# Evolutionary Prisoner's Dilemma Tournaments

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The prisoner's dilemma is a two-player game in which each player (prisoner) can either "cooperate" (stay silent) or "defect" (betray the other prisoner). If both players cooperate, they each get a reward ; if both defect, they get a punishment payoff ; if one player defects and the other cooperates, the defecting player gets a temptation payoff , while the cooperating player receives a sucker payoff . In the standard form of the game,

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Contributed by: Jiusi Li (April 2020)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

All of the strategies implemented in this Demonstration are "memory-one" strategies; that is, the action of a player in a given round depends only on the actions by the two players in the previous round. Memory-one strategies can be described by a strategy vector whose components represent the probabilities that the player cooperates given a particular pair of actions by the two players in the previous round. For example, in a strategy with vector the player always cooperates if both players cooperated in the previous round, cooperates with 50% probability if one player cooperated and the other player defected in the previous round, and cooperates with 25% probability if both players defected in the previous round.

Memory-one strategies can be modeled by a Markov chain on the four states CC, CD, DC, DD, and the stationary distribution of this Markov chain determines the average payoffs for the two players over a large number of games [2, 4]. The stationary distribution is an eigenvector with eigenvalue 1 of the transition matrix of the Markov chain.

Controls

All-c: the player always cooperates. Strategy vector .

All-d: the player always defects. Strategy vector .

TFT (tit for tat [1]): the player cooperates if the opponent cooperated in the previous round, and defects if the opponent defected in the previous round. Strategy vector .

GTFT (generous tit for tat [3]): the player cooperates after every instance of the opponent’s cooperations and after 25% of the opponent’s defections. Strategy vector .

SET-2 (an example of an equalizer strategy [3, 4]): this strategy forces the opponent’s payoff to be 2 regardless of what strategy the opponent uses. Strategy vector: .

EXTORT-2 (an example of an extortionate strategy [3, 4]): a strategy that guarantees the player a higher or equal payoff no matter what the opponent plays. Strategy vector: .

GEN-2 (generous zero-determinant strategy [3]): strategy vector .

Spiteful [3]: the player cooperates if both players cooperated in the previous round and defects otherwise. Strategy vector .

Pavlov [3]: the player cooperates if both players made the same decision in the previous move, and defects otherwise. Strategy vector .

Random: a random strategy, given by a vector whose components are random numbers in .

References

[1] R. M. Axelrod, *The Evolution of Cooperation*, New York: Basic Books, 1984.

[2] S. Kuhn. "Prisoner's Dilemma." The Stanford Encyclopedia of Philosophy. (Apr 14, 2020) plato.stanford.edu/entries/prisoner-dilemma.

[3] P. Mathieu and J.-P. Delahaye, "New Winning Strategies for the Iterated Prisoner's Dilemma," *Journal of Artificial Societies and Social Simulation*, 20(4), 2017 pp. 1–12. doi:10.18564/jasss.3517.

[4] W. H. Press and F. J. Dyson, "Iterated Prisoner’s Dilemma Contains Strategies That Dominate Any Evolutionary Opponent," *Proceedings of the National Academy of Sciences*, 109(26), 2012 pp. 10409–10413. doi:10.1073/pnas.1206569109.

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