelrod.Classifiers["stochastic"](boring_joss)
False
A classifier value defined on the instance overrides the value defined for the class.
There are currently three important dimensions that help identify if a strategy obeys axelrod’s original tournament rules.
inspects_source
 does the strategy ‘read’ any source code that it would not normally have access to. An example of this isGeller
.manipulates_source
 does the strategy ‘write’ any source code that it would not normally be able to. An example of this isMind Bender
.manipulates_state
 does the strategy ‘change’ any attributes that it would not normally be able to. An example of this isMind Reader
.
These dimensions are currently relevant to the obey_axelrod function which checks if a strategy obeys Axelrod’s original rules.
Writing tests for the new strategy¶
To write tests you either need to create a file called test_<library>.py
where <library>.py
is the name of the file you have created or similarly
add tests to the test file that is already present in the
axelrod/tests/strategies/
directory.
Typically we want to test the following:
 That the strategy behaves as intended on the first move and subsequent moves, triggering any expected actions
 That the strategy initializes correctly
A TestPlayer
class has been written that has
a member function versus_test
which can be used to test how the player
plays against a given opponent.
It takes an optional keyword
argument seed
(useful and necessary for stochastic strategies,
None
by default):
self.versus_test(opponent=axelrod.MockPlayer(actions=[C, D]),
expected_actions=[(D, C), (C, D), (C, C)], seed=None)
In this case the player is tested against an opponent that will cycle through
C, D
. The expected_actions
are the actions played by both
the tested player and the opponent in the match. In this case we see that the
player is expected to play D, C, C
against C, D, C
.
Note that you can either user a MockPlayer
that will cycle through a
given sequence or you can use another strategy from the Axelrod library.
The function versus_test
also accepts a dictionary parameter of
attributes to check at the end of the match. For example this test checks
if the player’s internal variable opponent_class
is set to
"Cooperative"
:
actions = [(C, C)] * 6
self.versus_test(axelrod.Cooperator(), expected_actions=actions
attrs={"opponent_class": "Cooperative"})
Note here that instead of passing a sequence of actions as an opponent we are passing an actual player from the axelrod library.
The function versus_test
also accepts a dictionary parameter of match
attributes that dictate the knowledge of the players. For example this test
assumes that players do not know the length of the match:
actions = [(C, C), (C, D), (D, C), (C, D)]
self.versus_test(axelrod.Alternator(), expected_actions=actions,
match_attributes={"length": float("inf")})
The function versus_test
also accepts a dictionary parameter of
keyword arguments that dictate how the player is initiated. For example this
tests how the player plays when initialised with p=1
:
actions = [(C, C), (C, D), (C, C), (C, D)]
self.versus_test(axelrod.Alternator(), expected_actions=actions,
init_kwargs={"p": 1})
As an example, the tests for TitForTat are as follows:
import axelrod
from test_player import TestPlayer
C, D = axelrod.Action.C, axelrod.Action.D
class TestTitForTat(TestPlayer):
"""
Note that this test is referred to in the documentation as an example on
writing tests. If you modify the tests here please also modify the
documentation.
"""
name = "Tit For Tat"
player = axelrod.TitForTat
expected_classifier = {
'memory_depth': 1,
'stochastic': False,
'makes_use_of': set(),
'inspects_source': False,
'manipulates_source': False,
'manipulates_state': False
}
def test_strategy(self):
self.first_play_test(C)
self.second_play_test(rCC=C, rCD=D, rDC=C, rDD=D)
# Play against opponents
actions = [(C, C), (C, D), (D, C), (C, D)]
self.versus_test(axelrod.Alternator(), expected_actions=actions)
actions = [(C, C), (C, C), (C, C), (C, C)]
self.versus_test(axelrod.Cooperator(), expected_actions=actions)
actions = [(C, D), (D, D), (D, D), (D, D)]
self.versus_test(axelrod.Defector(), expected_actions=actions)
# This behaviour is independent of knowledge of the Match length
actions = [(C, C), (C, D), (D, C), (C, D)]
self.versus_test(axelrod.Alternator(), expected_actions=actions,
match_attributes={"length": float("inf")})
# We can also test against random strategies
actions = [(C, D), (D, D), (D, C), (C, C)]
self.versus_test(axelrod.Random(), expected_actions=actions,
seed=0)
actions = [(C, C), (C, D), (D, D), (D, C)]
self.versus_test(axelrod.Random(), expected_actions=actions,
seed=1)
# If you would like to test against a sequence of moves you should use
# a MockPlayer
opponent = axelrod.MockPlayer(actions=[C, D])
actions = [(C, C), (C, D), (D, C), (C, D)]
self.versus_test(opponent, expected_actions=actions)
opponent = axelrod.MockPlayer(actions=[C, C, D, D, C, D])
actions = [(C, C), (C, C), (C, D), (D, D), (D, C), (C, D)]
self.versus_test(opponent, expected_actions=actions)
There are other examples of using this testing framework in
axelrod/tests/strategies/test_titfortat.py
.
The expected_classifier
dictionary tests that the classification of the
strategy is as expected (the tests for this is inherited in the init
method). Please be sure to classify new strategies according to the already
present dimensions but if you create a new dimension you do not need to re
classify all the other strategies (but feel free to! :)), but please do add it
to the default_classifier
in the axelrod/player.py
parent class.
Contributing to the library¶
All contributions (docs, tests, etc) are very welcome, if there is a specific functionality that you would like to add then please open an issue (or indeed take a look at the ones already there and jump in the conversation!).
If you want to work on documentation please keep in mind that doctests are encouraged to help keep the documentation up to date.
Running tests¶
Basic test runners¶
The project has an extensive test suite which is run each time a new contribution is made to the repository. If you want to check that all the tests pass before you submit a pull request you can run the tests yourself:
$ python m unittest discover
If you are developing new tests for the suite, it is useful to run a single test file so that you don’t have to wait for the entire suite each time. For example, to run only the tests for the Grudger strategy:
$ python m unittest axelrod.tests.strategies.test_grudger
The test suite is divided into three categories: strategy tests, unit tests and integration tests. Each can be run individually:
$ python m unittest discover s axelrod.tests.strategies
$ python m unittest discover s axelrod.tests.unit
$ python m unittest discover s axelrod.tests.integration
Testing coverage of tests¶
The library has 100% test coverage. This can be tested using the Python
coverage
package. Once installed (pip install coverage
), to run
the tests and check the coverage for the entire library:
$ coverage run source=axelrod m unittest discover
You can then view a report of the coverage:
$ coverage report m
You can also run the coverage on a subset of the tests. For example, to run the tests with coverage for the Grudger strategy:
$ coverage run source=axelrod m unittest axelrod.tests.strategies.test_grudger
Testing the documentation¶
The documentation is doctested, to run those tests you can run the script:
$ python doctests.py
You can also run the doctests on any given file. For example, to run the
doctests for the docs/tutorials/getting_started/match.rst
file:
$ python m doctest docs/tutorials/getting_started/match.rst
Type checking¶
The library makes use of type hinting, this can be checked using the Python
mypy
package. Once installed (pip install mypy
), to run the type checker:
$ python run_mypy.py
You can also run the type checker on a given file. For example, to run the type checker on the Grudger strategy:
$ mypy ignoremissingimports followimports skip axelrod/strategies/grudger.py
Discussion¶
This section is a discussion of various aspects of Game Theory related to the Axelrod library.
Contents:
Background to Axelrod’s Tournament¶
Another nice write up of Axelrod’s work and this tournament on github was put together by Artem Kaznatcheev here.
The Prisoner’s Dilemma¶
The Prisoner’s dilemma is the simple two player game shown below:
Cooperate  Defect  

Cooperate  (3,3)  (0,5) 
Defect  (5,0)  (1,1) 
If both players cooperate they will each go to prison for 2 years and receive an equivalent utility of 3. If one cooperates and the other defects: the defector does not go to prison and the cooperator goes to prison for 5 years, the cooperator receives a utility of 0 and the defector a utility of 5. If both defect: they both go to prison for 4 years and receive an equivalent utility of 1.
Note
Years in prison doesn’t equal to utility directly. The formula is U = 5  Y for Y in [0, 5], where U
is the utility, Y
are years in prison. The reason is to follow the original Axelrod’s scoring.
By simply investigating the best responses against both possible actions of each player it is immediate to see that the Nash equilibrium for this game is for both players to defect.
The Iterated Prisoner’s Dilemma¶
We can use the basic Prisoner’s Dilemma as a stage game in a repeated game. Players now aim to maximise the utility (corresponding to years in prison) over a repetition of the game. Strategies can take in to account both players history and so can take the form:
“I will cooperate unless you defect 3 times in a row at which point I will defect forever.”
Axelrod ran such a tournament (twice) and invited strategies from anyone who would contribute. The tournament was a round robin and the winner was the strategy who had the lowest total amount of time in prison.
This tournament has been used to study how cooperation can evolve from a very simple set of rules. This is mainly because the winner of both tournaments was ‘tit for tat’: a strategy that would never defect first (referred to as a ‘nice’ strategy).
Strategy Archetypes¶
Finite State Machines¶
A finite state machine (FSM) is a general computation model. In the context of Axelrod, it’s a set of states and “transitions.” A transition for a given state/previousopponentaction combination says how the strategy will respond, both in what action it will take and in what state it will transitions to. That is a transition will specify that in state \(a\), the strategy will respond to action \(X\) by taking action \(Y\) and moving to state \(b\) (which will tell us which transitions to use in later moves). We may write this transition \((a, X, b, Y)\). For Axelrod, a FSM must have a full set of transitions, which specifies a unique response for each state/previousopponentaction combination.
See [Harper2017] for a moredetailed explanation.
Representing a strategy as a finite state machine has been useful in some research (see [Harper2017] or [Ashlock2006b]). Though it’s theoretically possible to represent all strategies as FSMs, this is impractical for most strategies. However, some strategies lend themselves naturally to a FSM representation. For example, for the Iterated Prisoner’s Dilemma, we could consider a strategy that cooperates (C) until the opponent defects (D) twice in a row, then defect forever thereafter. (We’ll call this strategy grudger_2 for the example.) We could call state 1, the state where the opponent hasn’t started a defect streak; state 2, the state where the opponent is on a 1defect streak; and state 3, the state where the opponent has defected twice in a row at some point. Then the transitions would be:
>>> from axelrod import Action
>>> C, D = Action.C, Action.D
>>> grudger_2_transitions = (
... (1, C, 1, C),
... (1, D, 2, C),
... (2, C, 1, C),
... (2, D, 3, D),
... (3, C, 3, D),
... (3, D, 3, D)
... )
The Axelrod library includes a FSM metastrategy player, which will you let you specify a player’s strategy by this transition matrix, along with an initial state and initial action. The syntax for this is:
>>> from axelrod.strategies.finite_state_machines import FSMPlayer
>>> grudger_2 = FSMPlayer(transitions=grudger_2_transitions,
... initial_state=1, initial_action=C)
The library also includes the functionality to compute the memory from the set of transitions. In the grudger_2 example, the memory would be 2. Because either the strategy’s own previous move was a defect (in which case, continue to defect) or we just need to check if the last two opponent moves were defects or not. Though this function takes the transitions in a slightly different format:
>>> transition_dict = {
... (t[0], t[1]): (t[2], t[3]) for t in grudger_2_transitions
... }
>>> from axelrod.compute_finite_state_machine_memory import *
>>> get_memory_from_transitions(transitions=transition_dict,
... initial_state=1)
2
Overview of past Tournaments¶
Axelrod’s first tournament¶
Axelrod’s first tournament is described in his 1980 paper entitled ‘Effective choice in the Prisoner’s Dilemma’ [Axelrod1980]. This tournament included 14 strategies (plus a random “strategy”) and they are listed below, (ranked in the order in which they appeared).
An indication is given as to whether or not this strategy is implemented in the
axelrod
library. If this strategy is not implemented please do send us a
pull request.
Name  Author  Axelrod Library Name 

Tit For Tat  Anatol Rapoport  TitForTat 
Tideman and Chieruzzi  T Nicolaus Tideman and Paula Chieruzz  TidemanAndChieruzzi 
Nydegger  Rudy Nydegger  Nydegger 
Grofman  Bernard Grofman  Grofman 
Shubik  Martin Shubik  Shubik 
Stein and Rapoport  Stein and Anatol Rapoport  SteinAndRapoport 
Grudger  James W Friedman  Grudger 
Davis  Morton Davis  Davis 
Graaskamp  Jim Graaskamp  Graaskamp 
FirstByDowning  Leslie Downing  RevisedDowning 
Feld  Scott Feld  Feld 
Joss  Johann Joss  Joss 
Tullock  Gordon Tullock  Tullock 
(Name withheld)  Unknown  UnnamedStrategy 
Random  Unknownd  Random 
Axelrod’s second tournament¶
The code for Axelrod’s second touranment was originally published by the University of Michigan Center for the Study of Complex Systems and is now available from Robert Axelrod’s personal website subject to a disclaimer which states:
“All materials in this archive are copyright (c) 1996, Robert Axelrod, unless otherwise noted. You are free to download these materials and use them without restriction.”
The AxelrodPython organisation has published a modified version of the original code. In the following table, links to original code point to the AxelrodPython repository.
Original Code  Author  Axelrod Library Name 

GRASR  Unknown  Not Implemented 
K31R  Gail Grisell  GoByMajority 
K32R  Charles Kluepfel  SecondByKluepfel 
K33R  Harold Rabbie  Not Implemented 
K34R  James W Friedman  Grudger 
K35R  Abraham Getzler  Not Implemented 
K36R  Roger Hotz  Not Implemented 
K37R  George Lefevre  Not Implemented 
K38R  Nelson Weiderman  Not Implemented 
K39R  Tom Almy  Not Implemented 
K40R  Robert Adams  Not Implemented 
K41R  Herb Weiner  SecondByWeiner 
K42R  Otto Borufsen  SecondByBorufsen 
K43R  R D Anderson  Not Implemented 
K44R  William Adams  SecondByWmAdams 
K45R  Michael F McGurrin  Not Implemented 
K46R  Graham J Eatherley  SecondByEatherley 
K47R  Richard Hufford  SecondByRichardHufford 
K48R  George Hufford  Not Implemented 
K49R  Rob Cave  SecondByCave 
K50R  Rik Smoody  Not Implemented 
K51R  John Willaim Colbert  Not Implemented 
K52R  David A Smith  Not Implemented 
K53R  Henry Nussbacher  Not Implemented 
K54R  William H Robertson  Not Implemented 
K55R  Steve Newman  Not Implemented 
K56R  Stanley F Quayle  Not Implemented 
K57R  Rudy Nydegger  Not Implemented 
K58R  Glen Rowsam  SecondByRowsam 
K59R  Leslie Downing  RevisedDowning 
K60R  Jim Graaskamp and Ken Katzen  SecondByGraaskampKatzen 
K61R  Danny C Champion  SecondByChampion 
K62R  Howard R Hollander  Not Implemented 
K63R  George Duisman  Not Implemented 
K64R  Brian Yamachi  SecondByYamachi 
K65R  Mark F Batell  Not Implemented 
K66R  Ray Mikkelson  Not Implemented 
K67R  Craig Feathers  SecondByTranquilizer 
K68R  Fransois Leyvraz  SecondByLeyvraz 
K69R  Johann Joss  Not Implemented 
K70R  Robert Pebly  Not Implemented 
K71R  James E Hall  Not Implemented 
K72R  Edward C White Jr  SecondByWhite 
K73R  George Zimmerman  Not Implemented 
K74R  Edward Friedland  Not Implemented 
K74RXX  Edward Friedland  Not Implemented 
K75R  Paul D Harrington  SecondByHarrington 
K76R  David Gladstein  SecondByGladstein 
K77R  Scott Feld  Not Implemented 
K78R  Fred Mauk  Not Implemented 
K79R  Dennis Ambuehl and Kevin Hickey  Not Implemented 
K80R  Robyn M Dawes and Mark Batell  Not Implemented 
K81R  Martyn Jones  Not Implemented 
K82R  Robert A Leyland  Not Implemented 
K83R  Paul E Black  SecondByWhite 
K84R  T Nicolaus Tideman and Paula Chieruzzi  SecondByTidemanChieruzzi 
K85R  Robert B Falk and James M Langsted  Not Implemented 
K86R  Bernard Grofman  Not Implemented 
K87R  E E H Schurmann  Not Implemented 
K88R  Scott Appold  SecondByAppold 
K89R  Gene Snodgrass  Not Implemented 
K90R  John Maynard Smith  TitFor2Tats 
K91R  Jonathan Pinkley  Not Implemented 
K92R  Anatol Rapoport  TitForTat 
K93R  Unknown  Not Implemented 
KPAVLOVC  Unknown  WinStayLoseShift 
KRANDOMC  Unknown  Random 
KTF2TC  Unknown  TitFor2Tats 
KTITFORTATC  Unknown  TitForTat 
Stewart and Plotkin’s Tournament (2012)¶
In 2012, Alexander Stewart and Joshua Plotkin ran a variant of Axelrod’s tournament with 19 strategies to test the effectiveness of the then newly discovered ZeroDeterminant strategies.
The paper is identified as doi: 10.1073/pnas.1208087109 and referred to as [Stewart2012] below. Unfortunately the details of the tournament and the implementation of strategies is not clear in the manuscript. We can, however, make reasonable guesses to the implementation of many strategies based on their names and classical definitions.
The following classical strategies are included in the library:
S&P Name  Long Name  Axelrod Library Name 

ALLC  Always Cooperate  Cooperator 
ALLD  Always Defect  Defector 
EXTORT2  Extort2  ZDExtort2 
HARD_MAJO  Hard majority  HardGoByMajority 
HARD_JOSS  Hard Joss  Joss 
HARD_TFT  Hard tit for tat  HardTitForTat 
HARD_TF2T  Hard tit for 2 tats  HardTitFor2Tats 
TFT  TitForTat  TitForTat 
GRIM  Grim  Grudger 
GTFT  Generous TitForTat  GTFT 
TF2T  TitForTwoTats  TitFor2Tats 
WSLS  WinStayLoseShift  WinStayLoseShift 
RANDOM  Random  Random 
ZDGTFT2  ZDGTFT2  ZDGTFT2 
ALLC, ALLD, TFT and RANDOM are defined above. The remaining classical strategies are defined below. The tournament also included two Zero Determinant strategies, both implemented in the library. The full table of strategies and results is available online.
Memory one strategies¶
In 2012 Press and Dyson [Press2012] showed interesting results with regards to so called memory one strategies. Stewart and Plotkin implemented a number of these. A memory one strategy is simply a probabilistic strategy that is defined by 4 parameters. These four parameters dictate the probability of cooperating given 1 of 4 possible outcomes of the previous round:
 \(P(C\,\,CC) = p_1\)
 \(P(C\,\,CD) = p_2\)
 \(P(C\,\,DC) = p_3\)
 \(P(C\,\,DD) = p_4\)
The memory one strategy class is used to define a number of strategies below.
GTFT¶
GenerousTitForTat plays TitForTat with occasional forgiveness, which prevents cycling defections against itself.
GTFT is defined as a memoryone strategy as follows:
 \(P(C\,\,CC) = 1\)
 \(P(C\,\,CD) = p\)
 \(P(C\,\,DC) = 1\)
 \(P(C\,\,DD) = p\)
where \(p = \min\left(1  \frac{TR}{RS}, \frac{RP}{TP}\right)\).
GTFT came 2nd in average score and 18th in wins in S&P’s tournament.
TF2T¶
TitForTwoTats is like TitForTat but only retaliates after two defections rather than one. This is not a memoryone strategy.
TF2T came 3rd in average score and last (?) in wins in S&P’s tournament.
WSLS¶
WinStayLoseShift is a strategy that shifts if the highest payoff was not earned in the previous round. WSLS is also known as “WinStayLoseSwitch” and “Pavlov”. It can be seen as a memoryone strategy as follows:
 \(P(C\,\,CC) = 1\)
 \(P(C\,\,CD) = 0\)
 \(P(C\,\,DC) = 0\)
 \(P(C\,\,DD) = 1\)
WSLS came 7th in average score and 13th in wins in S&P’s tournament.
RANDOM¶
Random is a strategy that was defined in Axelrod’s first tournament, note that this is also a memoryone strategy:
 \(P(C\,\,CC) = 0.5\)
 \(P(C\,\,CD) = 0.5\)
 \(P(C\,\,DC) = 0.5\)
 \(P(C\,\,DD) = 0.5\)
RANDOM came 8th in average score and 8th in wins in S&P’s tournament.
ZDGTFT2¶
This memoryone strategy is defined by the following four conditional probabilities based on the last round of play:
 \(P(C\,\,CC) = 1\)
 \(P(C\,\,CD) = 1/8\)
 \(P(C\,\,DC) = 1\)
 \(P(C\,\,DD) = 1/4\)
This strategy came 1st in average score and 16th in wins in S&P’s tournament.
EXTORT2¶
This memoryone strategy is defined by the following four conditional probabilities based on the last round of play:
 \(P(C\,\,CC) = 8/9\)
 \(P(C\,\,CD) = 1/2\)
 \(P(C\,\,DC) = 1/3\)
 \(P(C\,\,DD) = 0\)
This strategy came 18th in average score and 2nd in wins in S&P’s tournament.
GRIM¶
Grim is not defined in [Stewart2012] but it is defined elsewhere as follows. GRIM (also called “Grim trigger”), cooperates until the opponent defects and then always defects thereafter. In the library this strategy is called Grudger.
GRIM came 10th in average score and 11th in wins in S&P’s tournament.
HARD_JOSS¶
HARD_JOSS is not defined in [Stewart2012] but is otherwise defined as a strategy that plays like TitForTat but cooperates only with probability \(0.9\). This is a memoryone strategy with the following probabilities:
 \(P(C\,\,CC) = 0.9\)
 \(P(C\,\,CD) = 0\)
 \(P(C\,\,DC) = 1\)
 \(P(C\,\,DD) = 0\)
HARD_JOSS came 16th in average score and 4th in wins in S&P’s tournament.
HARD_JOSS as described above is implemented in the library as Joss and is the same as the Joss strategy from Axelrod’s first tournament.
HARD_MAJO¶
HARD_MAJO is not defined in [Stewart2012] and is presumably the same as “Go by Majority”, defined as follows: the strategy defects on the first move, defects if the number of defections of the opponent is greater than or equal to the number of times it has cooperated, and otherwise cooperates,
HARD_MAJO came 13th in average score and 5th in wins in S&P’s tournament.
HARD_TFT¶
Hard TFT is not defined in [Stewart2012] but is elsewhere defined as follows. The strategy cooperates on the first move, defects if the opponent has defected on any of the previous three rounds, and otherwise cooperates.
HARD_TFT came 12th in average score and 10th in wins in S&P’s tournament.
HARD_TF2T¶
Hard TF2T is not defined in [Stewart2012] but is elsewhere defined as follows. The strategy cooperates on the first move, defects if the opponent has defected twice (successively) of the previous three rounds, and otherwise cooperates.
HARD_TF2T came 6th in average score and 17th in wins in S&P’s tournament.
Calculator¶
This strategy is not unambiguously defined in [Stewart2012] but is defined elsewhere. Calculator plays like Joss for 20 rounds. On the 21 round, Calculator attempts to detect a cycle in the opponents history, and defects unconditionally thereafter if a cycle is found. Otherwise Calculator plays like TFT for the remaining rounds.
Prober¶
PROBE is not unambiguously defined in [Stewart2012] but is defined elsewhere as Prober. The strategy starts by playing D, C, C on the first three rounds and then defects forever if the opponent cooperates on rounds two and three. Otherwise Prober plays as TitForTat would.
Prober came 15th in average score and 9th in wins in S&P’s tournament.
Prober2¶
PROBE2 is not unambiguously defined in [Stewart2012] but is defined elsewhere as Prober2. The strategy starts by playing D, C, C on the first three rounds and then cooperates forever if the opponent played D then C on rounds two and three. Otherwise Prober2 plays as TitForTat would.
Prober2 came 9th in average score and 12th in wins in S&P’s tournament.
Prober3¶
PROBE3 is not unambiguously defined in [Stewart2012] but is defined elsewhere as Prober3. The strategy starts by playing D, C on the first two rounds and then defects forever if the opponent cooperated on round two. Otherwise Prober3 plays as TitForTat would.
Prober3 came 17th in average score and 7th in wins in S&P’s tournament.
HardProber¶
HARD_PROBE is not unambiguously defined in [Stewart2012] but is defined elsewhere as HardProber. The strategy starts by playing D, D, C, C on the first four rounds and then defects forever if the opponent cooperates on rounds two and three. Otherwise Prober plays as TitForTat would.
HardProber came 5th in average score and 6th in wins in S&P’s tournament.
NaiveProber¶
NAIVE_PROBER is a modification of Tit For Tat strategy which with a small probability randomly defects. Default value for a probability of defection is 0.1.
Beaufils et al.’s tournament (1997)¶
In 1997, [Beaufils1997] the authors used a tournament to describe a new strategy of their called “Gradual”. The description given in the paper of “Gradual” is:
This strategy acts as titfortat, except when it is time to forgive and remember the past. It uses cooperation on the first move and then continues to do so as long as the other player cooperates. Then after the first defection of the other player, it defects one time and cooperates two times; after the second defection of the opponent, it defects two times and cooperates two times, … after the nth defection it reacts with n consecutive defections and then calms down its opponent with two cooperations.
This is the only description of the strategy however the paper does include a table of results of the tournament. The scores of “Gradual” against the opponents (including itself) are:
Name  Name used in [Beaufils1997]  Score (1000 turns) 

Cooperator  coop  3000 
Defector  def  915 
Random  rand  2815 
Tit For Tat  tft  3000 
Grudger  spite  3000 
Cycler DDC  p_nst  2219 
Cycler CCD  p_kn  3472 
Go By Majority  sft_mj  3000 
Suspicious Tit For Tat  mist  2996 
Prober  prob  2999 
Gradual  grad  3000 
Win Stay Lose Shift  pav  3000 
The following code reproduces the above:
>>> import axelrod as axl
>>> players = [axl.Cooperator(),
... axl.Defector(),
... axl.Random(),
... axl.TitForTat(),
... axl.Grudger(),
... axl.CyclerDDC(),
... axl.CyclerCCD(),
... axl.GoByMajority(),
... axl.SuspiciousTitForTat(),
... axl.Prober(),
... axl.OriginalGradual(),
... axl.WinStayLoseShift(),
... ]
>>> turns = 1000
>>> tournament = axl.Tournament(players, turns=turns, repetitions=1, seed=75)
>>> results = tournament.play(progress_bar=False)
>>> for average_score_per_turn in results.payoff_matrix[2]:
... print(round(average_score_per_turn * turns, 1))
3000.0
915.0
2763.0
3000.0
3000.0
2219.0
3472.0
3000.0
2996.0
2999.0
3000.0
3000.0
The OriginalGradual
strategy implemented has the following description:
A player that punishes defections with a growing number of defections but after punishing for punishment_limit number of times enters a calming state and cooperates no matter what the opponent does for two rounds.
The punishment_limit is incremented whenever the opponent defects and the strategy is not in either calming or punishing state.
Note that a different version of Gradual appears in [CRISTALSMAC2018]. This was brought to the attention of the maintainers of the library by one of the authors of [Beaufils1997] and is documented here https://github.com/AxelrodPython/Axelrod/issues/1294.
The strategy implemented in [CRISTALSMAC2018] and defined here as Gradual
has the following description:
Similar to OriginalGradual, this is a player that punishes defections with a growing number of defections but after punishing for punishment_limit number of times enters a calming state and cooperates no matter what the opponent does for two rounds.
This version of Gradual is an update of OriginalGradual and the difference is that the punishment_limit is incremented whenever the opponent defects (regardless of the state of the player).
This highlights the importance of best practice and reproducible computational research. Both strategies implemented in this library are fully tested and documented clearly and precisely.
Play Contexts and Generic Prisoner’s Dilemma¶
There are four possible round outcomes:
 Mutual cooperation: \((C, C)\)
 Defection: \((C, D)\) or \((D, C)\)
 Mutual defection: \((D, D)\)
Each of these corresponds to one particular set of payoffs in the following generic Prisoner’s dilemma:
Cooperate  Defect  

Cooperate  (R,R)  (S,T) 
Defect  (T,S)  (P,P) 
For the above to constitute a Prisoner’s dilemma, the following must hold: \(T>R>P>S\).
These payoffs are commonly referred to as:
 \(R\): the Reward payoff (default value in the library: 3)
 \(P\): the Punishment payoff (default value in the library: 1)
 \(S\): the Sucker payoff (default value in the library: 0)
 \(T\): the Temptation payoff (default value in the library: 5)
A particular Prisoner’s Dilemma is often described by the 4tuple: \((R, P, S, T)\):
>>> import axelrod
>>> axelrod.game.DefaultGame.RPST()
(3, 1, 0, 5)
Community¶
Contents:
Part of the team¶
If you’re reading this you’re probably interested in contributing to and/or using the Axelrod library! Firstly: thank you and welcome!
We are proud of the library and the environment that surrounds it. A primary goal of the project is to cultivate an open and welcoming community, considerate and respectful to newcomers to python and game theory.
The Axelrod library has been a first contribution to open source software for many, and this is in large part due to the fact that we all aim to help and encourage all levels of contribution. If you’re a beginner, that’s awesome! You’re very welcome and don’t hesitate to ask for help.
With regards to any contribution, please do not feel the need to wait until your contribution is perfectly polished and complete: we’re happy to offer early feedback, help with git, and anything else that you need to have a positive experience.
If you are using the library for your own work and there’s anything in the documentation that is unclear: we want to know so that we can fix it. We also want to help so please don’t hesitate to get in touch.
Communication¶
There are various ways of communicating with the team:
 Gitter: a web based chat client, you can talk directly to the users and maintainers of the library.
 Irc: we have an irc channel. It’s #axelrodpython on freenode.
 Email forum.
 Issues: you are also very welcome to open an issue on github
 Twitter. This account periodically tweets out random match and tournament results; you’re welcome to get in touch through twitter as well.
Code of Conduct¶
Our Pledge¶
In the interest of fostering an open and welcoming environment, we as contributors and maintainers pledge to making participation in our project and our community a harassmentfree experience for everyone, regardless of age, body size, disability, ethnicity, gender identity and expression, level of experience, nationality, personal appearance, race, religion, or sexual identity and orientation.
Our Standards¶
Examples of behavior that contributes to creating a positive environment include:
 Using welcoming and inclusive language
 Being respectful of differing viewpoints and experiences
 Gracefully accepting constructive criticism
 Focusing on what is best for the community
 Showing empathy towards other community members
Examples of unacceptable behavior by participants include:
 The use of sexualized language or imagery and unwelcome sexual attention or advances
 Trolling, insulting/derogatory comments, and personal or political attacks
 Public or private harassment
 Publishing others’ private information, such as a physical or electronic address, without explicit permission
 Other conduct which could reasonably be considered inappropriate in a professional setting
Our Responsibilities¶
Project maintainers are responsible for clarifying the standards of acceptable behavior and are expected to take appropriate and fair corrective action in response to any instances of unacceptable behavior.
Project maintainers have the right and responsibility to remove, edit, or reject comments, commits, code, wiki edits, issues, and other contributions that are not aligned to this Code of Conduct, or to ban temporarily or permanently any contributor for other behaviors that they deem inappropriate, threatening, offensive, or harmful.
Scope¶
This Code of Conduct applies both within project spaces and in public spaces when an individual is representing the project or its community. Examples of representing a project or community include using an official project email address, posting via an official social media account, or acting as an appointed representative at an online or offline event. Representation of a project may be further defined and clarified by project maintainers.
Enforcement¶
Instances of abusive, harassing, or otherwise unacceptable behavior may be reported by contacting a member of the core team. All complaints will be reviewed and investigated and will result in a response that is deemed necessary and appropriate to the circumstances. The project team is obligated to maintain confidentiality with regard to the reporter of an incident. Further details of specific enforcement policies may be posted separately.
Project maintainers who do not follow or enforce the Code of Conduct in good faith may face temporary or permanent repercussions as determined by other members of the project’s leadership.
Attribution¶
This Code of Conduct is adapted from the Contributor Covenant, version 1.4, available at http://contributorcovenant.org/version/1/4
Reference¶
This section is the reference guide for the various components of the library.
Bibliography¶
This is a collection of various bibliographic items referenced in the documentation.
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[Axelrod1984]  The Evolution of Cooperation. Basic Books. ISBN 0465021212. 
[Axelrod1995]  Wu, J. and Axelrod, R. (1995). How to cope with noise in the Iterated prisoner’s dilemma, Journal of Conflict Resolution, 39(1), pp. 183–189. doi: 10.1177/0022002795039001008. 
[Banks1990]  Banks, J. S., & Sundaram, R. K. (1990). Repeated games, finite automata, and complexity. Games and Economic Behavior, 2(2), 97–117. http://doi.org/10.1016/08998256(90)90024O 
[Bendor1993]  Bendor, Jonathan. “Uncertainty and the Evolution of Cooperation.” The Journal of Conflict Resolution, 37(4), 709–734. 
[Beaufils1997]  Beaufils, B. & Delahaye, J. & Mathieu, P. (1997). Our Meeting With Gradual: A Good Strategy For The Iterated Prisoner’s Dilemma. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.4041 
[Berg2015]  Berg, P. Van Den, & Weissing, F. J. (2015). The importance of mechanisms for the evolution of cooperation. Proceedings of the Royal Society BBiological Sciences, 282. 
[CRISTALSMAC2018]  CRISTAL Lab, SMAC Team, Lille University (2018). IPD : the Iterated Prisoner’s Dilemma. https://github.com/cristalsmac/ipd 
[Downing1975]  Downing, Leslie L. “The Prisoner’s Dilemma game as a problemsolving phenomenon: An outcome maximization interpretation.” Simulation & Games 6.4 (1975): 366391. 
[Eckhart2015]  Eckhart Arnold (2016) CoopSim v0.9.9 beta 6. https://github.com/jecki/CoopSim/ 
[Frean1994]  Frean, Marcus R. “The Prisoner’s Dilemma without Synchrony.” Proceedings: Biological Sciences, vol. 257, no. 1348, 1994, pp. 75–79. www.jstor.org/stable/50253. 
[Harper2017]  Harper, M., Knight, V., Jones, M., Koutsovoulos, G., Glynatsi, N. E., & Campbell, O. (2017) Reinforcement learning produces dominant strategies for the Iterated Prisoner’s Dilemma. PloS one. https://doi.org/10.1371/journal.pone.0188046 
[Hauert2002]  Hauert, Christoph, and Olaf Stenull. “Simple adaptive strategy wins the prisoner’s dilemma.” Journal of Theoretical Biology 218.3 (2002): 261272. 
[Hilbe2013]  Hilbe, C., Nowak, M.A. and Traulsen, A. (2013). Adaptive dynamics of extortion and compliance, PLoS ONE, 8(11), p. e77886. doi: 10.1371/journal.pone.0077886. 
[Hilbe2017]  Hilbe, C., MartinezVaquero, L. A., Chatterjee K., Nowak M. A. (2017). Memoryn strategies of direct reciprocity, Proceedings of the National Academy of Sciences May 2017, 114 (18) 47154720; doi: 10.1073/pnas.1621239114. 
[Kuhn2017]  Kuhn, Steven, “Prisoner’s Dilemma”, The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2017/entries/prisonerdilemma/ 
[Kraines1989]  Kraines, David, and Vivian Kraines. “Pavlov and the prisoner’s dilemma.” Theory and decision 26.1 (1989): 4779. doi:10.1007/BF00134056 
[LessWrong2011]  Zoo of Strategies (2011) LessWrong. Available at: http://lesswrong.com/lw/7f2/prisoners_dilemma_tournament_results/ 
[Li2007]  Li, J, How to Design a Strategy to Win an IPD Tournament, in Kendall G., Yao X. and Chong S. (eds.) The iterated prisoner’s dilemma: 20 years on. World Scientific, chapter 4, pp. 2940, 2007. 
[Li2009]  Li, J. & Kendall, G. (2009). A Strategy with Novel Evolutionary Features for the Iterated Prisoner’s Dilemma. Evolutionary Computation 17(2): 257–274. 
[Li2011]  Li, J., Hingston, P., Member, S., & Kendall, G. (2011). Engineering Design of Strategies for Winning Iterated Prisoner ’ s Dilemma Competitions, 3(4), 348–360. 
[Li2014]  Li, J. and Kendall, G. (2014). The Effect of Memory Size on the Evolutionary Stability of Strategies in Iterated Prisoner’s Dilemma. IEEE Transactions on Evolutionary Computation, 18(6) 819826 
[LiS2014]  Li, Siwei. (2014). Strategies in the Stochastic Iterated Prisoner’s Dilemma. Available at: http://math.uchicago.edu/~may/REU2014/REUPapers/Li,Siwei.pdf 
[Luis2008]  Luis R. Izquierdo and Segismundo S. Izquierdo (2008). Dynamics of the BushMosteller Learning Algorithm in 2x2 Games, Reinforcement Learning, Cornelius Weber, Mark Elshaw and Norbert Michael Mayer (Ed.), InTech, DOI: 10.5772/5282. Available from: https://www.intechopen.com/books/reinforcement_learning/dynamics_of_the_bushmosteller_learning_algorithm_in_2x2_games 
[Mathieu2015]  Mathieu, P. and Delahaye, J. (2015). New Winning Strategies for the Iterated Prisoner’s Dilemma. Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. 
[Mittal2009]  Mittal, S., & Deb, K. (2009). Optimal strategies of the iterated prisoner’s dilemma problem for multiple conflicting objectives. IEEE Transactions on Evolutionary Computation, 13(3), 554–565. https://doi.org/10.1109/TEVC.2008.2009459 
[Nachbar1992]  Nachbar J., Evolution in the finitely repeated prisoner’s dilemma, Journal of Economic Behavior & Organization, 19(3): 307326, 1992. 
[NC2019]  https://github.com/ncase/trust (Accessed: 30 October 2019) 
[Nowak1989]  Nowak, Martin, and Karl Sigmund. “Gamedynamical aspects of the prisoner’s dilemma.” Applied Mathematics and Computation 30.3 (1989): 191213. 
[Nowak1990]  Nowak, M., & Sigmund, K. (1990). The evolution of stochastic strategies in the Prisoner’s Dilemma. Acta Applicandae Mathematica. https://link.springer.com/article/10.1007/BF00049570 
[Nowak1992]  Nowak, M.., & May, R. M. (1992). Evolutionary games and spatial chaos. Nature. http://doi.org/10.1038/359826a0 
[Nowak1993]  Nowak, M., & Sigmund, K. (1993). A strategy of winstay, loseshift that outperforms titfortat in the Prisoner’s Dilemma game. Nature, 364(6432), 56–58. http://doi.org/10.1038/364056a0 
[Ohtsuki2006]  Ohtsuki, Hisashi, et al. “A simple rule for the evolution of cooperation on graphs and social networks.” Nature 441.7092 (2006): 502. 
[PD2017]  http://www.prisonersdilemma.com/competition.html (Accessed: 6 June 2017) 
[Press2012]  Press, W. H., & Dyson, F. J. (2012). Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent. Proceedings of the National Academy of Sciences, 109(26), 10409–10413. http://doi.org/10.1073/pnas.1206569109 
[Prison1998]  LIFL (1998) PRISON. Available at: http://www.lifl.fr/IPD/ipd.frame.html (Accessed: 19 September 2016). 
[Robson1990]  Robson, Arthur J. “Efficiency in evolutionary games: Darwin, Nash and the secret handshake.” Journal of theoretical Biology 144.3 (1990): 379396. 
[Roemheld2013]  Roemheld, Lars. “Evolutionary Extortion and Mischief: Zero Determinant strategies in iterated 2x2 games”. Available at: https://arxiv.org/abs/1308.2576 
[SingerClark2014]  SingerClark, T. (2014). Morality Metrics On Iterated Prisoner’s Dilemma Players. 
[Shakarian2013]  Shakarian, P., Roos, P. & Moores, G. A Novel Analytical Method for Evolutionary Graph Theory Problems. 
[Slany2007]  Slany W. and Kienreich W., On some winning strategies for the iterated prisoner’s dilemma, in Kendall G., Yao X. and Chong S. (eds.) The iterated prisoner’s dilemma: 20 years on. World Scientific, chapter 8, pp. 171204, 2007. 
[Stewart2012]  Stewart, a. J., & Plotkin, J. B. (2012). Extortion and cooperation in the Prisoner’s Dilemma. Proceedings of the National Academy of Sciences, 109(26), 10134–10135. http://doi.org/10.1073/pnas.1208087109 
[Szabo2007]  Szabó, G., & Fáth, G. (2007). Evolutionary games on graphs. Physics Reports, 446(46), 97–216. http://doi.org/10.1016/j.physrep.2007.04.004 
[Gaudesi2016]  Gaudesi, Marco, et al. “Exploiting evolutionary modeling to prevail in iterated prisoner’s dilemma tournaments.” IEEE Transactions on Computational Intelligence and AI in Games 8.3 (2016): 288300. 
[Tzafestas2000]  Tzafestas, E. (2000). Toward adaptive cooperative behavior. From Animals to Animals: Proceedings of the 6th International Conference on the Simulation of Adaptive Behavior {(SAB2000)}, 2, 334–340. 
Strategy index¶
Here are the docstrings of all the strategies in the library.

class
axelrod.strategies.adaptive.
Adaptive
(initial_plays: List[axelrod.action.Action] = None)[source]¶ Start with a specific sequence of C and D, then play the strategy that has worked best, recalculated each turn.
Names:
 Adaptive: [Li2011]

class
axelrod.strategies.adaptor.
AbstractAdaptor
(delta: Dict[Tuple[axelrod.action.Action, axelrod.action.Action], float], perr: float = 0.01)[source]¶ An adaptive strategy that updates an internal state based on the last round of play. Using this state the player Cooperates with a probability derived from the state.
 s, float:
 the internal state, initially 0
 perr, float:
 an error threshold for misinterpreted moves
 delta, a dictionary of floats:
 additive update values for s depending on the last round’s outcome
Names:
 Adaptor: [Hauert2002]

class
axelrod.strategies.adaptor.
AdaptorBrief
[source]¶ An Adaptor trained on short interactions.
Names:
 AdaptorBrief: [Hauert2002]

class
axelrod.strategies.adaptor.
AdaptorLong
[source]¶ An Adaptor trained on long interactions.
Names:
 AdaptorLong: [Hauert2002]

class
axelrod.strategies.alternator.
Alternator
[source]¶ A player who alternates between cooperating and defecting.
Names
 Alternator: [Axelrod1984]
 Periodic player CD: [Mittal2009]

class
axelrod.strategies.ann.
ANN
(num_features: int, num_hidden: int, weights: List[float] = None)[source]¶ Artificial Neural Network based strategy.
A single layer neural network based strategy, with the following features: * Opponent’s first move is C * Opponent’s first move is D * Opponent’s second move is C * Opponent’s second move is D * Player’s previous move is C * Player’s previous move is D * Player’s second previous move is C * Player’s second previous move is D * Opponent’s previous move is C * Opponent’s previous move is D * Opponent’s second previous move is C * Opponent’s second previous move is D * Total opponent cooperations * Total opponent defections * Total player cooperations * Total player defections * Round number
Original Source: https://gist.github.com/mojones/550b32c46a8169bb3cd89d917b73111a#fileannstrategytestL60
Names
 Artificial Neural Network based strategy: Original name by Martin Jones

class
axelrod.strategies.ann.
EvolvableANN
(num_features: int, num_hidden: int, weights: List[float] = None, mutation_probability: float = None, mutation_distance: int = 5, seed: int = None)[source]¶ Evolvable version of ANN.

class
axelrod.strategies.ann.
EvolvedANN
[source]¶ A strategy based on a pretrained neural network with 17 features and a hidden layer of size 10.
Trained using the axelrod_dojo version: 0.0.8 Training data is archived at doi.org/10.5281/zenodo.1306926
Names:
 Evolved ANN: Original name by Martin Jones.

class
axelrod.strategies.ann.
EvolvedANN5
[source]¶ A strategy based on a pretrained neural network with 17 features and a hidden layer of size 5.
Trained using the axelrod_dojo version: 0.0.8 Training data is archived at doi.org/10.5281/zenodo.1306931
Names:
 Evolved ANN 5: Original name by Marc Harper.

class
axelrod.strategies.ann.
EvolvedANNNoise05
[source]¶ A strategy based on a pretrained neural network with a hidden layer of size 5, trained with noise=0.05.
Trained using the axelrod_dojo version: 0.0.8 Training data i archived at doi.org/10.5281/zenodo.1314247.
Names:
 Evolved ANN Noise 5: Original name by Marc Harper.

axelrod.strategies.ann.
activate
(bias: List[float], hidden: List[float], output: List[float], inputs: <Mock name='mock.ndarray' id='140582027384592'>) → float[source]¶  Compute the output of the neural network:
 output = relu(inputs * hidden_weights + bias) * output_weights

axelrod.strategies.ann.
compute_features
(player: axelrod.player.Player, opponent: axelrod.player.Player) → <Mock name='mock.ndarray' id='140582027384592'>[source]¶ Compute history features for Neural Network: * Opponent’s first move is C * Opponent’s first move is D * Opponent’s second move is C * Opponent’s second move is D * Player’s previous move is C * Player’s previous move is D * Player’s second previous move is C * Player’s second previous move is D * Opponent’s previous move is C * Opponent’s previous move is D * Opponent’s second previous move is C * Opponent’s second previous move is D * Total opponent cooperations * Total opponent defections * Total player cooperations * Total player defections * Round number

axelrod.strategies.ann.
split_weights
(weights: List[float], num_features: int, num_hidden: int) → Tuple[List[List[float]], List[float], List[float]][source]¶ Splits the input vector into the the NN bias weights and layer parameters.

class
axelrod.strategies.apavlov.
APavlov2006
[source]¶ APavlov attempts to classify its opponent as one of five strategies: Cooperative, ALLD, STFT, PavlovD, or Random. APavlov then responds in a manner intended to achieve mutual cooperation or to defect against uncooperative opponents.
Names:
 Adaptive Pavlov 2006: [Li2007]

class
axelrod.strategies.apavlov.
APavlov2011
[source]¶ APavlov attempts to classify its opponent as one of four strategies: Cooperative, ALLD, STFT, or Random. APavlov then responds in a manner intended to achieve mutual cooperation or to defect against uncooperative opponents.
Names:
 Adaptive Pavlov 2011: [Li2011]

class
axelrod.strategies.appeaser.
Appeaser
[source]¶ A player who tries to guess what the opponent wants.
Switch the classifier every time the opponent plays D. Start with C, switch between C and D when opponent plays D.
Names:
 Appeaser: Original Name by Jochen Müller

class
axelrod.strategies.averagecopier.
AverageCopier
[source]¶ The player will cooperate with probability p if the opponent’s cooperation ratio is p. Starts with random decision.
Names:
 Average Copier: Original name by Geraint Palmer

class
axelrod.strategies.averagecopier.
NiceAverageCopier
[source]¶ Same as Average Copier, but always starts by cooperating.
Names:
 Average Copier: Original name by Owen Campbell
Strategies submitted to Axelrod’s first tournament. All strategies in this module are prefixed by FirstBy to indicate that they were submitted in Axelrod’s First tournament by the given author.
Note that these strategies are implemented from the descriptions presented in:
Axelrod, R. (1980). Effective Choice in the Prisoner’s Dilemma. Journal of Conflict Resolution, 24(1), 3–25.
These descriptions are not always clear and/or precise and when assumptions have been made they are explained in the strategy docstrings.

class
axelrod.strategies.axelrod_first.
FirstByAnonymous
[source]¶ Submitted to Axelrod’s first tournament by a graduate student whose name was withheld.
The description written in [Axelrod1980] is:
> “This rule has a probability of cooperating, P, which is initially 30% and > is updated every 10 moves. P is adjusted if the other player seems random, > very cooperative, or very uncooperative. P is also adjusted after move 130 > if the rule has a lower score than the other player. Unfortunately, the > complex process of adjustment frequently left the probability of cooperation > in the 30% to 70% range, and therefore the rule appeared random to many > other players.”
Given the lack of detail this strategy is implemented based on the final sentence of the description which is to have a cooperation probability that is uniformly random in the 30 to 70% range.
Names:
 (Name withheld): [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByDavis
(rounds_to_cooperate: int = 10)[source]¶ Submitted to Axelrod’s first tournament by Morton Davis.
The description written in [Axelrod1980] is:
> “A player starts by cooperating for 10 rounds then plays Grudger, > defecting if at any point the opponent has defected.”
This strategy came 8th in Axelrod’s original tournament.
Names:
 Davis: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByDowning
[source]¶ Submitted to Axelrod’s first tournament by Downing
The description written in [Axelrod1980] is:
> “This rule selects its choice to maximize its own longterm expected payoff on > the assumption that the other rule cooperates with a fixed probability which > depends only on whether the other player cooperated or defected on the previous > move. These two probabilities estimates are continuously updated as the game > progresses. Initially, they are both assumed to be .5, which amounts to the > pessimistic assumption that the other player is not responsive. This rule is > based on an outcome maximization interpretation of human performances proposed > by Downing (1975).”
The Downing (1975) paper is “The Prisoner’s Dilemma Game as a ProblemSolving Phenomenon” [Downing1975] and this is used to implement the strategy.
There are a number of specific points in this paper, on page 371:
> “[…] In these strategies, O’s [the opponent’s] response on trial N is in some way dependent or contingent on S’s [the subject’s] response on trial N 1. All varieties of these lagone matching strategies can be defined by two parameters: the conditional probability that O will choose C following C by S, P(C_o  C_s) and the conditional probability that O will choose C following D by S, P(C_o, D_s).”
Throughout the paper the strategy (S) assumes that the opponent (O) is playing a reactive strategy defined by these two conditional probabilities.
The strategy aims to maximise the long run utility against such a strategy and the mechanism for this is described in Appendix A (more on this later).
One final point from the main text is, on page 372:
> “For the various lagone matching strategies of O, the maximizing strategies of S will be 100% C, or 100% D, or for some strategies all S strategies will be functionally equivalent.”
This implies that the strategy S will either always cooperate or always defect (or be indifferent) dependent on the opponent’s defining probabilities.
To understand the particular mechanism that describes the strategy S, we refer to Appendix A of the paper on page 389.
The stated goal of the strategy is to maximize (using the notation of the paper):
EV_TOT = #CC(EV_CC) + #CD(EV_CD) + #DC(EV_DC) + #DD(EV_DD)This differs from the more modern literature where #CC, #CD, #DC and #DD would imply that counts of both players playing C and C, or the first playing C and the second D etc… In this case the author uses an argument based on the sequence of plays by the player (S) so #CC denotes the number of times the player plays C twice in a row.
On the second page of the appendix, figure 4 (page 390) identifies an expression for EV_TOT. A specific term is made to disappear in the case of T  R = P  S (which is not the case for the standard (R, P, S, T) = (3, 1, 0, 5)):
> “Where (t  r) = (p  s), EV_TOT will be a function of alpha, beta, t, r, p, s and N are known and V which is unknown.
V is the total number of cooperations of the player S (this is noted earlier in the abstract) and as such the final expression (with only V as unknown) can be used to decide if V should indicate that S always cooperates or not.
This final expression is used to show that EV_TOT is linear in the number of cooperations by the player thus justifying the fact that the player will always cooperate or defect.
All of the above details are used to give the following interpretation of the strategy:
1. On any given turn, the strategy will estimate alpha = P(C_o  C_s) and beta = P(C_o  D_s). 2. The strategy will calculate the expected utility of always playing C OR always playing D against the estimated probabilities. This corresponds to:
In the case of the player always cooperating:
P_CC = alpha and P_CD = 1  alpha
In the case of the player always defecting:
P_DC = beta and P_DD = 1  beta
Using this we have:
E_C = alpha R + (1  alpha) S E_D = beta T + (1  beta) PThus at every turn, the strategy will calculate those two values and cooperate if E_C > E_D and will defect if E_C < E_D.
In the case of E_C = E_D, the player will alternate from their previous move. This is based on specific sentence from Axelrod’s original paper:
> “Under certain circumstances, DOWNING will even determine that the best > strategy is to alternate cooperation and defection.”
One final important point is the early game behaviour of the strategy. It has been noted that this strategy was implemented in a way that assumed that alpha and beta were both 1/2:
> “Initially, they are both assumed to be .5, which amounts to the > pessimistic assumption that the other player is not responsive.”
Note that if alpha = beta = 1 / 2 then:
E_C = alpha R + alpha S E_D = alpha T + alpha PAnd from the defining properties of the Prisoner’s Dilemma (T > R > P > S) this gives: E_D > E_C. Thus, the player opens with a defection in the first two rounds. Note that from the Axelrod publications alone there is nothing to indicate defections on the first two rounds, although a defection in the opening round is clear. However there is a presentation available at http://www.sci.brooklyn.cuny.edu/~sklar/teaching/f05/alife/notes/azharipdOct19th.pdf That clearly states that Downing defected in the first two rounds, thus this is assumed to be the behaviour. Interestingly, in future tournaments this strategy was revised to not defect on the opening two rounds.
It is assumed that these first two rounds are used to create initial estimates of beta = P(C_o  D_s) and we will use the opening play of the player to estimate alpha = P(C_o  C_s). Thus we assume that the opponents first play is a response to a cooperation “before the match starts”.
So for example, if the plays are:
[(D, C), (D, C)]
Then the opponent’s first cooperation counts as a cooperation in response to the non existent cooperation of round 0. The total number of cooperations in response to a cooperation is 1. We need to take in to account that extra phantom cooperation to estimate the probability alpha=P(C_o  C_s) as 1 / 1 = 1.
This is an assumption with no clear indication from the literature.
– This strategy came 10th in Axelrod’s original tournament.
Names:
 Downing: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByFeld
(start_coop_prob: float = 1.0, end_coop_prob: float = 0.5, rounds_of_decay: int = 200)[source]¶ Submitted to Axelrod’s first tournament by Scott Feld.
The description written in [Axelrod1980] is:
> “This rule starts with tit for tat and gradually lowers its probability of > cooperation following the other’s cooperation to .5 by the two hundredth > move. It always defects after a defection by the other.”
This strategy plays Tit For Tat, always defecting if the opponent defects but cooperating when the opponent cooperates with a gradually decreasing probability until it is only .5. Note that the description does not clearly indicate how the cooperation probability should drop. This implements a linear decreasing function.
This strategy came 11th in Axelrod’s original tournament.
Names:
 Feld: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByGraaskamp
(alpha: float = 0.05)[source]¶ Submitted to Axelrod’s first tournament by James Graaskamp.
The description written in [Axelrod1980] is:
> “This rule plays tit for tat for 50 moves, defects on move 51, and then > plays 5 more moves of tit for tat. A check is then made to see if the player > seems to be RANDOM, in which case it defects from then on. A check is also > made to see if the other is TIT FOR TAT, ANALOGY (a program from the > preliminary tournament), and its own twin, in which case it plays tit for > tat. Otherwise it randomly defects every 5 to 15 moves, hoping that enough > trust has been built up so that the other player will not notice these > defections.:
This is implemented as:
 Plays Tit For Tat for the first 50 rounds;
 Defects on round 51;
 Plays 5 further rounds of Tit For Tat;
 A check is then made to see if the opponent is playing randomly in which case it defects for the rest of the game. This is implemented with a chi squared test.
 The strategy also checks to see if the opponent is playing Tit For Tat or a clone of itself. If so it plays Tit For Tat. If not it cooperates and randomly defects every 5 to 15 moves.
Note that there is no information about ‘Analogy’ available thus Step 5 is a “best possible” interpretation of the description in the paper. Furthermore the test for the clone is implemented as checking that both players have played the same moves for the entire game. This is unlikely to be the original approach but no further details are available.
This strategy came 9th in Axelrod’s original tournament.
Names:
 Graaskamp: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByGrofman
[source]¶ Submitted to Axelrod’s first tournament by Bernard Grofman.
The description written in [Axelrod1980] is:
> “If the players did different things on the previous move, this rule > cooperates with probability 2/7. Otherwise this rule always cooperates.”This strategy came 4th in Axelrod’s original tournament.
Names:
 Grofman: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByJoss
(p: float = 0.9)[source]¶ Submitted to Axelrod’s first tournament by Johann Joss.
The description written in [Axelrod1980] is:
> “This rule cooperates 90% of the time after a cooperation by the other. It > always defects after a defection by the other.”
This strategy came 12th in Axelrod’s original tournament.
Names:
 Joss: [Axelrod1980]
 Hard Joss: [Stewart2012]

class
axelrod.strategies.axelrod_first.
FirstByNydegger
[source]¶ Submitted to Axelrod’s first tournament by Rudy Nydegger.
The description written in [Axelrod1980] is:
> “The program begins with tit for tat for the first three moves, except > that if it was the only one to cooperate on the first move and the only one > to defect on the second move, it defects on the third move. After the third > move, its choice is determined from the 3 preceding outcomes in the > following manner. Let A be the sum formed by counting the other’s defection > as 2 points and one’s own as 1 point, and giving weights of 16, 4, and 1 to > the preceding three moves in chronological order. The choice can be > described as defecting only when A equals 1, 6, 7, 17, 22, 23, 26, 29, 30, > 31, 33, 38, 39, 45, 49, 54, 55, 58, or 61. Thus if all three preceding moves > are mutual defection, A = 63 and the rule cooperates. This rule was > designed for use in laboratory experiments as a stooge which had a memory > and appeared to be trustworthy, potentially cooperative, but not gullible > (Nydegger, 1978).”
The program begins with tit for tat for the first three moves, except that if it was the only one to cooperate on the first move and the only one to defect on the second move, it defects on the third move. After the third move, its choice is determined from the 3 preceding outcomes in the following manner.
\[A = 16 a_1 + 4 a_2 + a_3\]Where \(a_i\) is dependent on the outcome of the previous \(i\) th round. If both strategies defect, \(a_i=3\), if the opponent only defects: \(a_i=2\) and finally if it is only this strategy that defects then \(a_i=1\).
Finally this strategy defects if and only if:
\[A \in \{1, 6, 7, 17, 22, 23, 26, 29, 30, 31, 33, 38, 39, 45, 49, 54, 55, 58, 61\}\]Thus if all three preceding moves are mutual defection, A = 63 and the rule cooperates. This rule was designed for use in laboratory experiments as a stooge which had a memory and appeared to be trustworthy, potentially cooperative, but not gullible.
This strategy came 3rd in Axelrod’s original tournament.
Names:
 Nydegger: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstByShubik
[source]¶ Submitted to Axelrod’s first tournament by Martin Shubik.
The description written in [Axelrod1980] is:
> “This rule cooperates until the other defects, and then defects once. If > the other defects again after the rule’s cooperation is resumed, the rule > defects twice. In general, the length of retaliation is increased by one for > each departure from mutual cooperation. This rule is described with its > strategic implications in Shubik (1970). Further treatment of its is given > in Taylor (1976).
There is some room for interpretation as to how the strategy reacts to a defection on the turn where it starts to cooperate once more. In Shubik (1970) the strategy is described as:
> “I will play my move 1 to begin with and will continue to do so, so long > as my information shows that the other player has chosen his move 1. If my > information tells me he has used move 2, then I will use move 2 for the > immediate k subsequent periods, after which I will resume using move 1. If > he uses his move 2 again after I have resumed using move 1, then I will > switch to move 2 for the k + 1 immediately subsequent periods … and so > on, increasing my retaliation by an extra period for each departure from the > (1, 1) steady state.”
This is interpreted as:
The player cooperates, if when it is cooperating, the opponent defects it defects for k rounds. After k rounds it starts cooperating again and increments the value of k if the opponent defects again.
This strategy came 5th in Axelrod’s original tournament.
Names:
 Shubik: [Axelrod1980]

class
axelrod.strategies.axelrod_first.
FirstBySteinAndRapoport
(alpha: float = 0.05)[source]¶ Submitted to Axelrod’s first tournament by William Stein and Amnon Rapoport.
The description written in [Axelrod1980] is:
> “This rule plays tit for tat except that it cooperates on the first four > moves, it defects on the last two moves, and every fifteen moves it checks > to see if the opponent seems to be playing randomly. This check uses a > chisquared test of the other’s transition probabilities and also checks for > alternating moves of CD and DC.
This is implemented as follows:
 It cooperates for the first 4 moves.
 It defects on the last 2 moves.
 Every 15 moves it makes use of a chisquared test to check if the opponent is playing randomly. If so it defects.
This strategy came 6th in Axelrod’s original tournament.
Names:
 SteinAndRapoport: [Axelrod1980]

original_class
¶ alias of
FirstBySteinAndRapoport

class
axelrod.strategies.axelrod_first.
FirstByTidemanAndChieruzzi
[source]¶ Submitted to Axelrod’s first tournament by Nicolas Tideman and Paula Chieruzzi.
The description written in [Axelrod1980] is:
> “This rule begins with cooperation and tit for tat. However, when the > other player finishes his second run of defec tions, an extra punishment is > instituted, and the number of punishing defections is increased by one with > each run of the other’s defections. The other player is given a fresh start > if he is 10 or more points behind, if he has not just started a run of > defections, if it has been at least 20 moves since a fresh start, if there > are at least 10 moves remaining, and if the number of defections differs > from a 5050 random generator by at least 3.0 standard deviations. A fresh > start involves two cooperations and then play as if the game had just > started. The program defects automatically on the last two moves.”
This is interpreted as:
1. Every run of defections played by the opponent increases the number of defections that this strategy retaliates with by 1.
 The opponent is given a ‘fresh start’ if:
 it is 10 points behind this strategy
 and it has not just started a run of defections
 and it has been at least 20 rounds since the last ‘fresh start’
 and there are more than 10 rounds remaining in the match
 and the total number of defections differs from a 5050 random sample by at least 3.0 standard deviations.
A ‘fresh start’ is a sequence of two cooperations followed by an assumption that the game has just started (everything is forgotten).
The strategy defects on the last two moves.
This strategy came 2nd in Axelrod’s original tournament.
Names:
 TidemanAndChieruzzi: [Axelrod1980]

original_class
¶ alias of
FirstByTidemanAndChieruzzi

class
axelrod.strategies.axelrod_first.
FirstByTullock
[source]¶ Submitted to Axelrod’s first tournament by Gordon Tullock.
The description written in [Axelrod1980] is:
> “This rule cooperates on the first eleven moves. It then cooperates 10% > less than the other player has cooperated on the preceding ten moves. This > rule is based on an idea developed in Overcast and Tullock (1971). Professor > Tullock was invited to specify how the idea could be implemented, and he did > so out of scientific interest rather than an expectation that it would be a > likely winner.”
This is interpreted as:
Cooperates for the first 11 rounds then randomly cooperates 10% less often than the opponent has in the previous 10 rounds.
This strategy came 13th in Axelrod’s original tournament.
Names:
 Tullock: [Axelrod1980]
Strategies from Axelrod’s second tournament. All strategies in this module are prefixed by SecondBy to indicate that they were submitted in Axelrod’s Second tournament by the given author.

class
axelrod.strategies.axelrod_second.
SecondByAppold
[source]¶ Strategy submitted to Axelrod’s second tournament by Scott Appold (K88R) and came in 22nd in that tournament.
Cooperates for first four turns.
After four turns, will cooperate immediately following the first time the opponent cooperates (starting with the opponent’s fourth move). Otherwise will cooperate with probability equal to:
 If this strategy defected two turns ago, the portion of the time (historically) that the opponent followed a defection with a cooperation.
 If this strategy cooperated two turns ago, the portion of the time (historically) that the opponent followed a cooperation with a cooperation. The opponent’s first move is counted as a response to a cooperation.
Names:
 Appold: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByBlack
[source]¶ Strategy submitted to Axelrod’s second tournament by Paul E Black (K83R) and came in fifteenth in that tournament.
The strategy Cooperates for the first five turns. Then it calculates the number of opponent defects in the last five moves and Cooperates with probability prob_coop`[`number_defects], where:
prob_coop[number_defects] = 1  (number_defects^ 2  1) / 25
Names:
 Black: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByBorufsen
[source]¶ Strategy submitted to Axelrod’s second tournament by Otto Borufsen (K32R), and came in third in that tournament.
This player keeps track of the the opponent’s responses to own behavior:
 cd_count counts: Opponent cooperates as response to player defecting.
 cc_count counts: Opponent cooperates as response to player cooperating.
The player has a defect mode and a normal mode. In defect mode, the player will always defect. In normal mode, the player obeys the following ranked rules:
 If in the last three turns, both the player/opponent defected, then cooperate for a single turn.
 If in the last three turns, the player/opponent acted differently from each other and they’re alternating, then change next defect to cooperate. (Doesn’t block third rule.)
 Otherwise, do titfortat.
Start in normal mode, but every 25 turns starting with the 27th turn, reevaluate the mode. Enter defect mode if any of the following conditions hold:
 Detected random: Opponent cooperated 718 times since last mode evaluation (or start) AND less than 70% of opponent cooperation was in response to player’s cooperation, i.e. cc_count / (cc_count+cd_count) < 0.7
 Detect defective: Opponent cooperated fewer than 3 times since last mode evaluation.
When switching to defect mode, defect immediately. The first two rules for normal mode require that last three turns were in normal mode. When starting normal mode from defect mode, defect on first move.
Names:
 Borufsen: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByCave
[source]¶ Strategy submitted to Axelrod’s second tournament by Rob Cave (K49R), and came in fourth in that tournament.
First look for overlydefective or apparently random opponents, and defect if found. That is any opponent meeting one of:
 turn > 39 and percent defects > 0.39
 turn > 29 and percent defects > 0.65
 turn > 19 and percent defects > 0.79
Otherwise, respond to cooperation with cooperation. And respond to defections with either a defection (if opponent has defected at least 18 times) or with a random (50/50) choice. [Cooperate on first.]
Names:
 Cave: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByChampion
[source]¶ Strategy submitted to Axelrod’s second tournament by Danny Champion.
This player cooperates on the first 10 moves and plays Tit for Tat for the next 15 more moves. After 25 moves, the program cooperates unless all the following are true: the other player defected on the previous move, the other player cooperated less than 60% and the random number between 0 and 1 is greater that the other player’s cooperation rate.
Names:
 Champion: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByColbert
[source]¶ Strategy submitted to Axelrod’s second tournament by William Colbert (K51R) and came in eighteenth in that tournament.
In the first eight turns, this strategy Coopearates on all but the sixth turn, in which it Defects. After that, the strategy responds to an opponent Cooperation with a single Cooperation, and responds to a Defection with a chain of responses: Defect, Defect, Cooperate, Cooperate. During this chain, the strategy ignores opponent’s moves.
Names:
 Colbert: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByEatherley
[source]¶ Strategy submitted to Axelrod’s second tournament by Graham Eatherley.
A player that keeps track of how many times in the game the other player defected. After the other player defects, it defects with a probability equal to the ratio of the other’s total defections to the total moves to that point.
Names:
 Eatherley: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByGetzler
[source]¶ Strategy submitted to Axelrod’s second tournament by Abraham Getzler (K35R) and came in eleventh in that tournament.
Strategy Defects with probability flack, where flack is calculated as the sum over opponent Defections of 0.5 ^ (turns ago Defection happened).
Names:
 Getzler: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByGladstein
[source]¶ Submitted to Axelrod’s second tournament by David Gladstein.
This strategy is also known as Tester and is based on the reverse engineering of the Fortran strategies from Axelrod’s second tournament.
This strategy is a TFT variant that defects on the first round in order to test the opponent’s response. If the opponent ever defects, the strategy ‘apologizes’ by cooperating and then plays TFT for the rest of the game. Otherwise, it defects as much as possible subject to the constraint that the ratio of its defections to moves remains under 0.5, not counting the first defection.
Names:
 Gladstein: [Axelrod1980b]
 Tester: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByGraaskampKatzen
[source]¶ Strategy submitted to Axelrod’s second tournament by Jim Graaskamp and Ken Katzen (K60R), and came in sixth in that tournament.
Play TitforTat at first, and track own score. At select checkpoints, check for a high score. Switch to Default Mode if:
 On move 11, score < 23
 On move 21, score < 53
 On move 31, score < 83
 On move 41, score < 113
 On move 51, score < 143
 On move 101, score < 293
Once in Defect Mode, defect forever.
Names:
 GraaskampKatzen: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByGrofman
[source]¶ Submitted to Axelrod’s second tournament by Bernard Grofman.
This strategy has 3 phases:
 First it cooperates on the first two rounds
 For rounds 37 inclusive, it plays the same as the opponent’s last move
 Thereafter, it applies the following logic, looking at its memory of the last 8* rounds (ignoring the most recent round).
 If its own previous move was C and the opponent has defected less than 3 times in the last 8* rounds, cooperate
 If its own previous move was C and the opponent has defected 3 or more times in the last 8* rounds, defect
 If its own previous move was D and the opponent has defected only once or not at all in the last 8* rounds, cooperate
 If its own previous move was D and the opponent has defected more than once in the last 8* rounds, defect
The code looks at the first 7 of the last 8 rounds, ignoring the most recent round.
Names:  Grofman’s strategy: [Axelrod1980b]  K86R: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByHarrington
[source]¶ Strategy submitted to Axelrod’s second tournament by Paul Harrington (K75R) and came in eighth in that tournament.
This strategy has three modes: Normal, Fairweather, and Defect. These mode names were not present in Harrington’s submission.
In Normal and Fairweather modes, the strategy begins by:
 Update history
 Try to detect random opponent if turn is multiple of 15 and >=30.
 Check if burned flag should be raised.
 Check for Fairweather opponent if turn is 38.
Updating history means to increment the correct cell of the move_history. move_history is a matrix where the columns are the opponent’s previous move and the rows are indexed by the combo of this player’s and the opponent’s moves two turns ago. [The upperleft cell must be all Cooperations, but otherwise order doesn’t matter.] After we enter Defect mode, move_history won’t be used again.
If the turn is a multiple of 15 and >=30, then attempt to detect random. If random is detected, enter Defect mode and defect immediately. If the player was previously in Defect mode, then do not reenter. The random detection logic is a modified Pearson’s Chi Squared test, with some additional checks. [More details in detect_random docstrings.]
Some of this player’s moves are marked as “generous.” If this player made a generous move two turns ago and the opponent replied with a Defect, then raise the burned flag. This will stop certain generous moves later.
The player mostly plays TitforTat for the first 36 moves, then defects on the 37th move. If the opponent cooperates on the first 36 moves, and defects on the 37th move also, then enter Fairweather mode and cooperate this turn. Entering Fairweather mode is extremely rare, since this can only happen if the opponent cooperates for the first 36 then defects unprovoked on the 37th. (That is, this player’s first 36 moves are also Cooperations, so there’s nothing really to trigger an opponent Defection.)
Next in Normal Mode:
 Check for defect and parity streaks.
 Check if cooperations are scheduled.
 Otherwise,
 If turn < 37, TitforTat.
 If turn = 37, defect, mark this move as generous, and schedule two more cooperations**.
 If turn > 37, then if burned flag is raised, then TitforTat. Otherwise, TitforTat with probability 1  prob. And with probability prob, defect, schedule two cooperations, mark this move as generous, and increase prob by 5%.
** Scheduling two cooperations means to set more_coop flag to two. If in Normal mode and no streaks are detected, then the player will cooperate and lower this flag, until hitting zero. It’s possible that the flag can be overwritten. Notable on the 37th turn defect, this is set to two, but the 38th turn Fairweather check will set this.
If the opponent’s last twenty moves were defections, then defect this turn. Then check for a parity streak, by flipping the parity bit (there are two streaks that get tracked which are something like odd and even turns, but this flip bit logic doesn’t get run every turn), then incrementing the parity streak that we’re pointing to. If the parity streak that we’re pointing to is then greater than parity_limit then reset the streak and cooperate immediately. parity_limit is initially set to five, but after it has been hit eight times, it decreases to three. The parity streak that we’re pointing to also gets incremented if in normal mode and we defect but not on turn 38, unless we are defecting as the result of a defect streak. Note that the parity streaks resets but the defect streak doesn’t.
If more_coop >= 1, then we cooperate and lower that flag here, in Normal mode after checking streaks. Still lower this flag if cooperating as the result of a parity streak or in Fairweather mode.
Then use the logic based on turn from above.
In FairWeather mode after running the code from above, check if opponent defected last turn. If so, exit FairWeather mode, and proceed THIS TURN with Normal mode. Otherwise cooperate.
In Defect mode, update the exit_defect_meter (originally zero) by incrementing if opponent defected last turn and decreasing by three otherwise. If exit_defect_meter is then 11, then set mode to Normal (for future turns), cooperate and schedule two more cooperations. [Note that this move is not marked generous.]
Names:
 Harrington: [Axelrod1980b]

calculate_chi_squared
(turn)[source]¶ Pearson’s Chi Squared statistic = sum[ (E_iO_i)^2 / E_i ], where O_i are the observed matrix values, and E_i is calculated as number (of defects) in the row times the number in the column over (total number in the matrix minus 1). Equivalently, we expect we expect (for an independent distribution) the total number of recorded turns times the portion in that row times the portion in that column.
In this function, the statistic is nonstandard in that it excludes summands where E_i <= 1.

detect_parity_streak
(last_move)[source]¶ Switch which parity_streak we’re pointing to and incerement if the opponent’s last move was a Defection. Otherwise reset the flag. Then return true if and only if the parity_streak is at least parity_limit.
This is similar to detect_streak with alternating streaks, except that these streaks get incremented elsewhere as well.

detect_random
(turn)[source]¶ We check if the topleft cell of the matrix (corresponding to all Cooperations) has over 80% of the turns. In which case, we label nonrandom.
Then we check if over 75% or under 25% of the opponent’s turns are Defections. If so, then we label as nonrandom.
Otherwise we calculates a modified Pearson’s Chi Squared statistic on self.history, and returns True (is random) if and only if the statistic is less than or equal to 3.

detect_streak
(last_move)[source]¶ Return true if and only if the opponent’s last twenty moves are defects.

class
axelrod.strategies.axelrod_second.
SecondByKluepfel
[source]¶ Strategy submitted to Axelrod’s second tournament by Charles Kluepfel (K32R).
This player keeps track of the the opponent’s responses to own behavior:
 cd_count counts: Opponent cooperates as response to player defecting.
 dd_count counts: Opponent defects as response to player defecting.
 cc_count counts: Opponent cooperates as response to player cooperating.
 dc_count counts: Opponent defects as response to player cooperating.
After 26 turns, the player then tries to detect a random player. The player decides that the opponent is random if cd_counts >= (cd_counts+dd_counts)/2  0.75*sqrt(cd_counts+dd_counts) AND cc_counts >= (dc_counts+cc_counts)/2  0.75*sqrt(dc_counts+cc_counts). If the player decides that they are playing against a random player, then they will always defect.
Otherwise respond to recent history using the following set of rules:
 If opponent’s last three choices are the same, then respond in kind.
 If opponent’s last two choices are the same, then respond in kind with probability 90%.
 Otherwise if opponent’s last action was to cooperate, then cooperate with probability 70%.
 Otherwise if opponent’s last action was to defect, then defect with probability 60%.
Names:
 Kluepfel: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByLeyvraz
[source]¶ Strategy submitted to Axelrod’s second tournament by Fransois Leyvraz (K68R) and came in twelfth in that tournament.
The strategy uses the opponent’s last three moves to decide on an action based on the following ordered rules.
 If opponent Defected last two turns, then Defect with prob 75%.
 If opponent Defected three turns ago, then Cooperate.
 If opponent Defected two turns ago, then Defect.
 If opponent Defected last turn, then Defect with prob 50%.
 Otherwise (all Cooperations), then Cooperate.
Names:
 Leyvraz: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByMikkelson
[source]¶ Strategy submitted to Axelrod’s second tournament by Ray Mikkelson (K66R) and came in twentieth in that tournament.
The strategy keeps track of a variable called credit, which determines if the strategy will Cooperate, in the sense that if credit is positive, then the strategy Cooperates. credit is initialized to 7. After the first turn, credit increments if the opponent Cooperated last turn, and decreases by two otherwise. credit is capped above by 8 and below by 7. [credit is assessed as postive or negative, after increasing based on opponent’s last turn.]
If credit is nonpositive within the first ten turns, then the strategy Defects and credit is set to 4. If credit is nonpositive later, then the strategy Defects if and only if (total # opponent Defections) / (turn#) is at least 15%. [Turn # starts at 1.]
Names:
 Mikkelson: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByRichardHufford
[source]¶ Strategy submitted to Axelrod’s second tournament by Richard Hufford (K47R) and came in sixteenth in that tournament.
The strategy tracks opponent “agreements”, that is whenever the opponent’s previous move is the some as this player’s move two turns ago. If the opponent’s first move is a Defection, this is counted as a disagreement, and otherwise an agreement. From the agreement counts, two measures are calculated:
 proportion_agree: This is the number of agreements (through opponent’s last turn) + 2 divided by the current turn number.
 last_four_num: The number of agreements in the last four turns. If there have been fewer than four previous turns, then this is number of agreement + (4  number of past turns).
We then use these measures to decide how to play, using these rules:
 If proportion_agree > 0.9 and last_four_num >= 4, then Cooperate.
 Otherwise if proportion_agree >= 0.625 and last_four_num >= 2, then TitforTat.
 Otherwise, Defect.
However, if the opponent has Cooperated the last streak_needed turns, then the strategy deviates from the usual strategy, and instead Defects. (We call such deviation an “aberration”.) In the turn immediately after an aberration, the strategy doesn’t override, even if there’s a streak of Cooperations. Two turns after an aberration, the strategy: Restarts the Cooperation streak (never looking before this turn); Cooperates; and changes streak_needed to:
floor(20.0 * num_abb_def / num_abb_coop) + 1
Here num_abb_def is 2 + the number of times that the opponent Defected in the turn after an aberration, and num_abb_coop is 2 + the number of times that the opponent Cooperated in response to an aberration.
Names:
 RichardHufford: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByRowsam
[source]¶ Strategy submitted to Axelrod’s second tournament by Glen Rowsam (K58R) and came in 21st in that tournament.
The strategy starts in Normal mode, where it cooperates every turn. Every six turns it checks the score per turn. [Rather the score of all previous turns divided by the turn number, which will be one more than the number of turns scored.] If this measure is less than 2.5 (the strategy is doing badly) and it increases distrust_points. distrust_points is a variable that starts at 0; if it ever exceeds 6 points, the strategy will enter Defect mode and defect from then on. It will increase distrust_points depending on the precise score per turn according to:
 5 points if score per turn is less than 1.0
 3 points if score per turn is less than 1.5, but at least 1.0
 2 points if score per turn is less than 2.0, but at least 1.5
 1 points if score per turn is less than 2.5, but at least 2.0
If distrust_points are increased, then the strategy defects on that turn, then cooperates and defects on the next two turns. [Unless distrust_points exceeds 6 points, then it will enter Defect mode immediately.]
Every 18 turns in Normal mode, the strategy will decrement distrust_score if it’s more than 3. This represents a wearing off effect of distrust.
Names:
 Rowsam: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByTester
[source]¶ Submitted to Axelrod’s second tournament by David Gladstein.
This strategy is a TFT variant that attempts to exploit certain strategies. It defects on the first move. If the opponent ever defects, TESTER ‘apologies’ by cooperating and then plays TFT for the rest of the game. Otherwise TESTER alternates cooperation and defection.
This strategy came 46th in Axelrod’s second tournament.
Names:
 Tester: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByTidemanAndChieruzzi
[source]¶ Strategy submitted to Axelrod’s second tournament by T. Nicolaus Tideman and Paula Chieruzzi (K84R) and came in ninth in that tournament.
This strategy Cooperates if this player’s score exceeds the opponent’s score by at least score_to_beat. score_to_beat starts at zero and increases by score_to_beat_inc every time the opponent’s last two moves are a Cooperation and Defection in that order. score_to_beat_inc itself increase by 5 every time the opponent’s last two moves are a Cooperation and Defection in that order.
Additionally, the strategy executes a “fresh start” if the following hold:
 The strategy would Defect by score (difference less than score_to_beat)
 The opponent did not Cooperate and Defect (in order) in the last two turns.
 It’s been at least 10 turns since the last fresh start. Or since the match started if there hasn’t been a fresh start yet.
A “fresh start” entails two Cooperations and resetting scores, scores_to_beat and scores_to_beat_inc.
Names:
 TidemanAndChieruzzi: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByTranquilizer
[source]¶ Submitted to Axelrod’s second tournament by Craig Feathers
Description given in Axelrod’s “More Effective Choice in the Prisoner’s Dilemma” paper: The rule normally cooperates but is ready to defect if the other player defects too often. Thus the rule tends to cooperate for the first dozen or two moves if the other player is cooperating, but then it throws in a defection. If the other player continues to cooperate, then defections become more frequent. But as long as Tranquilizer is maintaining an average payoff of at least 2.25 points per move, it will never defect twice in succession and it will not defect more than onequarter of the time.
This implementation is based on the reverse engineering of the Fortran strategy K67R from Axelrod’s second tournament. Reversed engineered by: Owen Campbell, Will Guo and Mansour Hakem.
The strategy starts by cooperating and has 3 states.
At the start of the strategy it updates its states:
It counts the number of consecutive defections by the opponent.
If it was in state 2 it moves to state 0 and calculates the following quantities two_turns_after_good_defection_ratio and two_turns_after_good_defection_ratio_count.
Formula for:
two_turns_after_good_defection_ratio:
self.two_turns_after_good_defection_ratio = ( ((self.two_turns_after_good_defection_ratio * self.two_turns_after_good_defection_ratio_count) + (3  (3 * self.dict[opponent.history[1]])) + (2 * self.dict[self.history[1]])  ((self.dict[opponent.history[1]] * self.dict[self.history[1]]))) / (self.two_turns_after_good_defection_ratio_count + 1) )
two_turns_after_good_defection_ratio_count = two_turns_after_good_defection_ratio + 1
If it was in state 1 it moves to state 2 and calculates the following quantities one_turn_after_good_defection_ratio and one_turn_after_good_defection_ratio_count.
Formula for:
one_turn_after_good_defection_ratio:
self.one_turn_after_good_defection_ratio = ( ((self.one_turn_after_good_defection_ratio * self.one_turn_after_good_defection_ratio_count) + (3  (3 * self.dict[opponent.history[1]])) + (2 * self.dict[self.history[1]])  (self.dict[opponent.history[1]] * self.dict[self.history[1]])) / (self.one_turn_after_good_defection_ratio_count + 1) )
one_turn_after_good_defection_ratio_count:
one_turn_after_good_defection_ratio_count = one_turn_after_good_defection_ratio + 1
If after this it is in state 1 or 2 then it cooperates.
If it is in state 0 it will potentially perform 1 of the 2 following stochastic tests:
1. If average score per turn is greater than 2.25 then it calculates a value of probability:
probability = ( (.95  (((self.one_turn_after_good_defection_ratio) + (self.two_turns_after_good_defection_ratio)  5) / 15)) + (1 / (((len(self.history))+1) ** 2))  (self.dict[opponent.history[1]] / 4) )
and will cooperate if a random sampled number is less than that value of probability. If it does not cooperate then the strategy moves to state 1 and defects.
2. If average score per turn is greater than 1.75 but less than 2.25 then it calculates a value of probability:
probability = ( (.25 + ((opponent.cooperations + 1) / ((len(self.history)) + 1)))  (self.opponent_consecutive_defections * .25) + ((current_score[0]  current_score[1]) / 100) + (4 / ((len(self.history)) + 1)) )
and will cooperate if a random sampled number is less than that value of probability. If not, it defects.
If none of the above holds the player simply plays tit for tat.
Tranquilizer came in 27th place in Axelrod’s second torunament.
Names:
 Tranquilizer: [Axelrod1980]

class
axelrod.strategies.axelrod_second.
SecondByWeiner
[source]¶ Strategy submitted to Axelrod’s second tournament by Herb Weiner (K41R), and came in seventh in that tournament.
Play TitforTat with a chance for forgiveness and a defective override.
The chance for forgiveness happens only if forgive_flag is raised (flag discussed below). If raised and turn is greater than grudge, then override TitforTat with Cooperation. grudge is a variable that starts at 0 and increments 20 with each forgiven Defect (a Defect that is overriden through the forgiveness logic). forgive_flag is lower whether logic is overriden or not.
The variable defect_padding increments with each opponent Defect, but resets to zero with each opponent Cooperate (or forgive_flag lowering) so that it roughly counts Defects between Cooperates. Whenever the opponent Cooperates, if defect_padding (before reseting) is odd, then we raise forgive_flag for next turn.
Finally a defective override is assessed after forgiveness. If five or more of the opponent’s last twelve actions are Defects, then Defect. This will overrule a forgiveness, but doesn’t undo the lowering of forgiveness_flag. Note that “last twelve actions” doesn’t count the most recent action. Actually the original code updates history after checking for defect override.
Names:
 Weiner: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByWhite
[source]¶ Strategy submitted to Axelrod’s second tournament by Edward C White (K72R) and came in thirteenth in that tournament.
 Cooperate in the first ten turns.
 If the opponent Cooperated last turn then Cooperate.
 Otherwise Defect if and only if:
 floor(log(turn)) * opponent Defections >= turn
Names:
 White: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByWmAdams
[source]¶ Strategy submitted to Axelrod’s second tournament by William Adams (K44R), and came in fifth in that tournament.
Count the number of opponent defections after their first move, call c_defect. Defect if c_defect equals 4, 7, or 9. If c_defect > 9, then defect immediately after opponent defects with probability = (0.5)^(c_defect1). Otherwise cooperate.
Names:
 WmAdams: [Axelrod1980b]

class
axelrod.strategies.axelrod_second.
SecondByYamachi
[source]¶ Strategy submitted to Axelrod’s second tournament by Brian Yamachi (K64R) and came in seventeenth in that tournament.
The strategy keeps track of play history through a variable called count_them_us_them, which is a dict indexed by (X, Y, Z), where X is an opponent’s move and Y and Z are the following moves by this player and the opponent, respectively. Each turn, we look at our opponent’s move two turns ago, call X, and our move last turn, call Y. If (X, Y, C) has occurred more often (or as often) as (X, Y, D), then Cooperate. Otherwise Defect. [Note that this reflects likelihood of Cooperations or Defections in opponent’s previous move; we don’t update count_them_us_them with previous move until next turn.]
Starting with the 41st turn, there’s a possibility to override this behavior. If portion_defect is between 45% and 55% (exclusive), then Defect, where portion_defect equals number of opponent defects plus 0.5 divided by the turn number (indexed by 1). When overriding this way, still record count_them_us_them as though the strategy didn’t override.
Names:
 Yamachi: [Axelrod1980b]

class
axelrod.strategies.backstabber.
BackStabber
[source]¶ Forgives the first 3 defections but on the fourth will defect forever. Defects on the last 2 rounds unconditionally.
Names:
 Backstabber: Original name by Thomas Campbell

original_class
¶ alias of
BackStabber

class
axelrod.strategies.backstabber.
DoubleCrosser
[source]¶ Forgives the first 3 defections but on the fourth will defect forever. Defects on the last 2 rounds unconditionally.
If 8 <= current round <= 180, if the opponent did not defect in the first 7 rounds, the player will only defect after the opponent has defected twice inarow.
Names:
 Double Crosser: Original name by Thomas Campbell

original_class
¶ alias of
DoubleCrosser

class
axelrod.strategies.better_and_better.
BetterAndBetter
[source]¶ Defects with probability of ‘(1000  current turn) / 1000’. Therefore it is less and less likely to defect as the round goes on.
 Names:
 Better and Better: [Prison1998]

class
axelrod.strategies.bush_mosteller.
BushMosteller
(c_prob: float = 0.5, d_prob: float = 0.5, aspiration_level_divider: float = 3.0, learning_rate: float = 0.5)[source]¶ A player that is based on Bush Mosteller reinforced learning algorithm, it decides what it will play only depending on its own previous payoffs.
The probability of playing C or D will be updated using a stimulus which represents a win or a loss of value based on its previous play’s payoff in the specified probability. The more a play will be rewarded through rounds, the more the player will be tempted to use it.
Names:
 Bush Mosteller: [Luis2008]

class
axelrod.strategies.calculator.
Calculator
[source]¶ Plays like (Hard) Joss for the first 20 rounds. If periodic behavior is detected, defect forever. Otherwise play TFT.
Names:
 Calculator: [Prison1998]

class
axelrod.strategies.cooperator.
Cooperator
[source]¶ A player who only ever cooperates.
Names:
 Cooperator: [Axelrod1984]
 ALLC: [Press2012]
 Always cooperate: [Mittal2009]

class
axelrod.strategies.cooperator.
TrickyCooperator
[source]¶ A cooperator that is trying to be tricky.
Names:
 Tricky Cooperator: Original name by Karol Langner

class
axelrod.strategies.cycler.
AntiCycler
[source]¶ A player that follows a sequence of plays that contains no cycles: CDD CD CCD CCCD CCCCD …
Names:
 Anti Cycler: Original name by Marc Harper

class
axelrod.strategies.cycler.
Cycler
(cycle: str = 'CCD')[source]¶ A player that repeats a given sequence indefinitely.
Names:
 Cycler: Original name by Marc Harper

class
axelrod.strategies.cycler.
CyclerCCCCCD
[source]¶ Cycles C, C, C, C, C, D
Names:
 Cycler CCCD: Original name by Marc Harper

class
axelrod.strategies.cycler.
CyclerCCCD
[source]¶ Cycles C, C, C, D
Names:
 Cycler CCCD: Original name by Marc Harper

class
axelrod.strategies.cycler.
CyclerCCCDCD
[source]¶ Cycles C, C, C, D, C, D
Names:
 Cycler CCCDCD: Original name by Marc Harper

class
axelrod.strategies.cycler.
CyclerCCD
[source]¶ Cycles C, C, D
Names:
 Cycler CCD: Original name by Marc Harper
 Periodic player CCD: [Mittal2009]

class
axelrod.strategies.cycler.
CyclerDC
[source]¶ Cycles D, C
Names:
 Cycler DC: Original name by Marc Harper

class
axelrod.strategies.cycler.
CyclerDDC
[source]¶ Cycles D, D, C
Names:
 Cycler DDC: Original name by Marc Harper
 Periodic player DDC: [Mittal2009]

class
axelrod.strategies.cycler.
EvolvableCycler
(cycle: str = None, cycle_length: int = None, mutation_probability: float = 0.2, mutation_potency: int = 1, seed: int = None)[source]¶ Evolvable version of Cycler.
The player class in this module does not obey standard rules of the IPD (as indicated by their classifier). We do not recommend putting a lot of time in to optimising it.

class
axelrod.strategies.darwin.
Darwin
[source]¶ A strategy which accumulates a record (the ‘genome’) of what the most favourable response in the previous round should have been, and naively assumes that this will remain the correct response at the same round of future trials.
This ‘genome’ is preserved between opponents, rounds and repetitions of the tournament. It becomes a characteristic of the type and so a single version of this is shared by all instances for each loading of the class.
As this results in information being preserved between tournaments, this is classified as a cheating strategy!
If no record yet exists, the opponent’s response from the previous round is returned.
Names:
 Darwin: Original name by Paul Slavin

class
axelrod.strategies.dbs.
DBS
(discount_factor=0.75, promotion_threshold=3, violation_threshold=4, reject_threshold=3, tree_depth=5)[source]¶ A strategy that learns the opponent’s strategy and uses symbolic noise detection for detecting whether anomalies in player’s behavior are deliberate or accidental. From the learned opponent’s strategy, a tree search is used to choose the best move.
Default values for the parameters are the suggested values in the article. When noise increases you can try to diminish violation_threshold and rejection_threshold.
Names
 Derived Belief Strategy: [Au2006]

compute_prob_rule
(outcome, alpha=1)[source]¶ Uses the game history to compute the probability of the opponent playing C, in the outcome situation (example: outcome = (C, C)). When alpha = 1, the results is approximately equal to the frequency of the occurrence of outcome C. alpha is a discount factor that gives more weight to recent events than earlier ones.
Parameters
outcome: tuple of two actions.Action alpha: int, optional. Discount factor. Default is 1.

should_demote
(r_minus, violation_threshold=4)[source]¶ Checks if the number of successive violations of a deterministic rule (in the opponent’s behavior) exceeds the userdefined violation_threshold.

should_promote
(r_plus, promotion_threshold=3)[source]¶ This function determines if the move r_plus is a deterministic behavior of the opponent, and then returns True, or if r_plus is due to a random behavior (or noise) which would require a probabilistic rule, in which case it returns False.
To do so it looks into the game history: if the k last times when the opponent was in the same situation than in r_plus it played the same thing then then r_plus is considered as a deterministic rule (where K is the userdefined promotion_threshold).
Parameters
 r_plus: tuple of (tuple of actions.Action, actions.Action)
 example: ((C, C), D) r_plus represents one outcome of the history, and the following move played by the opponent.
 promotion_threshold: int, optional
 Number of successive observations needed to promote an opponent behavior as a deterministic rule. Default is 3.

class
axelrod.strategies.dbs.
DeterministicNode
(action1, action2, depth)[source]¶ Nodes (C, C), (C, D), (D, C), or (D, D) with deterministic choice for siblings.

class
axelrod.strategies.dbs.
Node
[source]¶ Nodes used to build a tree for the treesearch procedure. The tree has Deterministic and Stochastic nodes, as the opponent’s strategy is learned as a probability distribution.

class
axelrod.strategies.dbs.
StochasticNode
(own_action, pC, depth)[source]¶ Node that have a probability pC to get to each sibling. A StochasticNode can be written (C, X) or (D, X), with X = C with a probability pC, else X = D.

axelrod.strategies.dbs.
create_policy
(pCC, pCD, pDC, pDD)[source]¶ Creates a dict that represents a Policy. As defined in the reference, a Policy is a set of (prev_move, p) where p is the probability to cooperate after prev_move, where prev_move can be (C, C), (C, D), (D, C) or (D, D).
Parameters
 pCC, pCD, pDC, pDD : float
 Must be between 0 and 1.

axelrod.strategies.dbs.
minimax_tree_search
(begin_node, policy, max_depth)[source]¶ Tree search function (minimax search procedure) for the tree (built by recursion) corresponding to the opponent’s policy, and solves it. Returns a tuple of two floats that are the utility of playing C, and the utility of playing D.

axelrod.strategies.dbs.
move_gen
(outcome, policy, depth_search_tree=5)[source]¶ Returns the best move considering opponent’s policy and last move, using treesearch procedure.

class
axelrod.strategies.defector.
Defector
[source]¶ A player who only ever defects.
Names:
 Defector: [Axelrod1984]
 ALLD: [Press2012]
 Always defect: [Mittal2009]

class
axelrod.strategies.defector.
TrickyDefector
[source]¶ A defector that is trying to be tricky.
Names:
 Tricky Defector: Original name by Karol Langner

class
axelrod.strategies.doubler.
Doubler
[source]¶ Cooperates except when the opponent has defected and the opponent’s cooperation count is less than twice their defection count.
Names:
 Doubler: [Prison1998]

class
axelrod.strategies.finite_state_machines.
EvolvableFSMPlayer
(transitions: tuple = None, initial_state: int = None, initial_action: axelrod.action.Action = None, num_states: int = None, mutation_probability: float = 0.1, seed: int = None)[source]¶ Abstract base class for evolvable finite state machine players.

crossover
(other)[source]¶ Optional method to allow Player to produce variants in combination with another player. Returns a new Player.

classmethod
normalize_transitions
(transitions: Sequence[Sequence[T_co]]) → Tuple[Tuple[Any, ...], ...][source]¶ Translate a list of lists to a tuple of tuples.

receive_vector
(vector)[source]¶ Read a serialized vector into the set of FSM parameters (less initial state). Then assign those FSM parameters to this class instance.
The vector has three parts. The first is used to define the next state (for each of the player’s states  for each opponents action).
The second part is the player’s next moves (for each state  for each opponent’s actions).
Finally, a probability to determine the player’s first move.


class
axelrod.strategies.finite_state_machines.
EvolvedFSM16
[source]¶ A 16 state FSM player trained with an evolutionary algorithm.
Names:
 Evolved FSM 16: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
EvolvedFSM16Noise05
[source]¶ A 16 state FSM player trained with an evolutionary algorithm with noisy matches (noise=0.05).
Names:
 Evolved FSM 16 Noise 05: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
EvolvedFSM4
[source]¶ A 4 state FSM player trained with an evolutionary algorithm.
Names:
 Evolved FSM 4: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
EvolvedFSM6
[source]¶ An 6 state FSM player trained with an evolutionary algorithm.
Evolved using axelroddojo version 0.0.8 and axelrod version 4.10.0, trained to maximize score against the short_run_time_strategies with 10 machine states for 500 generations, population size of 40, mutation rate at 0.1, bottleneck at 10, 200 turns, and 0 noise. The resulting strategy had only 6 states in its accessible component.
Names:
 Evolved FSM 6: Original name by Frederick Vincent & Dashiell Fryer

class
axelrod.strategies.finite_state_machines.
FSMPlayer
(transitions: Tuple[Tuple[int, axelrod.action.Action, int, axelrod.action.Action], ...] = ((1, C, 1, C), (1, D, 1, D)), initial_state: int = 1, initial_action: axelrod.action.Action = C)[source]¶ Abstract base class for finite state machine players.

class
axelrod.strategies.finite_state_machines.
Fortress3
[source]¶ Finite state machine player specified in http://DOI.org/10.1109/CEC.2006.1688322.
Note that the description in http://www.grahamkendall.com/papers/lhk2011.pdf is not correct.
Names:
 Fortress 3: [Ashlock2006b]

class
axelrod.strategies.finite_state_machines.
Fortress4
[source]¶ Finite state machine player specified in http://DOI.org/10.1109/CEC.2006.1688322.
Note that the description in http://www.grahamkendall.com/papers/lhk2011.pdf is not correct.
Names:
 Fortress 4: [Ashlock2006b]

class
axelrod.strategies.finite_state_machines.
Predator
[source]¶ Finite state machine player specified in http://DOI.org/10.1109/CEC.2006.1688322.
Names:
 Predator: [Ashlock2006b]

class
axelrod.strategies.finite_state_machines.
Pun1
[source]¶ FSM player described in [Ashlock2006].
Names:
 Pun1: [Ashlock2006]

class
axelrod.strategies.finite_state_machines.
Raider
[source]¶ FSM player described in http://DOI.org/10.1109/FOCI.2014.7007818.
Names
 Raider: [Ashlock2014]

class
axelrod.strategies.finite_state_machines.
Ripoff
[source]¶ FSM player described in http://DOI.org/10.1109/TEVC.2008.920675.
Names
 Ripoff: [Ashlock2008]

class
axelrod.strategies.finite_state_machines.
SimpleFSM
(transitions: tuple, initial_state: int)[source]¶ Simple implementation of a finite state machine that transitions between states based on the last round of play.

class
axelrod.strategies.finite_state_machines.
SolutionB1
[source]¶ FSM player described in http://DOI.org/10.1109/TCIAIG.2014.2326012.
Names
 Solution B1: [Ashlock2015]

class
axelrod.strategies.finite_state_machines.
SolutionB5
[source]¶ FSM player described in http://DOI.org/10.1109/TCIAIG.2014.2326012.
Names
 Solution B5: [Ashlock2015]

class
axelrod.strategies.finite_state_machines.
TF1
[source]¶ A FSM player trained to maximize Moran fixation probabilities.
Names:
 TF1: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
TF2
[source]¶ A FSM player trained to maximize Moran fixation probabilities.
Names:
 TF2: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
TF3
[source]¶ A FSM player trained to maximize Moran fixation probabilities.
Names:
 TF3: Original name by Marc Harper

class
axelrod.strategies.finite_state_machines.
Thumper
[source]¶ FSM player described in http://DOI.org/10.1109/TEVC.2008.920675.
Names
 Thumper: [Ashlock2008]

class
axelrod.strategies.finite_state_machines.
UsuallyCooperates
[source]¶ This strategy cooperates except after a C following a D.
Names:
 Usually Cooperates (UC): [Ashlock2009]

class
axelrod.strategies.finite_state_machines.
UsuallyDefects
[source]¶ This strategy defects except after a D following a C.
Names:
 Usually Defects (UD): [Ashlock2009]

class
axelrod.strategies.forgiver.
Forgiver
[source]¶ A player starts by cooperating however will defect if at any point the opponent has defected more than 10 percent of the time
Names:
 Forgiver: Original name by Thomas Campbell

class
axelrod.strategies.forgiver.
ForgivingTitForTat
[source]¶ A player starts by cooperating however will defect if at any point, the opponent has defected more than 10 percent of the time, and their most recent decision was defect.
Names:
 Forgiving Tit For Tat: Original name by Thomas Campbell
Stochastic variants of Lookup table basedstrategies, trained with particle swarm algorithms.
 For the original see:
 https://gist.github.com/GDKO/60c3d0fd423598f3c4e4

class
axelrod.strategies.gambler.
EvolvableGambler
(lookup_dict: dict = None, initial_actions: tuple = None, pattern: Any = None, parameters: axelrod.strategies.lookerup.Plays = None, mutation_probability: float = None, seed: int = None)[source]¶

class
axelrod.strategies.gambler.
Gambler
(lookup_dict: dict = None, initial_actions: tuple = None, pattern: Any = None, parameters: axelrod.strategies.lookerup.Plays = None)[source]¶ A stochastic version of LookerUp which will select randomly an action in some cases.
Names:
 Gambler: Original name by Georgios Koutsovoulos

class
axelrod.strategies.gambler.
PSOGambler1_1_1
[source]¶ A 1x1x1 PSOGambler trained with pyswarm.
Names:
 PSO Gambler 1_1_1: Original name by Marc Harper

class
axelrod.strategies.gambler.
PSOGambler2_2_2
[source]¶ A 2x2x2 PSOGambler trained with a particle swarm algorithm (implemented in pyswarm). Original version by Georgios Koutsovoulos.
Names:
 PSO Gambler 2_2_2: Original name by Marc Harper

class
axelrod.strategies.gambler.
PSOGambler2_2_2_Noise05
[source]¶ A 2x2x2 PSOGambler trained with pyswarm with noise=0.05.
Names:
 PSO Gambler 2_2_2 Noise 05: Original name by Marc Harper

class
axelrod.strategies.gambler.
PSOGamblerMem1
[source]¶ A 1x1x0 PSOGambler trained with pyswarm. This is the ‘optimal’ memory one strategy trained against the set of short run time strategies in the Axelrod library.
Names:
 PSO Gambler Mem1: Original name by Marc Harper

class
axelrod.strategies.gambler.
ZDMem2
[source]¶ A memory two generalization of a zero determinant player.
Names:
 ZDMem2: Original name by Marc Harper
 Unnamed [LiS2014]

class
axelrod.strategies.gobymajority.
GoByMajority
(memory_depth: Union[int, float] = inf, soft: bool = True)[source]¶ Submitted to Axelrod’s second tournament by Gail Grisell. It came 23rd and was written in 10 lines of BASIC.
A player examines the history of the opponent: if the opponent has more defections than cooperations then the player defects.
In case of equal number of defections and cooperations this player will Cooperate. Passing the soft=False keyword argument when initialising will create a HardGoByMajority which Defects in case of equality.
An optional memory attribute will limit the number of turns remembered (by default this is 0)
Names:
 Go By Majority: [Axelrod1984]
 Grisell: [Axelrod1980b]
 Soft Majority: [Mittal2009]

strategy
(opponent: axelrod.player.Player) → axelrod.action.Action[source]¶ This is affected by the history of the opponent.
As long as the opponent cooperates at least as often as they defect then the player will cooperate. If at any point the opponent has more defections than cooperations in memory the player defects.

class
axelrod.strategies.gobymajority.
GoByMajority10
[source]¶ GoByMajority player with a memory of 10.
Names:
 Go By Majority 10: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
GoByMajority20
[source]¶ GoByMajority player with a memory of 20.
Names:
 Go By Majority 20: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
GoByMajority40
[source]¶ GoByMajority player with a memory of 40.
Names:
 Go By Majority 40: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
GoByMajority5
[source]¶ GoByMajority player with a memory of 5.
Names:
 Go By Majority 5: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
HardGoByMajority
(memory_depth: Union[int, float] = inf)[source]¶ A player examines the history of the opponent: if the opponent has more defections than cooperations then the player defects. In case of equal number of defections and cooperations this player will Defect.
An optional memory attribute will limit the number of turns remembered (by default this is 0)
Names:
 Hard Majority: [Mittal2009]

class
axelrod.strategies.gobymajority.
HardGoByMajority10
[source]¶ HardGoByMajority player with a memory of 10.
Names:
 Hard Go By Majority 10: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
HardGoByMajority20
[source]¶ HardGoByMajority player with a memory of 20.
Names:
 Hard Go By Majority 20: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
HardGoByMajority40
[source]¶ HardGoByMajority player with a memory of 40.
Names:
 Hard Go By Majority 40: Original name by Karol Langner

class
axelrod.strategies.gobymajority.
HardGoByMajority5
[source]¶ HardGoByMajority player with a memory of 5.
Names:
 Hard Go By Majority 5: Original name by Karol Langner

class
axelrod.strategies.gradualkiller.
GradualKiller
[source]¶ It begins by defecting in the first five moves, then cooperates two times. It then defects all the time if the opponent has defected in move 6 and 7, else cooperates all the time. Initially designed to stop Gradual from defeating TitForTat in a 3 Player tournament.
Names
 Gradual Killer: [Prison1998]

original_class
¶ alias of
GradualKiller

class
axelrod.strategies.grudger.
Aggravater
[source]¶ Grudger, except that it defects on the first 3 turns
Names
 Aggravater: Original name by Thomas Campbell

class
axelrod.strategies.grudger.
EasyGo
[source]¶ A player starts by defecting however will cooperate if at any point the opponent has defected.
Names:
 Easy Go: [Prison1998]
 Reverse Grudger (RGRIM): [Li2011]
 Fool Me Forever: [Harper2017]

class
axelrod.strategies.grudger.
ForgetfulGrudger
[source]¶ A player starts by cooperating however will defect if at any point the opponent has defected, but forgets after mem_length matches.
Names:
 Forgetful Grudger: Original name by Geraint Palmer

class
axelrod.strategies.grudger.
GeneralSoftGrudger
(n: int = 1, d: int = 4, c: int = 2)[source]¶ A generalization of the SoftGrudger strategy. SoftGrudger punishes by playing: D, D, D, D, C, C. after a defection by the opponent. GeneralSoftGrudger only punishes after its opponent defects a specified amount of times consecutively. The punishment is in the form of a series of defections followed by a ‘penance’ of a series of consecutive cooperations.
Names:
 General Soft Grudger: Original Name by J. Taylor Smith

class
axelrod.strategies.grudger.
Grudger
[source]¶ A player starts by cooperating however will defect if at any point the opponent has defected.
This strategy came 7th in Axelrod’s original tournament.
Names:
 Friedman’s strategy: [Axelrod1980]
 Grudger: [Li2011]
 Grim: [Berg2015]
 Grim Trigger: [Banks1990]
 Spite: [Beaufils1997]
 Spiteful: [Mathieu2015]
 Vengeful: [Ashlock2009]

class
axelrod.strategies.grudger.
GrudgerAlternator
[source]¶ A player starts by cooperating until the first opponents defection, then alternates DC.
Names:
 c_then_per_dc: [Prison1998]
 Grudger Alternator: Original name by Geraint Palmer

class
axelrod.strategies.grudger.
OppositeGrudger
[source]¶ A player starts by defecting however will cooperate if at any point the opponent has cooperated.
Names:
 Opposite Grudger: Original name by Geraint Palmer

class
axelrod.strategies.grudger.
SoftGrudger
[source]¶ A modification of the Grudger strategy. Instead of punishing by always defecting: punishes by playing: D, D, D, D, C, C. (Will continue to cooperate afterwards).
 Soft Grudger (SGRIM): [Li2011]

class
axelrod.strategies.grudger.
SpitefulCC
[source]¶ Behaves like Grudger after cooperating for 2 turns
Names:
 spiteful_cc: [Mathieu2015]

class
axelrod.strategies.grumpy.
Grumpy
(starting_state: str = 'Nice', grumpy_threshold: int = 10, nice_threshold: int = 10)[source]¶ A player that defects after a certain level of grumpiness. Grumpiness increases when the opponent defects and decreases when the opponent cooperates.
Names:
 Grumpy: Original name by Jason Young

strategy
(opponent: axelrod.player.Player) → axelrod.action.Action[source]¶ A player that gets grumpier the more the opposition defects, and nicer the more they cooperate.
Starts off Nice, but becomes grumpy once the grumpiness threshold is hit. Won’t become nice once that grumpy threshold is hit, but must reach a much lower threshold before it becomes nice again.

class
axelrod.strategies.handshake.
Handshake
(initial_plays: List[axelrod.action.Action] = None)[source]¶ Starts with C, D. If the opponent plays the same way, cooperate forever, else defect forever.
Names:
 Handshake: [Robson1990]

class
axelrod.strategies.hmm.
EvolvableHMMPlayer
(transitions_C=None, transitions_D=None, emission_probabilities=None, initial_state=0, initial_action=C, num_states=None, mutation_probability=None, seed: int = None)[source]¶ Evolvable version of HMMPlayer.

crossover
(other)[source]¶ Optional method to allow Player to produce variants in combination with another player. Returns a new Player.

receive_vector
(vector)[source]¶ Read a serialized vector into the set of HMM parameters (less initial state). Then assign those HMM parameters to this class instance.
Assert that the vector has the right number of elements for an HMMParams class with self.num_states.
Assume the first num_states^2 entries are the transitions_C matrix. The next num_states^2 entries are the transitions_D matrix. Then the next num_states entries are the emission_probabilities vector. Finally the last entry is the initial_action.


class
axelrod.strategies.hmm.
EvolvedHMM5
[source]¶ An HMMbased player with five hidden states trained with an evolutionary algorithm.
Names:
 Evolved HMM 5: Original name by Marc Harper

class
axelrod.strategies.hmm.
HMMPlayer
(transitions_C=None, transitions_D=None, emission_probabilities=None, initial_state=0, initial_action=C)[source]¶ Abstract base class for Hidden Markov Model players.
Names
 HMM Player: Original name by Marc Harper

class
axelrod.strategies.hmm.
SimpleHMM
(transitions_C, transitions_D, emission_probabilities, initial_state)[source]¶ Implementation of a basic Hidden Markov Model. We assume that the transition matrix is conditioned on the opponent’s last action, so there are two transition matrices. Emission distributions are stored as Bernoulli probabilities for each state. This is essentially a stochastic FSM.

axelrod.strategies.hmm.
is_stochastic_matrix
(m, ep=1e08) → bool[source]¶ Checks that the matrix m (a list of lists) is a stochastic matrix.

axelrod.strategies.hmm.
mutate_row
(row, mutation_probability, rng)[source]¶ , crossover_lists_of_lists Given a row of probabilities, randomly change each entry with probability mutation_probability (a value between 0 and 1). If changing, then change by a value randomly (uniformly) chosen from [0.25, 0.25] bounded by 0 and 100%.

class
axelrod.strategies.hunter.
AlternatorHunter
[source]¶ A player who hunts for alternators.
Names:
 Alternator Hunter: Original name by Karol Langner

class
axelrod.strategies.hunter.
CooperatorHunter
[source]¶ A player who hunts for cooperators.
Names:
 Cooperator Hunter: Original name by Karol Langner

class
axelrod.strategies.hunter.
CycleHunter
[source]¶ Hunts strategies that play cyclically, like any of the Cyclers, Alternator, etc.
Names:
 Cycle Hunter: Original name by Marc Harper

class
axelrod.strategies.hunter.
DefectorHunter
[source]¶ A player who hunts for defectors.
Names:
 Defector Hunter: Original name by Karol Langner

class
axelrod.strategies.hunter.
EventualCycleHunter
[source]¶ Hunts strategies that eventually play cyclically.
Names:
 Eventual Cycle Hunter: Original name by Marc Harper

class
axelrod.strategies.hunter.
MathConstantHunter
[source]¶ A player who hunts for mathematical constant players.
Names:
Math Constant Hunter: Original name by Karol Langner

strategy
(opponent: axelrod.player.Player) → axelrod.action.Action[source]¶ Check whether the number of cooperations in the first and second halves of the history are close. The variance of the uniform distribution (1/4) is a reasonable delta but use something lower for certainty and avoiding false positives. This approach will also detect a lot of random players.


class
axelrod.strategies.hunter.
RandomHunter
[source]¶ A player who hunts for random players.
Names:
 Random Hunter: Original name by Karol Langner

class
axelrod.strategies.inverse.
Inverse
[source]¶ A player who defects with a probability that diminishes relative to how long ago the opponent defected.
Names:
 Inverse: Original Name by Karol Langner

class
axelrod.strategies.lookerup.
EvolvableLookerUp
(lookup_dict: dict = None, initial_actions: tuple = None, pattern: Any = None, parameters: axelrod.strategies.lookerup.Plays = None, mutation_probability: float = None, seed: int = None)[source]¶

class
axelrod.strategies.lookerup.
EvolvedLookerUp1_1_1
[source]¶ A 1 1 1 Lookerup trained with an evolutionary algorithm.
Names:
 Evolved Lookerup 1 1 1: Original name by Marc Harper

class
axelrod.strategies.lookerup.
EvolvedLookerUp2_2_2
[source]¶ A 2 2 2 Lookerup trained with an evolutionary algorithm.
Names:
 Evolved Lookerup 2 2 2: Original name by Marc Harper

class
axelrod.strategies.lookerup.
LookerUp
(lookup_dict: dict = None, initial_actions: tuple = None, pattern: Any = None, parameters: axelrod.strategies.lookerup.Plays = None)[source]¶ This strategy uses a LookupTable to decide its next action. If there is not enough history to use the table, it calls from a list of self.initial_actions.
if self_depth=2, op_depth=3, op_openings_depth=5, LookerUp finds the last 2 plays of self, the last 3 plays of opponent and the opening 5 plays of opponent. It then looks those up on the LookupTable and returns the appropriate action. If 5 rounds have not been played (the minimum required for op_openings_depth), it calls from self.initial_actions.
LookerUp can be instantiated with a dictionary. The dictionary uses tuple(tuple, tuple, tuple) or Plays as keys. for example.
self_plays: depth=2
op_plays: depth=1
op_openings: depth=0:
{Plays((C, C), (C), ()): C, Plays((C, C), (D), ()): D, Plays((C, D), (C), ()): D, < example below Plays((C, D), (D), ()): D, Plays((D, C), (C), ()): C, Plays((D, C), (D), ()): D, Plays((D, D), (C), ()): C, Plays((D, D), (D), ()): D}
From the above table, if the player last played C, D and the opponent last played C (here the initial opponent play is ignored) then this round, the player would play D.
The dictionary must contain all possible permutations of C’s and D’s.
LookerUp can also be instantiated with pattern=str/tuple of actions, and:
parameters=Plays( self_plays=player_depth: int, op_plays=op_depth: int, op_openings=op_openings_depth: int)
It will create keys of len=2 ** (sum(parameters)) and map the pattern to the keys.
initial_actions is a tuple such as (C, C, D). A table needs initial actions equal to max(self_plays depth, opponent_plays depth, opponent_initial_plays depth). If provided initial_actions is too long, the extra will be ignored. If provided initial_actions is too short, the shortfall will be made up with C’s.
Some wellknown strategies can be expressed as special cases; for example Cooperator is given by the dict (All history is ignored and always play C):
{Plays((), (), ()) : C}
TitForTat is given by (The only history that is important is the opponent’s last play.):
{Plays((), (D,), ()): D, Plays((), (C,), ()): C}
LookerUp’s LookupTable defaults to TitForTat. The initial_actions defaults to playing C.
Names:
 Lookerup: Original name by Martin Jones

class
axelrod.strategies.lookerup.
LookupTable
(lookup_dict: dict)[source]¶ LookerUp and its children use this object to determine their next actions.
It is an object that creates a table of all possible plays to a specified depth and the action to be returned for each combination of plays. The “get” method returns the appropriate response. For the table containing:
.... Plays(self_plays=(C, C), op_plays=(C, D), op_openings=(D, C): D Plays(self_plays=(C, C), op_plays=(C, D), op_openings=(D, D): C ...
with: player.history[2:]=[C, C] and opponent.history[2:]=[C, D] and opponent.history[:2]=[D, D], calling LookupTable.get(plays=(C, C), op_plays=(C, D), op_openings=(D, D)) will return C.
Instantiate the table with a lookup_dict. This is {(self_plays_tuple, op_plays_tuple, op_openings_tuple): action, …}. It must contain every possible permutation with C’s and D’s of the above tuple. so:
good_dict = {((C,), (C,), ()): C, ((C,), (D,), ()): C, ((D,), (C,), ()): D, ((D,), (D,), ()): C} bad_dict = {((C,), (C,), ()): C, ((C,), (D,), ()): C, ((D,), (C,), ()): D}
LookupTable.from_pattern() creates an ordered list of keys for you and maps the pattern to the keys.:
LookupTable.from_pattern(pattern=(C, D, D, C), player_depth=0, op_depth=1, op_openings_depth=1 )
creates the dictionary:
{Plays(self_plays=(), op_plays=(C), op_openings=(C)): C, Plays(self_plays=(), op_plays=(C), op_openings=(D)): D, Plays(self_plays=(), op_plays=(D), op_openings=(C)): D, Plays(self_plays=(), op_plays=(D), op_openings=(D)): C,}
and then returns a LookupTable with that dictionary.

class
axelrod.strategies.lookerup.
Plays
(self_plays, op_plays, op_openings)¶ 
op_openings
¶ Alias for field number 2

op_plays
¶ Alias for field number 1

self_plays
¶ Alias for field number 0


class
axelrod.strategies.lookerup.
Winner12
[source]¶ A lookup table based strategy.
Names:
 Winner12: [Mathieu2015]

class
axelrod.strategies.lookerup.
Winner21
[source]¶ A lookup table based strategy.
Names:
 Winner21: [Mathieu2015]

axelrod.strategies.lookerup.
create_lookup_table_keys
(player_depth: int, op_depth: int, op_openings_depth: int) → list[source]¶ Returns a list of Plays that has all possible permutations of C’s and D’s for each specified depth. the list is in order, C < D sorted by ((player_tuple), (op_tuple), (op_openings_tuple)). create_lookup_keys(2, 1, 0) returns:
[Plays(self_plays=(C, C), op_plays=(C,), op_openings=()), Plays(self_plays=(C, C), op_plays=(D,), op_openings=()), Plays(self_plays=(C, D), op_plays=(C,), op_openings=()), Plays(self_plays=(C, D), op_plays=(D,), op_openings=()), Plays(self_plays=(D, C), op_plays=(C,), op_openings=()), Plays(self_plays=(D, C), op_plays=(D,), op_openings=()), Plays(self_plays=(D, D), op_plays=(C,), op_openings=()), Plays(self_plays=(D, D), op_plays=(D,), op_openings=())]

axelrod.strategies.lookerup.
get_last_n_plays
(player: axelrod.player.Player, depth: int) → tuple[source]¶ Returns the last N plays of player as a tuple.

axelrod.strategies.lookerup.
make_keys_into_plays
(lookup_table: dict) → dict[source]¶ Returns a dict where all keys are Plays.

class
axelrod.strategies.mathematicalconstants.
CotoDeRatio
[source]¶ The player will always aim to bring the ratio of cooperations to defections closer to the ratio as given in a sub class
Names:
 Co to Do Ratio: Original Name by Timothy Standen

class
axelrod.strategies.mathematicalconstants.
Golden
[source]¶ The player will always aim to bring the ratio of cooperations to defections closer to the golden mean
Names:
 Golden: Original Name by Timothy Standen

class
axelrod.strategies.mathematicalconstants.
Pi
[source]¶ The player will always aim to bring the ratio of cooperations to defections closer to the pi
Names:
 Pi: Original Name by Timothy Standen

class
axelrod.strategies.mathematicalconstants.
e
[source]¶ The player will always aim to bring the ratio of cooperations to defections closer to the e
Names:
 e: Original Name by Timothy Standen
Memory Two strategies.

class
axelrod.strategies.memorytwo.
AON2
[source]¶ AON2 a memory two strategy introduced in [Hilbe2017]. It belongs to the AONk (allornone) family of strategies. These strategies were designed to satisfy the three following properties:
1. Mutually Cooperative. A strategy is mutually cooperative if there are histories for which the strategy prescribes to cooperate, and if it continues to cooperate after rounds with mutual cooperation (provided the last k actions of the focal player were actually consistent).
2. Error correcting. A strategy is error correcting after at most k rounds if, after any history, it generally takes a group of players at most k + 1 rounds to reestablish mutual cooperation.
3. Retaliating. A strategy is retaliating for at least k rounds if, after rounds in which the focal player cooperated while the coplayer defected, the strategy responds by defecting the following k rounds.
In [Hilbe2017] the following vectors are reported as “equivalent” to AON2 with their respective selfcooperation rate (note that these are not the same):
1. [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], selfcooperation rate: 0.952 2. [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], selfcooperation rate: 0.951 3. [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], selfcooperation rate: 0.951 4. [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1], selfcooperation rate: 0.952
AON2 is implemented using vector 1 due its selfcooperation rate.
In essence it is a strategy that starts off by cooperating and will cooperate again only after the states (CC, CC), (CD, CD), (DC, DC), (DD, DD).
Names:
 AON2: [Hilbe2017]

class
axelrod.strategies.memorytwo.
DelayedAON1
[source]¶ Delayed AON1 a memory two strategy also introduced in [Hilbe2017] and belongs to the AONk family. Note that AON1 is equivalent to Win Stay Lose Shift.
In [Hilbe2017] the following vectors are reported as “equivalent” to Delayed AON1 with their respective selfcooperation rate (note that these are not the same):
1. [1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1], selfcooperation rate: 0.952 2. [1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1], selfcooperation rate: 0.970 3. [1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1], selfcooperation rate: 0.971
Delayed AON1 is implemented using vector 3 due its selfcooperation rate.
In essence it is a strategy that starts off by cooperating and will cooperate again only after the states (CC, CC), (CD, CD), (CD, DD), (DD, CD), (DC, DC) and (DD, DD).
Names:
 Delayed AON1: [Hilbe2017]

class
axelrod.strategies.memorytwo.
MEM2
[source]¶ A memorytwo player that switches between TFT, TFTT, and ALLD.
Note that the reference claims that this is a memory two strategy but in fact it is infinite memory. This is because the player plays as ALLD if ALLD has ever been selected twice, which can only be known if the entire history of play is accessible.
Names:
 MEM2: [Li2014]

class
axelrod.strategies.memorytwo.
MemoryTwoPlayer
(sixteen_vector: Optional[Tuple[float, ...]] = None, initial: Optional[Tuple[axelrod.action.Action, axelrod.action.Action]] = None)[source]¶ Uses a sixteenvector for strategies based on the 16 conditional probabilities P(X  I,J,K,L) where X, I, J, K, L in [C, D] and I, J are the players last two moves and K, L are the opponents last two moves. These conditional probabilities are the following: 1. P(CCC, CC) 2. P(CCC, CD) 3. P(CCC, DC) 4. P(CCC, DD) 5. P(CCD, CC) 6. P(CCD, CD) 7. P(CCD, DC) 8. P(CCD, DD) 9. P(CDC, CC) 10. P(CDC, CD) 11. P(CDC, DC) 12. P(CDC, DD) 13. P(CDD, CC) 14. P(CDD, CD) 15. P(CDD, DC) 16. P(CDD, DD) Cooperator is set as the default player if sixteen_vector is not given.
Names
 Memory Two: [Hilbe2017]
Memory One strategies. Note that there are Memory One strategies in other files, including titfortat.py and zero_determinant.py

class
axelrod.strategies.memoryone.
ALLCorALLD
[source]¶ This strategy is at the parameter extreme of the ZD strategies (phi = 0). It simply repeats its last move, and so mimics ALLC or ALLD after round one. If the tournament is noisy, there will be long runs of C and D.
For now starting choice is random of 0.6, but that was an arbitrary choice at implementation time.
Names:
 ALLC or ALLD: Original name by Marc Harper
 Repeat: [Akin2015]

class
axelrod.strategies.memoryone.
FirmButFair
[source]¶ A strategy that cooperates on the first move, and cooperates except after receiving a sucker payoff.
Names:
 Firm But Fair: [Frean1994]

class
axelrod.strategies.memoryone.
GTFT
(p: float = None)[source]¶ Generous Tit For Tat Strategy.
Names:
 Generous Tit For Tat: [Nowak1993]
 Naive peace maker: [Gaudesi2016]
 Soft Joss: [Gaudesi2016]

class
axelrod.strategies.memoryone.
MemoryOnePlayer
(four_vector: Tuple[float, float, float, float] = None, initial: axelrod.action.Action = C)[source]¶ Uses a fourvector for strategies based on the last round of play, (P(CCC), P(CCD), P(CDC), P(CDD)). WinStay LoseShift is set as the default player if four_vector is not given. Intended to be used as an abstract base class or to at least be supplied with a initializing four_vector.
Names
 Memory One: [Nowak1990]

class
axelrod.strategies.memoryone.
ReactivePlayer
(probabilities: Tuple[float, float])[source]¶ A generic reactive player. Defined by 2 probabilities conditional on the opponent’s last move: P(CC), P(CD).
Names:
 Reactive: [Nowak1989]

class
axelrod.strategies.memoryone.
SoftJoss
(q: float = 0.9)[source]¶ Defects with probability 0.9 when the opponent defects, otherwise emulates TitForTat.
Names:
 Soft Joss: [Prison1998]

class
axelrod.strategies.memoryone.
StochasticCooperator
[source]¶ Stochastic Cooperator.
Names:
 Stochastic Cooperator: [Adami2013]

class
axelrod.strategies.memoryone.
StochasticWSLS
(ep: float = 0.05)[source]¶ Stochastic WSLS, similar to Generous TFT. Note that this is not the same as Stochastic WSLS described in [Amaral2016], that strategy is a modification of WSLS that learns from the performance of other strategies.
Names:
 Stochastic WSLS: Original name by Marc Harper

class
axelrod.strategies.memoryone.
WinShiftLoseStay
(initial: axelrod.action.Action = D)[source]¶ WinShift LoseStay, also called Reverse Pavlov.
Names:
 WSLS: [Li2011]

class
axelrod.strategies.memoryone.
WinStayLoseShift
[source]¶ WinStay LoseShift, also called Pavlov.
Names:
 Win Stay Lose Shift: [Nowak1993]
 WSLS: [Stewart2012]
 Pavlov: [Kraines1989]

class
axelrod.strategies.meta.
MemoryDecay
(p_memory_delete: float = 0.1, p_memory_alter: float = 0.03, loss_value: float = 2, gain_value: float = 1, memory: list = None, start_strategy: axelrod.player.Player = <class 'axelrod.strategies.titfortat.TitForTat'>, start_strategy_duration: int = 15)[source]¶ A player utilizes the (default) Tit for Tat strategy for the first (default) 15 turns, at the same time memorizing the opponent’s decisions. After the 15 turns have passed, the player calculates a ‘net cooperation score’ (NCS) for their opponent, weighing decisions to Cooperate as (default) 1, and to Defect as (default) 2. If the opponent’s NCS is below 0, the player defects; otherwise, they cooperate.
The player’s memories of the opponent’s decisions have a random chance to be altered (i.e., a C decision becomes D or vice versa; default probability is 0.03) or deleted (default probability is 0.1).
It is possible to pass a different axelrod player class to change the initial player behavior.
Name: Memory Decay

class
axelrod.strategies.meta.
MetaHunter
[source]¶ A player who uses a selection of hunters.
Names
 Meta Hunter: Original name by Karol Langner

class
axelrod.strategies.meta.
MetaHunterAggressive
(team=None)[source]¶ A player who uses a selection of hunters.
Names
 Meta Hunter Aggressive: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaMajority
(team=None)[source]¶ A player who goes by the majority vote of all other nonmeta players.
Names:
 Meta Majority: Original name by Karol Langner

class
axelrod.strategies.meta.
MetaMajorityFiniteMemory
[source]¶ MetaMajority with the team of Finite Memory Players
Names
 Meta Majority Finite Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaMajorityLongMemory
[source]¶ MetaMajority with the team of Long (infinite) Memory Players
Names
 Meta Majority Long Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaMajorityMemoryOne
[source]¶ MetaMajority with the team of Memory One players
Names
 Meta Majority Memory One: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaMinority
(team=None)[source]¶ A player who goes by the minority vote of all other nonmeta players.
Names:
 Meta Minority: Original name by Karol Langner

class
axelrod.strategies.meta.
MetaMixer
(team=None, distribution=None)[source]¶ A player who randomly switches between a team of players. If no distribution is passed then the player will uniformly choose between sub players.
In essence this is creating a Mixed strategy.
Parameters
 team : list of strategy classes, optional
 Team of strategies that are to be randomly played If none is passed will select the ordinary strategies.
 distribution : list representing a probability distribution, optional
 This gives the distribution from which to select the players. If none is passed will select uniformly.
Names
 Meta Mixer: Original name by Vince Knight

class
axelrod.strategies.meta.
MetaPlayer
(team=None)[source]¶ A generic player that has its own team of players.
Names:
 Meta Player: Original name by Karol Langner

class
axelrod.strategies.meta.
MetaWinner
(team=None)[source]¶ A player who goes by the strategy of the current winner.
Names:
 Meta Winner: Original name by Karol Langner

class
axelrod.strategies.meta.
MetaWinnerDeterministic
[source]¶ Meta Winner with the team of Deterministic Players.
Names
 Meta Winner Deterministic: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaWinnerEnsemble
(team=None)[source]¶ A variant of MetaWinner that chooses one of the top scoring strategies at random against each opponent. Note this strategy is always stochastic regardless of the team, if team larger than 1, and the players are distinct.
Names:
 Meta Winner Ensemble: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaWinnerFiniteMemory
[source]¶ MetaWinner with the team of Finite Memory Players
Names
 Meta Winner Finite Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaWinnerLongMemory
[source]¶ MetaWinner with the team of Long (infinite) Memory Players
Names
 Meta Winner Long Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaWinnerMemoryOne
[source]¶ MetaWinner with the team of Memory One players
Names
 Meta Winner Memory Memory One: Original name by Marc Harper

class
axelrod.strategies.meta.
MetaWinnerStochastic
[source]¶ Meta Winner with the team of Stochastic Players.
Names
 Meta Winner Stochastic: Original name by Marc Harper

class
axelrod.strategies.meta.
NMWEDeterministic
[source]¶ Nice Meta Winner Ensemble with the team of Deterministic Players.
Names
 Nice Meta Winner Ensemble Deterministic: Original name by Marc Harper

class
axelrod.strategies.meta.
NMWEFiniteMemory
[source]¶ Nice Meta Winner Ensemble with the team of Finite Memory Players.
Names
 Nice Meta Winner Ensemble Finite Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
NMWELongMemory
[source]¶ Nice Meta Winner Ensemble with the team of Long Memory Players.
Names
 Nice Meta Winner Ensemble Long Memory: Original name by Marc Harper

class
axelrod.strategies.meta.
NMWEMemoryOne
[source]¶ Nice Meta Winner Ensemble with the team of Memory One Players.
Names
 Nice Meta Winner Ensemble Memory One: Original name by Marc Harper

class
axelrod.strategies.meta.
NMWEStochastic
[source]¶ Nice Meta Winner Ensemble with the team of Stochastic Players.
Names
 Nice Meta Winner Ensemble Stochastic: Original name by Marc Harper

class
axelrod.strategies.meta.
NiceMetaWinner
(team=None)¶ A player who goes by the strategy of the current winner.
Names:
 Meta Winner: Original name by Karol Langner

original_class
¶ alias of
MetaWinner

class
axelrod.strategies.meta.
NiceMetaWinnerEnsemble
(team=None)¶ A variant of MetaWinner that chooses one of the top scoring strategies at random against each opponent. Note this strategy is always stochastic regardless of the team, if team larger than 1, and the players are distinct.
Names:
 Meta Winner Ensemble: Original name by Marc Harper

original_class
¶ alias of
MetaWinnerEnsemble

class
axelrod.strategies.mutual.
Desperate
[source]¶ A player that only cooperates after mutual defection.
Names:
 Desperate: [Berg2015]

class
axelrod.strategies.mutual.
Hopeless
[source]¶ A player that only defects after mutual cooperation.
Names:
 Hopeless: [Berg2015]

class
axelrod.strategies.mutual.
Willing
[source]¶ A player that only defects after mutual defection.
Names:
 Willing: [Berg2015]

class
axelrod.strategies.negation.
Negation
[source]¶ A player starts by cooperating or defecting randomly if it’s their first move, then simply doing the opposite of the opponents last move thereafter.
Names:
 Negation: [PD2017]

class
axelrod.strategies.oncebitten.
FoolMeOnce
[source]¶ Forgives one D then retaliates forever on a second D.
Names:
 Fool me once: Original name by Marc Harper

class
axelrod.strategies.oncebitten.
ForgetfulFoolMeOnce
(forget_probability: float = 0.05)[source]¶ Forgives one D then retaliates forever on a second D. Sometimes randomly forgets the defection count, and so keeps a secondary count separate from the standard count in Player.
Names:
 Forgetful Fool Me Once: Original name by Marc Harper

class
axelrod.strategies.oncebitten.
OnceBitten
[source]¶ Cooperates once when the opponent defects, but if they defect twice in a row defaults to forgetful grudger for 10 turns defecting.
Names:
 Once Bitten: Original name by Holly Marissa

class
axelrod.strategies.prober.
CollectiveStrategy
[source]¶ Defined in [Li2009]. ‘It always cooperates in the first move and defects in the second move. If the opponent also cooperates in the first move and defects in the second move, CS will cooperate until the opponent defects. Otherwise, CS will always defect.’
Names:
 Collective Strategy: [Li2009]

class
axelrod.strategies.prober.
Detective
(initial_actions: List[axelrod.action.Action] = None)[source]¶ Starts with C, D, C, C, or with the given sequence of actions. If the opponent defects at least once in the first fixed rounds, play as TFT forever, else defect forever.
Names:
 Detective: [NC2019]

class
axelrod.strategies.prober.
HardProber
[source]¶ Plays D, D, C, C initially. Defects forever if opponent cooperated in moves 2 and 3. Otherwise plays TFT.
Names:
 Hard Prober: [Prison1998]

class
axelrod.strategies.prober.
NaiveProber
(p: float = 0.1)[source]¶ Like titfortat, but it occasionally defects with a small probability.
Names:
 Naive Prober: [Li2011]

class
axelrod.strategies.prober.
Prober
[source]¶ Plays D, C, C initially. Defects forever if opponent cooperated in moves 2 and 3. Otherwise plays TFT.
Names:
 Prober: [Li2011]

class
axelrod.strategies.prober.
Prober2
[source]¶ Plays D, C, C initially. Cooperates forever if opponent played D then C in moves 2 and 3. Otherwise plays TFT.
Names:
 Prober 2: [Prison1998]

class
axelrod.strategies.prober.
Prober3
[source]¶ Plays D, C initially. Defects forever if opponent played C in moves 2. Otherwise plays TFT.
Names:
 Prober 3: [Prison1998]

class
axelrod.strategies.prober.
Prober4
[source]¶ Plays C, C, D, C, D, D, D, C, C, D, C, D, C, C, D, C, D, D, C, D initially. Counts retaliating and provocative defections of the opponent. If the absolute difference between the counts is smaller or equal to 2, defects forever. Otherwise plays C for the next 5 turns and TFT for the rest of the game.
Names:
 Prober 4: [Prison1998]

class
axelrod.strategies.prober.
RemorsefulProber
(p: float = 0.1)[source]¶ Like Naive Prober, but it remembers if the opponent responds to a random defection with a defection by being remorseful and cooperating.
For reference see: [Li2011]. A more complete description is given in “The Selfish Gene” (https://books.google.co.uk/books?id=ekonDAAAQBAJ):
“Remorseful Prober remembers whether it has just spontaneously defected, and whether the result was prompt retaliation. If so, it ‘remorsefully’ allows its opponent ‘one free hit’ without retaliating.”
Names:
 Remorseful Prober: [Li2011]

class
axelrod.strategies.punisher.
InversePunisher
[source]¶ An inverted version of Punisher. The player starts by cooperating however will defect if at any point the opponent has defected, and forgets after mem_length matches, with 1 <= mem_length <= 20. This time mem_length is proportional to the amount of time the opponent has played C.
Names:
 Inverse Punisher: Original name by Geraint Palmer

class
axelrod.strategies.punisher.
LevelPunisher
[source]¶ A player starts by cooperating however, after 10 rounds will defect if at any point the number of defections by an opponent is greater than 20%.
Names:
 Level Punisher: [Eckhart2015]

class
axelrod.strategies.punisher.
Punisher
[source]¶ A player starts by cooperating however will defect if at any point the opponent has defected, but forgets after meme_length matches, with 1<=mem_length<=20 proportional to the amount of time the opponent has played D, punishing that player for playing D too often.
Names:
 Punisher: Original name by Geraint Palmer

class
axelrod.strategies.punisher.
TrickyLevelPunisher
[source]¶ A player starts by cooperating however, after 10, 50 and 100 rounds will defect if at any point the percentage of defections by an opponent is greater than 20%, 10% and 5% respectively.
Names:
 Tricky Level Punisher: [Eckhart2015]

class
axelrod.strategies.qlearner.
ArrogantQLearner
[source]¶ A player who learns the best strategies through the qlearning algorithm.
This Q learner jumps to quick conclusions and cares about the future.
Names:
 Arrogant Q Learner: Original name by Geraint Palmer

class
axelrod.strategies.qlearner.
CautiousQLearner
[source]¶ A player who learns the best strategies through the qlearning algorithm.
This Q learner is slower to come to conclusions and wants to look ahead more.
Names:
 Cautious Q Learner: Original name by Geraint Palmer

class
axelrod.strategies.qlearner.
HesitantQLearner
[source]¶ A player who learns the best strategies through the qlearning algorithm.
This Q learner is slower to come to conclusions and does not look ahead much.
Names:
 Hesitant Q Learner: Original name by Geraint Palmer

class
axelrod.strategies.qlearner.
RiskyQLearner
[source]¶ A player who learns the best strategies through the qlearning algorithm.
This Q learner is quick to come to conclusions and doesn’t care about the future.
Names:
 Risky Q Learner: Original name by Geraint Palmer

find_reward
(opponent: axelrod.player.Player) → Dict[axelrod.action.Action, Dict[axelrod.action.Action, Union[int, float]]][source]¶ Finds the reward gained on the last iteration

find_state
(opponent: axelrod.player.Player) → str[source]¶ Finds the my_state (the opponents last n moves + its previous proportion of playing C) as a hashable state

perform_q_learning
(prev_state: str, state: str, action: axelrod.action.Action, reward)[source]¶ Performs the qlearning algorithm

class
axelrod.strategies.rand.
Random
(p: float = 0.5)[source]¶ A player who randomly chooses between cooperating and defecting.
This strategy came 15th in Axelrod’s original tournament.
Names:
 Random: [Axelrod1980]
 Lunatic: [Tzafestas2000]

class
axelrod.strategies.resurrection.
DoubleResurrection
[source]¶ A player starts by cooperating and defects if the number of rounds played by the player is greater than five and the last five rounds are cooperations.
If the last five rounds were defections, the player cooperates.
Names:
 DoubleResurrection: [Eckhart2015]

class
axelrod.strategies.resurrection.
Resurrection
[source]¶ A player starts by cooperating and defects if the number of rounds played by the player is greater than five and the last five rounds are defections.
Otherwise, the strategy plays like Titfortat.
Names:
 Resurrection: [Eckhart2015]

class
axelrod.strategies.retaliate.
LimitedRetaliate
(retaliation_threshold: float = 0.1, retaliation_limit: int = 20)[source]¶ A player that cooperates unless the opponent defects and wins. It will then retaliate by defecting. It stops when either, it has beaten the opponent 10 times more often that it has lost or it reaches the retaliation limit (20 defections).
Names:
 Limited Retaliate: Original name by Owen Campbell

class
axelrod.strategies.retaliate.
LimitedRetaliate2
(retaliation_threshold: float = 0.08, retaliation_limit: int = 15)[source]¶ LimitedRetaliate player with a threshold of 8 percent and a retaliation limit of 15.
Names:
 Limited Retaliate 2: Original name by Owen Campbell

class
axelrod.strategies.retaliate.
LimitedRetaliate3
(retaliation_threshold: float = 0.05, retaliation_limit: int = 20)[source]¶ LimitedRetaliate player with a threshold of 5 percent and a retaliation limit of 20.
Names:
 Limited Retaliate 3: Original name by Owen Campbell

class
axelrod.strategies.retaliate.
Retaliate
(retaliation_threshold: float = 0.1)[source]¶ A player starts by cooperating but will retaliate once the opponent has won more than 10 percent times the number of defections the player has.
Names:
 Retaliate: Original name by Owen Campbell

class
axelrod.strategies.retaliate.
Retaliate2
(retaliation_threshold: float = 0.08)[source]¶ Retaliate player with a threshold of 8 percent.
Names:
 Retaliate 2: Original name by Owen Campbell

class
axelrod.strategies.retaliate.
Retaliate3
(retaliation_threshold: float = 0.05)[source]¶ Retaliate player with a threshold of 5 percent.
Names:
 Retaliate 3: Original name by Owen Campbell
Revised Downing implemented from the Fortran source code for the second of Axelrod’s tournaments.

class
axelrod.strategies.revised_downing.
RevisedDowning
[source]¶ Strategy submitted to Axelrod’s second tournament by Leslie Downing. (K59R).
Revised Downing attempts to determine if players are cooperative or not. If so, it cooperates with them.
This strategy is a revision of the strategy submitted by Downing to Axelrod’s first tournament.
Names:  Revised Downing: [Axelrod1980]

class
axelrod.strategies.sequence_player.
SequencePlayer
(generator_function: function, generator_args: Tuple = ())[source]¶ Abstract base class for players that use a generated sequence to determine their plays.
Names:
 Sequence Player: Original name by Marc Harper

class
axelrod.strategies.sequence_player.
ThueMorse
[source]¶ A player who cooperates or defects according to the ThueMorse sequence. The first few terms of the ThueMorse sequence are: 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 …
ThueMorse sequence: http://mathworld.wolfram.com/ThueMorseSequence.html
Names:
 Thue Morse: Original name by Geraint Palmer

class
axelrod.strategies.sequence_player.
ThueMorseInverse
[source]¶ A player who plays the inverse of the ThueMorse sequence.
Names:
 Inverse Thue Morse: Original name by Geraint Palmer

class
axelrod.strategies.shortmem.
ShortMem
[source]¶ A player starts by always cooperating for the first 10 moves.
From the tenth round on, the player analyzes the last ten actions, and compare the number of defects and cooperates of the opponent, based in percentage. If cooperation occurs 30% more than defection, it will cooperate. If defection occurs 30% more than cooperation, the program will defect. Otherwise, the program follows the TitForTat algorithm.
Names:
 ShortMem: [Andre2013]

class
axelrod.strategies.selfsteem.
SelfSteem
[source]¶ This strategy is based on the feeling with the same name. It is modeled on the sine curve(f = sin( 2* pi * n / 10 )), which varies with the current iteration.
If f > 0.95, ‘ego’ of the algorithm is inflated; always defects. If 0.95 > abs(f) > 0.3, rational behavior; follows TitForTat algortithm. If 0.3 > f > 0.3; random behavior. If f < 0.95, algorithm is at rock bottom; always cooperates.
Futhermore, the algorithm implements a retaliation policy, if the opponent defects; the sin curve is shifted. But due to lack of further information, this implementation does not include a sin phase change. Names:
 SelfSteem: [Andre2013]

class
axelrod.strategies.stalker.
Stalker
[source]¶ This is a strategy which is only influenced by the score. Its behavior is based on three values: the very_bad_score (all rounds in defection) very_good_score (all rounds in cooperation) wish_score (average between bad and very_good score)
It starts with cooperation.
 If current_average_score > very_good_score, it defects
 If current_average_score lies in (wish_score, very_good_score) it cooperates
 If current_average_score > 2, it cooperates
 If current_average_score lies in (1, 2)
 The remaining case, current_average_score < 1, it behaves randomly.
 It defects in the last round
Names:
 Stalker: [Andre2013]

class
axelrod.strategies.titfortat.
AdaptiveTitForTat
(rate: float = 0.5)[source]¶ ATFT  Adaptive Tit For Tat (Basic Model)
Algorithm
if (opponent played C in the last cycle) then world = world + r*(1world) else world = world + r*(0world) If (world >= 0.5) play C, else play D
Attributes
 world : float [0.0, 1.0], set to 0.5
 continuous variable representing the world’s image 1.0  total cooperation 0.0  total defection other values  something in between of the above updated every round, starting value shouldn’t matter as long as it’s >= 0.5
Parameters
 rate : float [0.0, 1.0], default=0.5
 adaptation rate  r in Algorithm above smaller value means more gradual and robust to perturbations behaviour
Names:
 Adaptive Tit For Tat: [Tzafestas2000]

class
axelrod.strategies.titfortat.
Alexei
[source]¶ Plays similar to TitforTat, but always defect on last turn.
Names:
 Alexei: [LessWrong2011]

class
axelrod.strategies.titfortat.
AntiTitForTat
[source]¶ A strategy that plays the opposite of the opponents previous move. This is similar to Bully, except that the first move is cooperation.
Names:
 Anti Tit For Tat: [Hilbe2013]
 Psycho (PSYC): [Ashlock2009]

class
axelrod.strategies.titfortat.
Bully
[source]¶ A player that behaves opposite to Tit For Tat, including first move.
Starts by defecting and then does the opposite of opponent’s previous move. This is the complete opposite of Tit For Tat, also called Bully in the literature.
Names:
 Reverse Tit For Tat: [Nachbar1992]

class
axelrod.strategies.titfortat.
ContriteTitForTat
[source]¶ A player that corresponds to Tit For Tat if there is no noise. In the case of a noisy match: if the opponent defects as a result of a noisy defection then ContriteTitForTat will become ‘contrite’ until it successfully cooperates.
Names:
 Contrite Tit For Tat: [Axelrod1995]

original_class
¶ alias of
ContriteTitForTat

class
axelrod.strategies.titfortat.
DynamicTwoTitsForTat
[source]¶ A player starts by cooperating and then punishes its opponent’s defections with defections, but with a dynamic bias towards cooperating based on the opponent’s ratio of cooperations to total moves (so their current probability of cooperating regardless of the opponent’s move (aka: forgiveness)).
Names:
 Dynamic Two Tits For Tat: Original name by Grant GarrettGrossman.

class
axelrod.strategies.titfortat.
EugineNier
[source]¶ Plays similar to TitforTat, but with two conditions: 1) Always Defect on Last Move 2) If other player defects five times, switch to all defects.
Names:
 Eugine Nier: [LessWrong2011]

original_class
¶ alias of
EugineNier

class
axelrod.strategies.titfortat.
Gradual
[source]¶ Similar to OriginalGradual, this is a player that punishes defections with a growing number of defections but after punishing for punishment_limit number of times enters a calming state and cooperates no matter what the opponent does for two rounds.
This version of Gradual is an update of OriginalGradual and the difference is that the punishment_limit is incremented whenever the opponent defects (regardless of the state of the player).
Note that this version of Gradual appears in [CRISTALSMAC2018] however this version of Gradual does not give the results reported in [Beaufils1997] which is the paper that first introduced the strategy. For a longer discussion of this see: https://github.com/AxelrodPython/Axelrod/issues/1294.
This version is based on https://github.com/cristalsmac/ipd/blob/master/src/strategies.py#L224
Names:
 Gradual: [CRISTALSMAC2018]

class
axelrod.strategies.titfortat.
HardTitFor2Tats
[source]¶ A variant of Tit For Two Tats that uses a longer history for retaliation.
Names:
 Hard Tit For Two Tats: [Stewart2012]

class
axelrod.strategies.titfortat.
HardTitForTat
[source]¶ A variant of Tit For Tat that uses a longer history for retaliation.
Names:
 Hard Tit For Tat: [PD2017]

class
axelrod.strategies.titfortat.
Michaelos
[source]¶ Plays similar to TitforTat with two exceptions: 1) Defect on last turn. 2) After own defection and opponent’s cooperation, 50 percent of the time, cooperate. The other 50 percent of the time, always defect for the rest of the game.
Names:
 Michaelos: [LessWrong2011]

class
axelrod.strategies.titfortat.
NTitsForMTats
(N: int = 3, M: int = 2)[source]¶ A parameterizable TitforTat, The arguments are: 1) M: the number of defection before retaliation 2) N: the number of retaliations
Names:
 N Tit(s) For M Tat(s): Original name by Marc Harper

class
axelrod.strategies.titfortat.
OmegaTFT
(deadlock_threshold: int = 3, randomness_threshold: int = 8)[source]¶ OmegaTFT modifies Tit For Tat in two ways:  checks for deadlock loops of alternating rounds of (C, D) and (D, C), and attempting to break them  uses a more sophisticated retaliation mechanism that is noise tolerant
Names:
 OmegaTFT: [Slany2007]

class
axelrod.strategies.titfortat.
OriginalGradual
[source]¶ A player that punishes defections with a growing number of defections but after punishing for punishment_limit number of times enters a calming state and cooperates no matter what the opponent does for two rounds.
The punishment_limit is incremented whenever the opponent defects and the strategy is not in either calming or punishing state.
Note that Gradual appears in [CRISTALSMAC2018] however that version of Gradual does not give the results reported in [Beaufils1997] which is the paper that first introduced the strategy. For a longer discussion of this see: https://github.com/AxelrodPython/Axelrod/issues/1294. This is why this strategy has been renamed to OriginalGradual.
Names:
 Gradual: [Beaufils1997]

class
axelrod.strategies.titfortat.
RandomTitForTat
(p: float = 0.5)[source]¶ A player starts by cooperating and then follows by copying its opponent (tit for tat style). From then on the player will switch between copying its opponent and randomly responding every other iteration.
Name:
 Random TitForTat: Original name by Zachary M. Taylor

class
axelrod.strategies.titfortat.
SlowTitForTwoTats2
[source]¶ A player plays C twice, then if the opponent plays the same move twice, plays that move, otherwise plays previous move.
Names:
 Slow Tit For Tat: [Prison1998]

class
axelrod.strategies.titfortat.
SneakyTitForTat
[source]¶ Tries defecting once and repents if punished.
Names:
 Sneaky Tit For Tat: Original name by Karol Langner

class
axelrod.strategies.titfortat.
SpitefulTitForTat
[source]¶ A player starts by cooperating and then mimics the previous action of the opponent until opponent defects twice in a row, at which point player always defects
Names:
 Spiteful Tit For Tat: [Prison1998]

class
axelrod.strategies.titfortat.
SuspiciousTitForTat
[source]¶ A variant of Tit For Tat that starts off with a defection.
Names:
 Suspicious Tit For Tat: [Hilbe2013]
 Mistrust: [Beaufils1997]

class
axelrod.strategies.titfortat.
TitFor2Tats
[source]¶ A player starts by cooperating and then defects only after two defects by opponent.
Submitted to Axelrod’s second tournament by John Maynard Smith; it came in 24th in that tournament.
Names:
 Tit for two Tats: [Axelrod1984]
 Slow tit for two tats: Original name by Ranjini Das
 JMaynardSmith: [Axelrod1980b]

class
axelrod.strategies.titfortat.
TitForTat
[source]¶ A player starts by cooperating and then mimics the previous action of the opponent.
This strategy was referred to as the ‘simplest’ strategy submitted to Axelrod’s first tournament. It came first.
Note that the code for this strategy is written in a fairly verbose way. This is done so that it can serve as an example strategy for those who might be new to Python.
Names:
 Rapoport’s strategy: [Axelrod1980]
 TitForTat: [Axelrod1980]

class
axelrod.strategies.titfortat.
TwoTitsForTat
[source]¶ A player starts by cooperating and replies to each defect by two defections.
Names:
 Two Tits for Tats: [Axelrod1984]

class
axelrod.strategies.verybad.
VeryBad
[source]¶ It cooperates in the first three rounds, and uses probability (it implements a memory, which stores the opponent’s moves) to decide for cooperating or defecting. Due to a lack of information as to what that probability refers to in this context, probability(P(X)) refers to (Count(X)/Total_Moves) in this implementation P(C) = Cooperations / Total_Moves P(D) = Defections / Total_Moves = 1  P(C)
Names:
 VeryBad: [Andre2013]

class
axelrod.strategies.worse_and_worse.
KnowledgeableWorseAndWorse
[source]¶ This strategy is based on ‘Worse And Worse’ but will defect with probability of ‘current turn / total no. of turns’.
 Names:
 Knowledgeable Worse and Worse: Original name by Adam Pohl

class
axelrod.strategies.worse_and_worse.
WorseAndWorse
[source]¶ Defects with probability of ‘current turn / 1000’. Therefore it is more and more likely to defect as the round goes on.
Source code available at the download tab of [Prison1998]
 Names:
 Worse and Worse: [Prison1998]

class
axelrod.strategies.worse_and_worse.
WorseAndWorse2
[source]¶ Plays as tit for tat during the first 20 moves. Then defects with probability (current turn  20) / current turn. Therefore it is more and more likely to defect as the round goes on.
 Names:
 Worse and Worse 2: [Prison1998]

class
axelrod.strategies.worse_and_worse.
WorseAndWorse3
[source]¶ Cooperates in the first turn. Then defects with probability no. of opponent defects / (current turn  1). Therefore it is more likely to defect when the opponent defects for a larger proportion of the turns.
 Names:
 Worse and Worse 3: [Prison1998]

class
axelrod.strategies.zero_determinant.
LRPlayer
(phi: float = 0.2, s: float = 0.1, l: float = 1)[source]¶ Abstraction for Linear Relation players. These players enforce a linear difference in stationary payoffs \(s (S_{xy}  l) = S_{yx}  l.\)
The parameter \(s\) is called the slope and the parameter \(l\) the baseline payoff. For extortionate strategies, the extortion factor \(\chi\) is the inverse of the slope \(s\).
For the standard prisoner’s dilemma where \(T > R > P > S\) and \(R > (T + S) / 2 > P\), a pair \((l, s)\) is enforceable iff
\begin{eqnarray} &P &<= l <= R \\ &s_{min} &= \min\left( \frac{T  l}{l  S}, \frac{l  S}{T  l}\right) <= s <= 1 \end{eqnarray}And also that there exists \(\phi\) such that
\begin{eqnarray} p_1 &= P(CCC) &= 1  \phi (1  s)(R  l) \\ p_2 &= P(CCD) &= 1  \phi (s(l  S) + (T  l)) \\ p_3 &= P(CDC) &= \phi ((l  S) + s(T  l)) \\ p_4 &= P(CDD) &= \phi (1  s)(l  P) \end{eqnarray}These conditions also force \(\phi >= 0\). For a given pair \((l, s)\) there may be multiple such \(\phi\).
This parameterization is Equation 14 in [Hilbe2013]. See Figure 2 of the article for a more indepth explanation. Other game parameters can alter the relations and bounds above.
Names:
 Linear Relation player: [Hilbe2013]

class
axelrod.strategies.zero_determinant.
ZDExtort2
(phi: float = 0.1111111111111111, s: float = 0.5)[source]¶ An Extortionate Zero Determinant Strategy with l=P.
Names:
 Extort2: [Stewart2012]

class
axelrod.strategies.zero_determinant.
ZDExtort2v2
(phi: float = 0.125, s: float = 0.5, l: float = 1)[source]¶ An Extortionate Zero Determinant Strategy with l=1.
Names:
 EXTORT2: [Kuhn2017]

class
axelrod.strategies.zero_determinant.
ZDExtort3
(phi: float = 0.11538461538461539, s: float = 0.3333333333333333, l: float = 1)[source]¶ An extortionate strategy from Press and Dyson’s paper witn an extortion factor of 3.
Names:
 ZDExtort3: Original name by Marc Harper
 Unnamed: [Press2012]

class
axelrod.strategies.zero_determinant.
ZDExtort4
(phi: float = 0.23529411764705882, s: float = 0.25, l: float = 1)[source]¶ An Extortionate Zero Determinant Strategy with l=1, s=1/4. TFT is the other extreme (with l=3, s=1)
Names:
 Extort 4: Original name by Marc Harper

class
axelrod.strategies.zero_determinant.
ZDExtortion
(phi: float = 0.2, s: float = 0.1, l: float = 1)[source]¶ An example ZD Extortion player.
Names:
 ZDExtortion: [Roemheld2013]

class
axelrod.strategies.zero_determinant.
ZDGTFT2
(phi: float = 0.25, s: float = 0.5)[source]¶ A Generous Zero Determinant Strategy with l=R.
Names:
 ZDGTFT2: [Stewart2012]

class
axelrod.strategies.zero_determinant.
ZDGen2
(phi: float = 0.125, s: float = 0.5, l: float = 3)[source]¶ A Generous Zero Determinant Strategy with l=3.
Names:
 GEN2: [Kuhn2017]

class
axelrod.strategies.zero_determinant.
ZDMischief
(phi: float = 0.1, s: float = 0.0, l: float = 1)[source]¶ An example ZD Mischief player.
Names:
 ZDMischief: [Roemheld2013]

class
axelrod.strategies.zero_determinant.
ZDSet2
(phi: float = 0.25, s: float = 0.0, l: float = 2)[source]¶ A Generous Zero Determinant Strategy with l=2.
Names:
 SET2: [Kuhn2017]
Glossary¶
There are a variety of terms used in the documentation and throughout the library. Here is an overview:
An action¶
An action is either C
or D
.
You can access these actions as follows but should not really have a reason to:
>>> import axelrod as axl
>>> axl.Action.C
C
>>> axl.Action.D
D
A play¶
A play is a single player choosing an action. In terms of code this is equivalent to:
>>> p1, p2 = axl.Cooperator(), axl.Defector()
>>> C = p1.strategy(p2) # This constitutes two 'plays' (p1 plays and p2 plays).
>>> D = p2.strategy(p1) # This constitutes two 'plays' (p1 plays and p2 plays).
A turn¶
A turn is a 1 shot interaction between two players. It is in effect a composition of two plays.
Each turn has four possible outcomes of a play: (C, C)
, (C, D)
,
(D, C)
, or (D, D)
.
A match¶
A match is a consecutive number of turns. The default number of turns used in the tournament is 200. Here is a single match between two players over 3 turns:
>>> p1, p2 = axl.Cooperator(), axl.Defector()
>>> match = axl.Match((p1, p2), turns=3)
>>> result = match.play()
>>> result
[(C, D), (C, D), (C, D)]
>>> p1.history, p2.history
([C, C, C], [D, D, D])
A win¶
A win is attributed to the player who has the higher total score at the end
of a match. For the example above, Defector
would win that match.
A strategy¶
A strategy is a set of instructions that dictate how to play given one’s own strategy and the strategy of an opponent. In the library these correspond to the strategy classes: TitForTat, Grudger, Cooperator etc…
When appropriate to do so this will be used interchangeable with A player.
A player¶
A player is a single agent using a given strategy. Players are the participants of tournament, usually they each represent one strategy but of course you can have multiple players choosing the same strategy. In the library these correspond to __instances__ of classes:
>>> p1, p2 = axl.Cooperator(), axl.Defector()
>>> p1
Cooperator
>>> p2
Defector
When appropriate to do so this will be used interchangeable with A strategy.
A round robin¶
A round robin is the set of all potential (order invariant) matches between a given collection of players.
A tournament¶
A tournament is a repetition of round robins so as to smooth out stochastic effects.
Noise¶
A match or tournament can be played with noise: this is the probability that indicates the chance of an action dictated by a strategy being swapped.