Elsevier

Applied Mathematics and Computation

Volume 341, 15 January 2019, Pages 402-407
Applied Mathematics and Computation

Coevolution of multi-game resolves social dilemma in network population

https://doi.org/10.1016/j.amc.2018.09.019Get rights and content

Highlight

  • We propose the coevolution setup of strategy and multi-game based on memory step.

  • For short memory step, the larger the value of sucker's payoff is, the higher frequency of cooperation will be.

  • For large memory step, middle sucker’s payoff leads to the highest frequency of cooperation.

Abstract

It is an open question to understand the emergence and maintenance of cooperation in nature and society. Aim to this issue, evolutionary game theory in networked population and its various derivations, like mixing game and multi-game, have proved an effective way to resolve the social dilemma. In this work, we propose the coevolution framework of strategy and multi-game: if a player, in prisoner's dilemma game, successively keeps its strategy constant for several times (referred as memory step), it will have opportunity to participate in snow drift game, which has lower dilemma strength than prisoner's dilemma game. Of particular, it is unveiled that for short memory step, the larger the value of sucker's payoff is, the higher frequency of cooperation will be. While for long memory step, middle sucker's payoff provides a best environment for cooperation. For all these findings, we also provide theoretical analysis, which guarantees further validation.

Introduction

Cooperation is abundant in society, ranging from bacteria community to human being society. For example, parasites can overcome the defense of host through mutual cooperation, and human capture food together in primitive society. While all these observations seem to be inconsistent with the prediction of natural selection [1], [2], [3], in which cooperators should vanish because of high private cost. Thus, elucidating the emergence of cooperation among selfish individuals represents one of the most important challenges in social science and behavioral science, and has attracted great research interest. During the past decades, evolutionary game theory has become a powerful theoretical framework to investigate this puzzle [4], [5], [6], [7], [8]. In particular, prisoner's dilemma game (PDG) and snowdrift game (SDG) are two well-known models [9], [10], [11], [12], [13], where two players must choose cooperation or defection at the same time. They both receive the reward R for mutual cooperation, mutual defection will lead to the punishment P. If a cooperator plays with a defector, the former one receives a sucker's payoff S, while the other gets the temptation to defect T. In PDG, the payoffs strictly satisfy the ranking T > R > P > S and 2R > T + S, which indicates that defection is always the best choice no matter what choice the opponent choose. While in SDG, the payoffs satisfy the ranking T > R > S > P, where the best option is to take the contrary action of your opponent (i.e., cooperation and defection consist the equilibrium state). Thus, compared with PDG, SDG shows lower dilemma strength [14].

In order to solve the social dilemma, a great number of mechanisms recently have been proposed in theoretical and experimental researches [9], [15], [16]. While Nowak attributed all these scenarios promoting cooperation to five key rules: kin selection, direct reciprocity, indirect reciprocity, group selection and spatial reciprocity [17], [18], [19], [20]. Among them, spatial reciprocity has attracted great attention, because of its broad application in statistical physics and mathematics [21]. In network population, each player only interacts with its nearest neighbors, and thus cooperator can survive by forming compact clusters [19]. After this seminal idea, the study of evolutionary has been encapsulated into variety of network topologies, such as Boolean network [22], small-world network [23], scale-free network [24], [25], [26], [27] and interdependent networks [28], [29]. Besides, many different factors have also been proposed based on different networks to explore the influence of these factors on evolution of cooperation, such as reputation [30], [31], punishment [32], migration [33], [34], [35], asymmetric [36] and aspiration [37], to name but a few. However, most majority of these mentioned works mainly focus on a simple game model, recent studies also investigate how cooperation survives in multigame or mixing game [38], [39], which means that each player chooses the given strategies, yet adopts different payoff matrices.

In spite of great progress of recent years, there still exists one case receiving little attention: games that agents play is not strictly constant, they usually change according to the environment. Inspired by this fact, we here consider a coevolution framework of multigame: if the individual in PDG successively keep strategy constant for several times (referred as memory step), it can convert to SDG next step (for avoiding higher dilemma strength). However, once the individual who plays SDG changes its strategy, it will turn to play PDG next time.

The rest of this paper is organized as follows. In Section 2, we present our evolutionary game model. Section 3 gives a description of numerical simulation results. Finally, we discuss the results and conclude the paper in Section 4.

Section snippets

Model

We first consider the interaction network, where each player occupies the nodes of L*L square lattice with periodic boundary conditions. Each node is initially designed as a cooperator (Sx) or a defector (Sy) with equal probability, which can be described as: Sx=(1,0)T,Sy=(0,1)T.

Besides, each player initially belongs to one of two populations (Ti) with the same probability. It is designed to play PDG once Ti = 1, and play SDG for Ti = 2. Following the previous literatures [8], [14], [23], we

Results

We start by examining the impact of memory steps M and sucker's payoff σ on the evolution of cooperation in Fig. 1. Fig. 1(a) shows how the stationary frequency of cooperation FC depends on the temptation to defection b for different values of σ with fixed memory M = 6. σ = 0 returns to the traditional version, where all the players play the weak PDG, and the frequency of cooperation decreases to zero quickly even when the value of b is small. When 0 < σ ≤ 1, two game models exist in the whole

Conclusion

Motivated by the fact that individual can help its offspring get competitive power to resist other population's invasion by transmitting his ability. In this paper, we have investigated the emergence of cooperation under the coevolution mechanism of strategy and multi-game base on memory step. Through MC simulations, we find that small memory step can promote cooperation effectively. Besides, the sucker's payoff which parameter affects the social dilemma promotes cooperation when memory step is

Acknowledgment

We acknowledge support from the National Natural Science Foundation of China (Nos. 31700393, 11671348 and 61672020), China Postdoctoral Science Foundation (No. 2013M542560 and 2015T81129), Shandong Province Higher Educational Science and Technology Program (No. J16LN61).

References (41)

  • DuW.B. et al.

    The effect of asymmetric payoff mechanism on evolution networked prisoner's dilemma game

    Physics A

    (2009)
  • R. Axelrod

    The Evolution of Cooperation

    (1984)
  • C. Darwin

    On the Origin of Species

    (1859)
  • M.A. Nowak

    Evolutionary Dynamics

    (2006)
  • M.A. Nowak et al.

    Evolutionary dynamics of biological games

    Science

    (2004)
  • R.L. Trivers

    The evolution of reciprocal altruism

    Q. Rev. Biol.

    (1971)
  • M.A. Nowak et al.

    The spatial dilemmas of evolution

    Intern. J. Bifur. Chaos

    (1993)
  • FuF. et al.

    Reputation-based partner choice promotes cooperation on graphs and social networks

    Phys. Rev. E

    (2008)
  • WuZ.X. et al.

    Effects of strategy-migration direction and noise in the evolutionary spatial prisoner's dilemma

    Phys. Rev. E

    (2009)
  • WangZ. et al.

    Onymity promotes cooperation in social dilemma experiments

    Sci. Adv.

    (2017)
  • Cited by (44)

    • The influence of quasi-cooperative strategy on social dilemma evolution

      2022, Chaos, Solitons and Fractals
      Citation Excerpt :

      Individuals have different behavioral characteristics during the evolutionary game. Introducing many potential promoting mechanisms in the evolutionary game, including the reputation [19,20], voluntary participation [21,22], reward and punishment [23–27], memory [28–32], extortion [33,34], aspiration [35–38], migration [39–43], and heterogeneity [44,45]. Cui et al. study the impact of heterogeneous distributions of abilities on the evolution of individual cooperation in the spatial prisoners dilemma game.

    View all citing articles on Scopus
    1

    These authors contributed equally.

    View full text