Persistence extends reciprocity

Persistence in repeated games encourages the evolution of spite

Social behavior is divided into four types: altruism, spite, mutualism, and selfishness. The former two are costly to the actor; therefore, from the perspective of natural selection, their existence can be regarded as mysterious. One potential setup which encourages the evolution of altruism and spite is repeated interaction. Players can behave conditionally based on their opponent's previous actions in the repeated interaction. On the one hand, the retaliatory strategy (who behaves altruistically when their opponent behaved altruistically and behaves non-altruistically when the opponent player behaved non-altruistically) is likely to evolve when players choose altruistic or selfish behavior in each round. On the other hand, the anti-retaliatory strategy (who is spiteful when the opponent was not spiteful and is not spiteful when the opponent player was spiteful) is likely to evolve when players opt for spiteful or mutualistic behavior in each round. These successful conditional behaviors can be favored by natural selection. Here, we notice that information on opponent players' actions is not always available. When there is no such information, players cannot determine their behavior according to their opponent's action. By investigating the case of altruism, a previous study (Kurokawa, 2017, Mathematical Biosciences, 286, 94-103) found that persistent altruistic strategies, which choose the same action as the own previous action, are favored by natural selection. How, then, should a spiteful conditional strategy behave when the player does not know what their opponent did? By studying the repeated game, we find that persistent spiteful strategies, which choose the same action as the own previous action, are favored by natural selection. Altruism and spite differ concerning whether retaliatory or anti-retaliatory strategies are favored by natural selection; however, they are identical concerning whether persistent strategies are favored by natural selection.

Effect of the group size on the evolution of cooperation when an exit option is present

The evolution of cooperation has been one of the main topics in evolutionary biology. If cooperators maintain interaction with cooperators and halt interaction with defectors, then cooperation can pay and can be favored by natural selection. This is called an exit option. Here, not only cooperation in dyadic interactions but also cooperation in sizable groups can be observed. Rivalry is about whether usage of the benefit by one individual reduces its availability to others or not. A common good is a rivalrous good, whereas a public good is a non-rivalrous good. In this paper, by analyzing n-player prisoner's dilemma games, we examine whether the effect of the group size on cooperation is positive or negative in the context of exit option. When goods are common goods, defectors always dominate cooperators when the group size is infinitely large. Thus, the group size has only negative effects on the evolution of cooperation when goods are common goods. In contrast, when goods are public goods, an increase in group size has positive effects as well as negative effects on the evolution of cooperation. In addition, we reveal that it has both positive and negative effects on the evolution of cooperation for cooperators to tolerate some defection and hope to keep the interaction.

Evolution of Groupwise Cooperation: Generosity, Paradoxical Behavior, and Non-Linear Payoff Functions

Evolution of cooperation by reciprocity has been studied using two-player and n-player repeated prisoner’s dilemma games. An interesting feature specific to the n-player case is that players can vary in generosity, or how many defections they tolerate in a given round of a repeated game. Reciprocators are quicker to detect defectors to withdraw further cooperation when less generous, and better at maintaining a long-term cooperation in the presence of rare defectors when more generous. A previous analysis on a stochastic evolutionary model of the n-player repeated prisoner’s dilemma has shown that the fixation probability of a single reciprocator in a population of defectors can be maximized for a moderate level of generosity. However, the analysis is limited in that it considers only tit-for-tat-type reciprocators within the conventional linear payoff assumption. Here we extend the previous study by removing these limitations and show that, if the games are repeated sufficiently many times, considering non-tit-for-tat type strategies does not alter the previous results, while the introduction of non-linear payoffs sometimes does. In particular, under certain conditions, the fixation probability is maximized for a “paradoxical” strategy, which cooperates in the presence of fewer cooperating opponents than in other situations in which it defects.

Which facilitates the evolution of cooperation more, retaliation or persistence?

The existence of cooperation in this world is a mysterious phenomenon. One of the mechanisms that explain the evolution of cooperation is repeated interaction. If interactions between the same individuals repeat and individuals cooperate conditionally, cooperation can evolve. A previous study pointed out that if individuals have persistence (i.e., imitate its "own" behavior in the last move), cooperation can evolve. However, retaliation and persistence are not mutually exclusive decisions, but rather a trade-off in the decision making process of individuals. Players can refer to the opponent's behavior and if the actor and the opponent opted for the different alternative in the last move, conditional cooperators have to give up either retaliation or persistence. The previous study also investigated this, and has revealed that the individual should give more importance to retaliation than to persistence. However, this study has assumed that the errors in perception are absent. In this world, errors in perception are present, and trying to imitate the opponent player can sometimes end in failure. And, it might be that imitating the focal player, which definitely ends in success, is more beneficial than trying to imitate the opponent player, which can end in failure especially when the error rate in recognition is large. Here, this paper uses evolutionarily stable strategy (ESS) analysis and analyzes the stability for reactive strategies against the invasion by unconditional defectors in the iterated prisoner's dilemma game. And our analysis reveals that even if we take errors in perception into consideration, retaliation facilitates the evolution of cooperation more than persistence unexpectedly. In addition, we analyze the stability for reactive cooperators against the invasion by a strategy other than unconditional defectors. Moreover, we also analyze the deterministic model in which unconditional cooperators, unconditional defectors, and the reactive strategy at the same time.