5,676 research outputs found

    Evolution of extortion in structured populations

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    Extortion strategies can dominate any opponent in an iterated prisoner’s dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionarily stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other hand, reveals counterintuitive solutions. Defectors and extortioners coarsen spontaneously, which allows cooperators to prevail even at prohibitively high temptations to defect. Here extortion strategies play the role of a Trojan horse. They may emerge among defectors by chance, and once they do, cooperators become viable as well. These results are independent of the interaction topology, and they highlight the importance of coarsening, checkerboard ordering, and best response updating in evolutionary games

    Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations

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    We study the evolution of cooperation in the spatial prisoner's dilemma game, where besides unconditional cooperation and defection, tit-for-tat, win-stay-lose-shift and extortion are the five competing strategies. While pairwise imitation fails to sustain unconditional cooperation and extortion regardless of game parametrization, myopic updating gives rise to the coexistence of all five strategies if the temptation to defect is sufficiently large or if the degree distribution of the interaction network is heterogeneous. This counterintuitive evolutionary outcome emerges as a result of an unexpected chain of strategy invasions. Firstly, defectors emerge and coarsen spontaneously among players adopting win-stay-lose-shift. Secondly, extortioners and players adopting tit-for-tat emerge and spread via neutral drift among the emerged defectors. And lastly, among the extortioners, cooperators become viable too. These recurrent evolutionary invasions yield a five-strategy phase that is stable irrespective of the system size and the structure of the interaction network, and they reveal the most unexpected mechanism that stabilizes extortion and cooperation in an evolutionary setting.Comment: 7 two-column pages, 5 figures; accepted for publication in Scientific Reports [related work available at http://arxiv.org/abs/1401.8294

    Zero-determinant strategies in finitely repeated games

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    Direct reciprocity is a mechanism for sustaining mutual cooperation in repeated social dilemma games, where a player would keep cooperation to avoid being retaliated by a co-player in the future. So-called zero-determinant (ZD) strategies enable a player to unilaterally set a linear relationship between the player's own payoff and the co-player's payoff regardless of the strategy of the co-player. In the present study, we analytically study zero-determinant strategies in finitely repeated (two-person) prisoner's dilemma games with a general payoff matrix. Our results are as follows. First, we present the forms of solutions that extend the known results for infinitely repeated games (with a discount factor w of unity) to the case of finitely repeated games (0 < w < 1). Second, for the three most prominent ZD strategies, the equalizers, extortioners, and generous strategies, we derive the threshold value of w above which the ZD strategies exist. Third, we show that the only strategies that enforce a linear relationship between the two players' payoffs are either the ZD strategies or unconditional strategies, where the latter independently cooperates with a fixed probability in each round of the game, proving a conjecture previously made for infinitely repeated games.Comment: 24 pages, 2 figure

    Comparing reactive and memory-one strategies of direct reciprocity

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    Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need to be for cooperation to evolve. Here, we study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player's previous move, and memory-one strategies, which take into account the own and the co-player's previous move. In both cases, we compare deterministic strategy and stochastic strategy spaces. For reactive strategies and small costs, we find that stochasticity benefits cooperation, because it allows for generous-tit-for-tat. For memory one strategies and small costs, we find that stochasticity does not increase the propensity for cooperation, because the deterministic rule of win-stay, lose-shift works best. For memory one strategies and large costs, however, stochasticity can augment cooperation.Comment: 18 pages, 7 figure

    Unbending strategies shepherd cooperation and suppress extortion in spatial populations

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    Evolutionary game dynamics on networks typically consider the competition among simple strategies such as cooperation and defection in the Prisoner's Dilemma and summarize the effect of population structure as network reciprocity. However, it remains largely unknown regarding the evolutionary dynamics involving multiple powerful strategies typically considered in repeated games, such as the zero-determinant (ZD) strategies that are able to enforce a linear payoff relationship between them and their co-players. Here, we consider the evolutionary dynamics of always cooperate (AllC), extortionate ZD (extortioners), and unbending players in lattice populations based on the commonly used death-birth updating. Out of the class of unbending strategies, we consider a particular candidate, PSO Gambler, a machine-learning-optimized memory-one strategy, which can foster reciprocal cooperation and fairness among extortionate players. We derive analytical results under weak selection and rare mutations, including pairwise fixation probabilities and long-term frequencies of strategies. In the absence of the third unbending type, extortioners can achieve a half-half split in equilibrium with unconditional cooperators for sufficiently large extortion factors. However, the presence of unbending players fundamentally changes the dynamics and tilts the system to favor unbending cooperation. Most surprisingly, extortioners cannot dominate at all regardless of how large their extortion factor is, and the long-term frequency of unbending players is maintained almost as a constant. Our analytical method is applicable to studying the evolutionary dynamics of multiple strategies in structured populations. Our work provides insights into the interplay between network reciprocity and direct reciprocity, revealing the role of unbending strategies in enforcing fairness and suppressing extortion.Comment: 21 pages, 6 figure

    Conformity enhances network reciprocity in evolutionary social dilemmas

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    The pursuit of highest payoffs in evolutionary social dilemmas is risky and sometimes inferior to conformity. Choosing the most common strategy within the interaction range is safer because it ensures that the payoff of an individual will not be much lower than average. Herding instincts and crowd behavior in humans and social animals also compel to conformity on their own right. Motivated by these facts, we here study the impact of conformity on the evolution of cooperation in social dilemmas. We show that an appropriate fraction of conformists within the population introduces an effective surface tension around cooperative clusters and ensures smooth interfaces between different strategy domains. Payoff-driven players brake the symmetry in favor of cooperation and enable an expansion of clusters past the boundaries imposed by traditional network reciprocity. This mechanism works even under the most testing conditions, and it is robust against variations of the interaction network as long as degree-normalized payoffs are applied. Conformity may thus be beneficial for the resolution of social dilemmas.Comment: 8 two-column pages, 5 figures; accepted for publication in Journal of the Royal Society Interfac

    Impact of Committed Minorities: Unveiling Critical Mass of Cooperation in the Iterated Prisoner's Dilemma Game

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    The critical mass effect is a prevailing topic in the study of complex systems. Recent research has shown that a minority of zealots can effectively drive widespread cooperation in social dilemma games. However, achieving a critical mass of cooperation in the prisoner's dilemma requires stricter conditions. The underlying mechanism behind this effect remains unclear, particularly in the context of repeated interactions. This paper aims to investigate the influence of a committed minority on cooperation in the Iterated Prisoner's Dilemma game, a widely studied model of repeated interactions between individuals facing a social dilemma. In contrast to previous findings, we identify tipping points for both well-mixed and structured populations. Our findings demonstrate that a committed minority of unconditional cooperators can induce full cooperation under weak imitation conditions. Conversely, a committed minority of conditional cooperators, who often employ Tit-for-Tat or extortion strategies, can promote widespread cooperation under strong imitation conditions. These results hold true across various network topologies and imitation rules, suggesting that critical mass effects may be a universal principle in social dilemma games. Additionally, we discover that an excessive density of committed conditional cooperators can hinder cooperation in structured populations. This research advances our understanding of the role of committed minorities in shaping social behavior and provides valuable insights into cooperation dynamics.Comment: 12 pages, 15 figure
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