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.

Strategies in the Axelrod’s first tournament:

Name Long name Axelrod Library Name
Tit For Tat Tit For Tat TitForTat
Tideman and Chieruzzi Tideman and Chieruzzi (authors’ names) Not Implemented
Nydegger Nydegger (author’s name) Nydegger
Grofman Grofman (author’s name) Grofman
Shubik Shubik (author’s name) Shubik
Stein and Rapoport Stein and Rapoport (authors’ names) Not Implemented
Grudger Grudger (by Friedman) Grudger
Davis Davis (author’s name) Davis
Graaskamp Graaskamp (author’s name) Not Implemented
Downing Downing (author’s name) RevisedDowning
Feld Feld (author’s name) Feld
Joss Joss (author’s name) Joss
Tullock Tullock (author’s name) Tullock
Unnamed Strategy Unnamed Strategy (by a Grad Student in Political Science) UnnamedStrategy
Random Random Random

Tit for Tat

This strategy was referred to as the ‘simplest’ strategy submitted. It begins by cooperating and then simply repeats the last moves made by the opponent.

Tit for Tat came 1st in Axelrod’s original tournament.

Tideman and Chieruzzi

Not implemented yet

This strategy begins by playing Tit For Tat and then things get slightly complicated:

  1. Every run of defections played by the opponent increases the number of defections that this strategy retaliates with by 1.
  2. 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 tournament
    • and the total number of defections differs from a 50-50 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).

This strategy came 2nd in Axelrod’s original tournament.

Nydegger

This strategy begins by playing Tit For Tat for the first 3 rounds with the following modifications:

If it is the only strategy to cooperate in the first round and the only strategy to defect on the second round then it defects on the 3 round (despite the fact that Tit For Tat would now cooperate).

After these first 3 rounds the next move is made depending on the previous 3 rounds. A score is given to these rounds according to the following calculation:

\[A = 16 a_1 + 4 a_2 + a_3\]

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:

\[A \in \{1, 6, 7, 17, 22, 23, 26, 29, 30, 31, 33, 38, 39, 45, 49, 54, 55, 58, 61\}\]

This strategy came 3rd in Axelrod’s original tournament.

Grofman

This is a pretty simple strategy: it cooperates on the first two rounds and returns the opponent’s last action for the next 5. For the rest of the game Grofman cooperates if both players selected the same action in the previous round, and otherwise cooperates randomly with probability \(\frac{2}{7}\).

This strategy came 4th in Axelrod’s original tournament.

Shubik

This strategy plays a modification of Tit For Tat. It starts by retaliating with a single defection but the number of defections increases by 1 each time the opponent defects when this strategy cooperates.

This strategy came 5th in Axelrod’s original tournament.

Stein and Rapoport

Not implemented yet

This strategy plays a modification of Tit For Tat.

  1. It cooperates for the first 4 moves.
  2. It defects on the last 2 moves.
  3. Every 15 moves it makes use of a chi-squared test to check if the opponent is playing randomly.

This strategy came 6th in Axelrod’s original tournament.

Grudger

This strategy cooperates until the opponent defects and then defects forever.

This strategy came 7th in Axelrod’s original tournament.

Davis

This strategy is a modification of Grudger. It starts by cooperating for the first 10 moves and then plays Grudger.

This strategy came 8th in Axelrod’s original tournament.

Graaskamp

Not implemented yet

This strategy follows the following rules:

  1. Play Tit For Tat for the first 50 rounds;
  2. Defects on round 51;
  3. Plays 5 further rounds of Tit For Tat;
  4. A check is then made to see if the opponent is playing randomly in which case it defects for the rest of the game;
  5. The strategy also checks to see if the opponent is playing Tit For Tat or another strategy from a preliminary tournament called ‘Analogy’. If so it plays Tit For Tat. If not it cooperates and randomly defects every 5 to 15 moves.

This strategy came 9th in Axelrod’s original tournament.

Downing

This strategy attempts to estimate the next move of the opponent by estimating the probability of cooperating given that they defected (\(p(C|D)\)) or cooperated on the previous round (\(p(C|C)\)). These probabilities are continuously updated during play and the strategy attempts to maximise the long term play. Note that the initial values are \(p(C|C)=p(C|D)=.5\).

Downing is implemented as RevisedDowning. Apparently in the first tournament the strategy was implemented incorrectly and defected on the first two rounds. This can be controlled by setting revised=True to prevent the initial defections.

This strategy came 10th in Axelrod’s original tournament.

Feld

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.

This strategy came 11th in Axelrod’s original tournament.

Joss

This strategy plays Tit For Tat, always defecting if the opponent defects but cooperating when the opponent cooperates with probability .9.

This strategy came 12th in Axelrod’s original tournament.

Tullock

This strategy cooperates for the first 11 rounds and then (randomly) cooperates 10% less often than the opponent has in the previous 10 rounds.

This strategy came 13th in Axelrod’s original tournament.

Unnamed Strategy

Apparently written by a grad student in political science whose name was withheld, this strategy cooperates with a given probability \(P\). This probability (which has initial value .3) is updated every 10 rounds based on whether the opponent seems to be random, very cooperative or very uncooperative. Furthermore, if after round 130 the strategy is losing then \(P\) is also adjusted.

Since the original code is not available and was apparently complicated, we have implemented this strategy based on published descriptions. The strategy cooperates with a random probability between 0.3 and 0.7.

This strategy came 14th in Axelrod’s original tournament.

Random

This strategy plays randomly (disregarding the history of play).

This strategy came 15th in Axelrod’s original tournament.

Axelrod’s second tournament

Work in progress.

EATHERLEY

This strategy was submitted by Graham Eatherley to Axelrod’s second tournament and generally cooperates unless the opponent defects, in which case Eatherley defects with a probability equal to the proportion of rounds that the opponent has defected.

This strategy came in Axelrod’s second tournament.

CHAMPION

This strategy was submitted by Danny Champion to Axelrod’s second tournament and operates in three phases. The first phase lasts for the first 1/20-th of the rounds and Champion always cooperates. In the second phase, lasting until 4/50-th of the rounds have passed, Champion mirrors its opponent’s last move. In the last phase, Champion cooperates unless - the opponent defected on the last round, and - the opponent has cooperated less than 60% of the rounds, and - a random number is greater than the proportion of rounds defected

TESTER

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.

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 Zero-Determinant 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
EXTORT-2 Extort-2 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 Tit-For-Tat TitForTat
GRIM Grim Grudger
GTFT Generous Tit-For-Tat GTFT
TF2T Tit-For-Two-Tats TitFor2Tats
WSLS Win-Stay-Lose-Shift WinStayLoseShift
RANDOM Random Random
ZDGTFT-2 ZDGTFT-2 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

Generous-Tit-For-Tat plays Tit-For-Tat with occasional forgiveness, which prevents cycling defections against itself.

GTFT is defined as a memory-one strategy as follows:

  • \(P(C\,|\,CC) = 1\)
  • \(P(C\,|\,CD) = p\)
  • \(P(C\,|\,DC) = 1\)
  • \(P(C\,|\,DD) = p\)

where \(p = \min\left(1 - \frac{T-R}{R-S}, \frac{R-P}{T-P}\right)\).

GTFT came 2nd in average score and 18th in wins in S&P’s tournament.

TF2T

Tit-For-Two-Tats is like Tit-For-Tat but only retaliates after two defections rather than one. This is not a memory-one strategy.

TF2T came 3rd in average score and last (?) in wins in S&P’s tournament.

WSLS

Win-Stay-Lose-Shift is a strategy that shifts if the highest payoff was not earned in the previous round. WSLS is also known as “Win-Stay-Lose-Switch” and “Pavlov”. It can be seen as a memory-one strategy as follows:

  • \(P(C\,|\,CC) = 1\)
  • \(P(C\,|\,CD) = 0\)
  • \(P(C\,|\,DC) = 0\)
  • \(P(C\,|\,DD) = 1\)

TF2T 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 memory-one 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.

ZDGTFT-2

This memory-one 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.

EXTORT-2

This memory-one 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 memory-one 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](http://www.prisoners-dilemma.com/strategies.html) 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.

Prober2 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.