Evolutionary cycles of cooperation and defection

Abstract

The main obstacle for the evolution of cooperation is that natural selection favors defection in most settings. In the repeated prisoner's dilemma, two individuals interact several times, and, in each round, they have a choice between cooperation and defection. We analyze the evolutionary dynamics of three simple strategies for the repeated prisoner's dilemma: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). We study mutationselection dynamics in finite populations. Despite ALLD being the only strict Nash equilibrium, we observe evolutionary oscillations among all three strategies. The population cycles from ALLD to TFT to ALLC and back to ALLD. Most surprisingly, the time average of these oscillations can be entirely concentrated on TFT. In contrast to the classical expectation, which is informed by deterministic evolutionary game theory of infinitely large populations, stochastic evolution of finite populations need not choose the strict Nash equilibrium and can therefore favor cooperation over defection.evolutionary dynamics ͉ finite population ͉ prisoner's dilemma ͉ reciprocity ͉ stochastic process I n the prisoner's dilemma, two players have the choice to cooperate or to defect. Both obtain payoff R for mutual cooperation, but a lower payoff P for mutual defection. If one individual defects, while the other cooperates, then the defector receives the highest payoff T whereas the cooperator receives the lowest payoff S. We have T Ͼ R Ͼ P Ͼ S. Defection dominates cooperation: in any mixed population, defectors have a higher fitness than cooperators.As is standard in repeated games, new strategies become possible when the game is repeated, and these strategies can lead to a wider range of equilibrium outcomes (1-8). In particular, in the infinitely repeated prisoner's dilemma, cooperation becomes an equilibrium outcome, but defection remains an equilibrium as well (9, 10). To select between these equilibria, the authors of refs. 11-13 looked at the replicator dynamic on a continuum population, and the authors of refs. 14 and 15 applied variants of evolutionary stability to repeated games with complexity costs. These solution concepts do not have explicit dynamics and are based on models with a continuum population.Our goal is to study explicit evolutionary dynamics in a large but finite population. To explicitly model evolutionary dynamics, the space of possible strategies must be restricted. In this article, we explore the evolutionary dynamics of three strategies: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). TFT cooperates in the first move and then does whatever the opponent did in the previous move. Ever since Axelrod's celebrated computer tournaments (16), TFT is a world champion in the repeated prisoner's dilemma, although it has some weaknesses and has at times been defeated by other strategies (11,12,17). For our purpose here, these weaknesses are not important. We conjecture that similar results hold for other reciprocal strategies, such as generoustit-for-...

Combination with anti-tit-for-tat remedies problems of tit-for-tat

“…Although TF2T was proposed as a remedy to avoid the inefficiency of TFT, its predictability opens another possibility of being exploited. Generous TFT avoids this dilemma between generosity and exploitability by introducing randomness in forgiving the co-player's defection [10,11,12,13]. However, its unpredictability might be a double-edged sword when it comes to public policy making, in which such random factors are not popular ideas [14].…”

“…All these difficulties are rooted in the error vulnerability of TFT. Worse is that this is not the only weakness: A TFT population is invaded via neutral drift by unconditional cooperators (AllC), which, in turn, opens the back door to AllD [11,12,13,19]. It would thus be desirable if some modification overcame these shortcomings while preserving the strengths of TFT.…”

Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

“…Second, the game-theoretic model in this paper and some previous studies are 2×2 games, in which the cooperative dilemma has not been removed. But recent studies revealed that cooperative dilemma will vanish when a third strategy is introduced (Imhof et al, 2005). It shows that the three strategies form cycle which is similar to the "rock-paper-scissors" game.…”

Evolutionary prisoner׳s dilemma game on graphs and social networks with external constraint

“…In the following, the evolutionary dynamics of three repeated strategies ALLD, TFT and ALLC (Nowak and Sigmund, 1989;Imhof et al, 2005) on a graph with the influence of external force are also examined. The following payoff matrix with only one parameter, bð1 ob o 2Þ (Nowak and May, 1992), will be used:…”