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).