Reentrant phase transitions and defensive alliances in social dilemmas with informed strategies

Knowing the strategy of an opponent in a competitive environment conveys obvious evolutionary advantages. But this information is costly, and the benefit of being informed may not necessarily offset the additional cost. Here we introduce social dilemmas with informed strategies, and we show that this gives rise to two cyclically dominant triplets that form defensive alliances. The stability of these two alliances is determined by the rotation velocity of the strategies within each triplet. A weaker strategy in a faster rotating triplet can thus overcome an individually stronger competitor. Fascinating spatial patterns favor the dominance of a single defensive alliance, but enable also the stable coexistence of both defensive alliances in very narrow regions of the parameter space. A continuous reentrant phase transition reveals before unseen complexity behind the stability of strategic alliances in evolutionary social dilemmas. Successful evolution of cooperation in the realm of social dilemmas is a grand challenge that continues to attract research across the social and natural sciences [1–6]. When individuals are torn between what is best for them and what is best for their society, cooperation can quickly come to play second fiddle to the pursuit of short-term personal benefits. Although studies in evolutionary game theory [7–11] have revealed fundamental rules that promote cooperation [12], the chasm behind the Darwinian “only the fittest survive” and the abundance of cooperation in human and animal societies remains overwhelming. The availability of information has been the holy grail in modern game theory since its inception [13]. If one knew the rules of the game and what strategy the opponent will play, then the best response would be guaranteed [14–16]. In repeated evolutionary settings with many players and under the reasonable assumption that information about the opponents is not free, however, the prospects are much less clear. In fact, recent human experiments have revealed that, contrary to expectations, costly information about the neighbors can impair the evolution of cooperation in social dilemmas [17]. Important unanswered questions are thus: (i) Under which conditions are informed strategies — players that invest into learning the strategies of other players — evolutionary stable? (ii) What are the properties of these stable solutions? and (iii) Which mechanisms are responsible for their stability? In this letter, we answer these questions by introducing social dilemmas with informed strategies. The following matrix describes the payoffs of the four competing strategies: C D IC ID C R S R S D T P 0 P IC R− ǫ 0− ǫ R− ǫ 0− ǫ ID T − ǫ 0− ǫ 0− ǫ 0− ǫ Between unconditional cooperators (C) and defectors (D), in agreement with the traditional formulation of a social dilemma [18], mutual cooperation yields the reward R,