Abstract Evolutionary game theory on spatial structures has received increasing attention during the past decades. However, the majority of these achievements focuses on single and static population structures, which is not fully consistent with the fact that real structures are composed of many interactive groups. These groups are interdependent on each other and present dynamical features, in which individuals mimic the strategy of neighbors and switch their partnerships continually. It is however unclear how the dynamical and interdependent interactions among groups affect the evolution of collective behaviors. In this work, we employ the prisoner's dilemma game to investigate how the dynamics of structure influences cooperation on interdependent populations, where populations are represented by group structures. It is found that the more robust the links between cooperators (or the more fragile the links between cooperators and defectors), the more prevalent of cooperation. Furthermore, theoretical analysis shows that the intra-group bias can favor cooperation, which is only possible when individuals are likely to attach neighbors within the same group. Yet, interestingly, cooperation can be even inhibited for large intra-group bias, allowing the moderate intra-group bias maximizes the cooperation level.

Citation: Tang C, Wang Z, Li X (2014) Moderate Intra-Group Bias Maximizes Cooperation on Interdependent Populations. PLoS ONE 9(2): e88412. https://doi.org/10.1371/journal.pone.0088412 Editor: Matjaž Perc, University of Maribor, Slovenia Received: December 4, 2013; Accepted: January 7, 2014; Published: February 12, 2014 Copyright: © 2014 Tang et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was partly supported by the National Key Basic Research and Development Program (No. 2010CB731403), the National Natural Science Foundation (No. 61273223), the Research Fund for the Doctoral Program of Higher Education (No. 20120071110029) and the Key Project of National Social Science Fund (No. 12&ZD18) of China. C.B.T. also acknowledges the financial support by the Academic New Artist Ministry of Education Doctoral Post Graduate (No. JFH1232003) of China. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Cooperation is a widely observed phenomenon in social science, biology and economics [1], [2]. However, cooperative behavior apparently contradicts the natural selection [3]: Selfish players always have a higher average fitness than that of cooperators, since selfish players enjoy the benefits from the cooperation of others without associated costs. Therefore, it has fascinated many interests from natural and social scientists to understand the emergence and the stability of cooperation. Within the interdisciplinary field of evolutionary game theory, this puzzle benefits from techniques of biology, economy, computer sciences, and physics [4], [5]. As a metaphor, the prisoner's dilemma (PD) game has attracted great attention in both theoretical and experimental studies to investigate the evolution of cooperation [6]–[12]. In a typical PD game, two players simultaneously decide whether they act as a cooperator ( ) or a defector ( ). s are willing to engage in cooperative tasks, while s prefer not to. They will receive the reward, , if both cooperate, and the punishment, , if both defect. However, if one player defects while the other decides to cooperate, the former will get the temptation, , while the latter will get the sucker's payoff, . Namely, the local interaction between and is given by the following payoff matrix: (1)where these payoffs satisfy and . It is obvious that players prefer to defect if they wish to maximize their own payoffs, whatever the opponent's decision. Various mechanisms to support the evolution of cooperation have been identified recently [13]–[18], such as direct reciprocity, indirect reciprocity, group selection and network reciprocity [19]. The most famous context is spatial games introduced by Nowak and May [20], where players are arranged on a spatially structured population and interact with their neighbors only. It is shown that cooperators could survive by means of forming compact clusters, which minimize the exploitation of defectors. In the line of this seminal achievement, the role of spatial game and its underlying promoted mechanisms in evolutionary games have been intensively explored, such as the mobility of players [21]–[23], different evolutionary time scales [24]–[27], social diversity [28], [29], heterogeneous ability and aspiration [30], [31] (for comprehensive reviews refer to Ref. [32]). Though large amounts of work upon spatial reciprocity are available, the main attention remains in an isolated and single structure. In human societies, empirical evidences have shown that the realistic structures are composed of many interactive groups, which interact with each other over time [33]–[40]. In this context, the evolution behavior traits have been considered underlying the interdependent populations, where populations are represented by group structures to account for different social types. Note that this framework is similar with previous studies that have addressed the structure on interdependent networks, in the sense that the success of one node in a given group not only depends on the nodes in the same group, but also replies on the states of other nodes in other groups. Taking some examples more specifically, in a recent paper [41], where the biased utility function on interdependent networks were implemented, it was shown that the stronger the bias in the utility function, the higher the level of public cooperation. While in [42], a replicator such as evolutionary game dynamics took place on interdependent populations, cooperative behaviors are fixed on the system (even if the system is well-mixed). Moreover, it was also a remarkable hint that only an intermediate density of sufficiently strong interactions between groups could lead the optimal resolution of social dilemmas [43], [44]. Aside from the effect of spatial structure and its various promoted mechanisms, the co-evolution of game models also attracts numerous attention [45]–[52], which not only reflects the evolving of strategies over time, but also characterizes the adaptive development of topologies and/or update rules. In particular, the interdependent populations in our real social life are dynamical and changing over time. Besides, the essence of evolutionary game theory on interdependent populations remains unclear, especially for the question how the structure of dynamical and interdependent populations affects the evolution of cooperation. Therefore, we introduce an intra-group bias based rewiring probability, and focus on co-evolution of strategy and structure to investigate the evolution of cooperation on interdependent populations. Within the fast rewiring process, we derive a simple rule quantitatively revealing how the link breaking probability and intra-group bias are chosen to stabilize cooperation. Interestingly, though cooperation is favored by intra-group bias conditionally, it is precluded for a large intra-group bias, which uncovers that the moderate intra-group bias maximizes the cooperation level.

Conclusions To sum up, we have established a microscopic model on the co-evolutionary dynamics of cooperation and interdependent populations. Under the assumption of fast structure evolving, we analytically arrived in the macro-dynamics at the population level: an extended replicator equation which incorporates both the interactions of groups and the strategy evolution. Based on this extended equation, it is shown that the less the fragile cooperator-cooperator links (or the more the fragile cooperator-defector links), the easier the emergence of cooperation. This result is consistent with previous findings that assortments of cooperators are likely to invade a defector population and escape from the exploitation of defector mutants [24], [25], [47], which paves the way for both emergence and stabilization of cooperation. Interestingly, we have revealed that the dynamical interactions on interdependent populations can greatly affect the evolution of cooperation: cooperation can only emerge when intra-group bias is big enough. This is intuitive in the sense that intra-group bias can lead to cooperation [35], [36], which indicates that it might be more likely to establish neighbors within the same group. However, counterintuitive results also arise: it is unlikely that the more possible individuals establishing neighbors within the same group, the higher cooperation level is. In fact, too large intra-group switching bias inhibits cooperation [42], [43]. This reminds us with the migration effect: Neither too large nor too small mutation rate benefits cooperation. Thus, it would be beneficial for cooperators to move from time to time in order to hunt/establish a paradise to live, since this accidental moving can help the cooperators to escape from the nasty environment consisting of mainly defectors. Yet large migration rates make the population approximately well-mixed destroying the cooperation clustering, which deters cooperation. The switching rate of attaching neighbors outside the group can be viewed as the migration rate, thus leads to moderate switching rate maximizing cooperation. Our work shed light on how the dynamic of interdependent have an impact on the cooperation. This insight might also be constructive to other collective behaviors such as swarming and coordination and opinion formation. Works along those lines are in progress.

Acknowledgments We thanks B. Wu and J.B. Wang for help in preparing the manuscript, and the anonymous reviewers for their constructive comments to help improve this paper.

Author Contributions Conceived and designed the experiments: CBT ZW XL. Performed the experiments: CBT ZW. Analyzed the data: CBT. Wrote the paper: CBT ZW XL.