Distinguishing the opponents in the prisoner dilemma in well-mixed populations arXiv:1001.1397v1 [physics.soc-ph] 9 Jan 2010 Lucas Wardil †∗ and Jafferson K. L. da Silva † $^{†}$Departamento de F´ısica, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte - MG, Brazil (Dated: October 29, 2018) Here we study the effects of adopting different strategies against different opponent instead of adopting the same strategy against all of them in the prisoner dilemma structured in well-mixed populations. We consider an evolutionary process in which strategies that provide reproductive success are imitated and players replace one of their worst interactions by the new one. We set individuals in a well-mixed population so that network reciprocity effect is excluded and we analyze both synchronous and asynchronous updates. As a consequence of the replacement rule, we show that mutual cooperation is never destroyed and the initial fraction of mutual cooperation is a lower bound for the level of cooperation. We show by simulation and mean-field analysis that for synchronous update cooperation dominates while for asynchronous update only cooperations associated to the initial mutual cooperations are maintained. As a side effect of the replacement rule, an “implicit punishment” mechanism comes up in a way that exploitations are always neutralized providing evolutionary stability for cooperation. PACS numbers: 87.23.-n, 89.65.-s, 02.50.Le Keywords: Cooperative dilemma was initially studied in the framework of classical game theory. Usually individu- als have two strategies: cooperation and defection. A cooperator provides a benefit to the opponent and pays a cost for that. A defector receives the benefits if the opponent is a cooperator. This defines a material pay- off. If individuals maximize their material payoff, it is well know that defection will dominate [1]. Departing from these initial ideas, evolutionary game theory has emerged and strategy evolution in populations was stud- ied. In this approach it is implicit assumed the principle of natural selection, where the payoff is equated to fit- ness and the fittest strategy survives [2]. In this context it was shown that the classical theory is recovered in the replicator equation, where population is considered to be well-mixed, that is, a population where everybody inter- acts with everybody [2]. Cooperation cannot be supported without extra mech- anisms [3]. Essentially two actions take place for cooper- ation survival: maintenance of mutual cooperation and prevention from exploitation [4]. Cooperators can be bet- ter off only if they meet each other so that their profits exceed defectors’ profits. If the individuals perceive that it is important what the opponents are doing in order to attend these two essential actions, reciprocal preferences can come up [5–8]. Reciprocity means that what an indi- vidual do depends on what others do to him/her directly or indirectly. Direct reciprocity means that I choose what to do against you depending on what you do to me. In- direct reciprocity means that my behavior toward you also depends on what you do to others. Another subtle Recently it was introduced the possibility of an “im- plicit punishment” without turn to extra individual pref- erences except the desire to maximize the own gain. This was accomplished by the adoption of different strategies against different opponents in the context of network reci- procity with synchronous update [18]. Instead of playing the same strategy against all of the neighbors, individu- als can choose one different strategy against each oppo- nent. If each player strategically updates their strategies possibly imitating a successful random neighbor and re- places the interaction that gives the worst payoff by the imitated strategy, it was shown for square lattices that cooperation was strongly supported, even for huge defec- way of reciprocity is network reciprocity. Individuals are set on the vertices of a network and interact only with their neighbors. In this context, cooperators form clus- ters of mutual cooperation and this mutualism is viewed as reciprocity [9–14]. But human behavior is not so sim- ple: individuals can adopt reciprocal strategies but, moti- vated by internal emotion, like anger against exploitation [15], they can punish defectors [15–17]. This would not be so intriguing, as it is just another way of reciprocal motives. But the important feature is that individuals usually input costs to defectors at their own expenses. This behavior is called altruistic punishment, because in- dividuals pay a cost to punish even if they never met the punished opponent again and because the punishment acts weaken the defectors and the entire population gets better off [15]. Reputation, rewards or repeated inter- action, as internal motives, they all interact with pun- ishment motives [5, 6]. Punishment involves some subtle questions and gives rise to another evolutionary puzzle: altruistic punishment, although seemingly usual, may be a maladaptive trait as the punishers get worst payoffs [8]. 1. Introduction