Publication: Generosity pays in the presence of direct reciprocity: a comprehensive study of 2x2 repeated games
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2012-04-18
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PLoS
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Abstract
By applying a technique previously developed to study ecosystem assembly [Capita´n et al., Phys. Rev. Lett. 103, 168101
(2009)] we study the evolutionary stable strategies of iterated 2|2 games. We focus on memory-one strategies, whose
probability to play a given action depends on the actions of both players in the previous time step. We find the
asymptotically stable populations resulting from all possible invasions of any known stable population. The results of this
invasion process are interpreted as transitions between different populations that occur with a certain probability. Thus the
whole process can be described as a Markov chain whose states are the different stable populations. With this approach we
are able to study the whole space of symmetric 2|2 games, characterizing the most probable results of evolution for the
different classes of games. Our analysis includes quasi-stationary mixed equilibria that are relevant as very long-lived
metastable states and is compared to the predictions of a fixation probability analysis. We confirm earlier results on the
success of the Pavlov strategy in a wide range of parameters for the iterated Prisoner’s Dilemma, but find that as the
temptation to defect grows there are many other possible successful strategies. Other regions of the diagram reflect the
equilibria structure of the underlying one-shot game, albeit often some non-expected strategies arise as well. We thus
provide a thorough analysis of iterated 2|2 games from which we are able to extract some general conclusions. Our most
relevant finding is that a great deal of the payoff parameter range can still be understood by focusing on win-stay, lose-shift
strategies, and that very ambitious ones, aspiring to obtaining always a high payoff, are never evolutionary stable
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Tit-for-tat, Prisoner's dilemma, Genetical evolution, Finite automata, Lose-shift, Win-stay, Cooperation, Rules, Chaos
Bibliographic citation
PLoS ONE, 7(4), e35135, pp. 1-12