Aspiration Can Promote Cooperation in Well-Mixed Populations As in Regular Graphs
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Aspiration Can Promote Cooperation in Well-Mixed Populations As in Regular Graphs Dhaker Kroumi1 Accepted: 25 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Classical studies on aspiration-based dynamics suggest that dissatisfied individuals switch their strategies without taking into account the success of others. The imitation-based dynamics allow individuals to imitate successful strategies without taking into account their own-satisfactions. In this article, we propose to study a dynamic based on aspiration, which takes into account imitation of successful strategies for dissatisfied individuals. Individuals compare their success to their aspired levels. This mechanism helps individuals with a minimum of self-satisfaction to maintain their strategies. Dissatisfied individuals will learn from their neighbors by choosing the successful strategies. We derive an exact expression of the fixation probability in well-mixed populations as in graph-structured populations. As a result, we show that weak selection favors the evolution of cooperation if the difference in aspired level exceeds some crucial value. Increasing the aspired level of cooperation should oppose cooperative behavior while increasing the aspired level of defection should promote cooperative behavior. We show that the cooperation level decreases as the connectivity increases. The best scenario for the cooperative evolution is a graph with a small connectivity, while the worst scenario is a well-mixed population. Keywords Fixation probability · Evolutionary game dynamics · Pair approximation · Cooperation · Imitation · Aspiration Mathematics Subject Classification Primary 92D25 · Secondary 60J70
1 Introduction Evolutionary game theory is the framework where the frequency of a strategy depends on the fitnesses of the different individuals in the population (Maynard Smith and Price [32], Maynard Smith [31], Hofbauer and Sigmund [13], Weibull [56], Samuelson [44], Cressman [3],
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Dhaker Kroumi [email protected] Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Dynamic Games and Applications
Vincent and Brown [55], Nowak [34]). Individuals interact and gain payoffs, which are seen as biological fitness or reproductive rates. The standard model, called the replicator equation, was formulated in an infinitely large well-mixed population where any two individuals have the same probability to interact (Taylor and Jonker [49], Zeeman [58], Hofbauer and Sigmund [14,15]). Suppose that there are n strategies {S1 , S2 , . . . , Sn }. The game is described by a payoff matrix A = {ai, j }i, j=1,...,n , where ai, j is the payoff of an Si -player if its partner is an S j -player. Let xi be the frequency of Si -players in the population. The dynamic is dxi = xi ( f i − f ), (1) dt n where f i = nj=1 x j ai, j and f = i=1 xi f i refer to the expected payoff on an Si -player and the average payoff in the population, respectively. Real popu
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