PHYSICAL REVIEW E 92, 012813 (2015) Payoff components and their effects in a spatial three-strategy evolutionary social dilemma Jeromos Vukov, 1 Levente Varga, 2 Benjamin Allen, 3 Martin A. Nowak, 3, 4 and Gy¨ orgy Szab´ o 5 1 Research Center for Educational and Network Studies, Centre for Social Sciences, Hungarian Academy of Sciences, P. O. Box 20, H-1250 Budapest, Hungary 2 Babes ¸-Bolyai University, Faculty of Physics, RO-400084 Cluj-Napoca, Romania 3 Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA 4 Department of Mathematics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA 5 Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary (Received 3 April 2015; published 17 July 2015) We study a three-strategy spatial evolutionary prisoner’s dilemma game with imitation and logit update rules. Players can follow the always-cooperating, always-defecting or the win-stay-lose-shift (WSLS) strategies and gain their payoff from games with their direct neighbors on a square lattice. The friendliness parameter of the WSLS strategy—characterizing its cooperation probability in the first round—tunes the cyclic component of the game determining whether the game can be characterized by a potential. We measured and calculated the phase diagrams of the system for a wide range of parameters. When the game is a potential game and the logit rule is applied, the theoretically predicted phase diagram agrees very well with the simulation results. Surprisingly, this phase diagram can be accurate even in the nonpotential case if there are only two surviving strategies in the stationary state; this result harmonizes with the fact that all 2 × 2 games are potential games. For the imitation dynamics, we found that the effects of spatiality combined with the presence of two cooperative strategies are so strong that they suppress even substantial changes in the payoff matrix, thus the phase diagrams are independent of the cyclic component’s intensity. At the same time, this type of strategy update mechanism supports the formation of cooperative clusters that results in a cooperative society in a wider parameter range compared to the logit dynamics. DOI: 10.1103/PhysRevE.92.012813 PACS number(s): 89.65.−s, 89.75.Fb, 87.23.Kg I. INTRODUCTION The mathematical explanation of the ubiquitous coopera- tion is one of the greatest challenges of our days. Evolutionary game theory [1–6] offers a suitable framework to study this conundrum. This interdisciplinary research tool incorporates the assets of mathematics, biology, physics, economy, sociol- ogy, and other disciplines and creates an ideal environment to investigate the evolutionary processes taking place in populations. In this paper, we examine the evolution of cooperation using a prisoner’s dilemma (PD) [6–8] model characterizing the sharpest social dilemma situation that can emerge in a scenario of conflicting interests. In a PD encounter, the two participating individuals have the option to choose between cooperation (C) and defection (D). Depending on their decisions, they earn different payoffs. Mutual cooperation (defection) results in the reward R (punishment P ), a defector encountering a co- operator earns the temptation T , and the exploited cooperator gets the sucker’s payoff S . Due to the T>R>P>S payoff ranking that characterizes the PD, the choice to defect results in higher earning than the choice to cooperate; however, if both players think along this argumentation, they end up with the second-worst payoff P instead of the second-best R that could be achieved by mutual cooperation. This outlines the dilemma. This reasoning shows that mutual defection is the Nash equilibrium in the one-shot PD. Cooperation needs additional mechanisms to emerge: One of these is to turn the interaction between the two players to a repeated game. In the iterated PD (IPD) setup, players play more rounds of one-shot PD games with each other and their success is measured by the total (or the average) payoff they acquire during this longer interaction sequence. Along this change, a plethora of possible strategies become accessible that can promote cooperation as, during the longer interaction, players can utilize the information gathered in the previous rounds. In our model, players can adopt three types of strategies: in addition to the simple always-cooperating AllC and always-defecting AllD, the win-stay lose-shift (WSLS) strategy is available for them. WSLS evaluates the outcome of the last round and if the earned payoff was below a given threshold, then the player changes his or her action for the next round. Players with this strategy possess a friendliness parameter that characterizes how cooperative they are in the first round. WSLS proved to be very effective in promoting cooperation in many cases due to its resilience against errors [9,10]. It is already clarified that the effect of memory [11], the competition between the aspiration levels [12], and the stochastic effects in systems with structured populations [13,14] can improve the efficiency of the WSLS strategy in the maintenance of cooperation. We have chosen these three strategies because the resulting 3 × 3 payoff matrix has peculiar properties. Recently, it was discovered [15] that all symmetric 3 × 3 games can be decomposed into the linear combination of a potential game matrix and a cyclic component. Potential games are convenient in the sense that they behave as many well-known physical systems: The random sequential application of the 1539-3755/2015/92(1)/012813(9) 012813-1 ©2015 American Physical Society