Evolution of cooperation on dynamical graphs

There are two key characteristic of animal and human societies: (1) degree heterogeneity, meaning that not all individual have the same number of associates; and (2) the interaction topology is not static, i.e. either individuals interact with different set of individuals at different times of their life, or at least they have different associations than their parents. Earlier works have shown that population structure is one of the mechanisms promoting cooperation. However, most studies had assumed that the interaction network can be described by a regular graph (homogeneous degree distribution). Recently there are an increasing number of studies employing degree heterogeneous graphs to model interaction topology. But mostly the interaction topology was assumed to be static. Here we investigate the fixation probability of the cooperator strategy in the prisoner’s dilemma, when interaction network is a random regular graph, a random graph or a scale-free graph and the interaction network is allowed to change. We show that the fixation probability of the cooperator strategy is lower when the interaction topology is described by a dynamical graph compared to a static graph. Even a limited network dynamics significantly decreases the fixation probability of cooperation, an effect that is mitigated stronger by degree heterogeneous networks topology than by a degree homogeneous one. We have also found that from the considered graph topologies the decrease of fixation probabilities due to graph dynamics is the lowest on scale-free graphs

Conformity Hinders the Evolution of Cooperation on Scale-Free Networks

We study the effects of conformity, the tendency of humans to imitate locally common behaviors, in the evolution of cooperation when individuals occupy the vertices of a graph and engage in the one-shot Prisoner's Dilemma or the Snowdrift game with their neighbors. Two different graphs are studied: rings (one-dimensional lattices with cyclic boundary conditions) and scale-free networks of the Barabasi-Albert type. The proposed evolutionary-graph model is studied both by means of Monte Carlo simulations and an extended pair-approximation technique. We find improved levels of cooperation when evolution is carried on rings and individuals imitate according to both the traditional pay-off bias and a conformist bias. More important, we show that scale-free networks are no longer powerful amplifiers of cooperation when fair amounts of conformity are introduced in the imitation rules of the players. Such weakening of the cooperation-promoting abilities of scale-free networks is the result of a less biased flow of information in scale-free topologies, making hubs more susceptible of being influenced by less-connected neighbors.Comment: 14 pages, 11 figure

Mutual Trust and Cooperation in the Evolutionary Hawks-Doves Game

Using a new dynamical network model of society in which pairwise interactions are weighted according to mutual satisfaction, we show that cooperation is the norm in the Hawks-Doves game when individuals are allowed to break ties with undesirable neighbors and to make new acquaintances in their extended neighborhood. Moreover, cooperation is robust with respect to rather strong strategy perturbations. We also discuss the empirical structure of the emerging networks, and the reasons that allow cooperators to thrive in the population. Given the metaphorical importance of this game for social interaction, this is an encouraging positive result as standard theory for large mixing populations prescribes that a certain fraction of defectors must always exist at equilibrium.Comment: 23 pages 12 images, to appea

Different reactions to adverse neighborhoods in games of cooperation

In social dilemmas, cooperation among randomly interacting individuals is often difficult to achieve. The situation changes if interactions take place in a network where the network structure jointly evolves with the behavioral strategies of the interacting individuals. In particular, cooperation can be stabilized if individuals tend to cut interaction links when facing adverse neighborhoods. Here we consider two different types of reaction to adverse neighborhoods, and all possible mixtures between these reactions. When faced with a gloomy outlook, players can either choose to cut and rewire some of their links to other individuals, or they can migrate to another location and establish new links in the new local neighborhood. We find that in general local rewiring is more favorable for the evolution of cooperation than emigration from adverse neighborhoods. Rewiring helps to maintain the diversity in the degree distribution of players and favors the spontaneous emergence of cooperative clusters. Both properties are known to favor the evolution of cooperation on networks. Interestingly, a mixture of migration and rewiring is even more favorable for the evolution of cooperation than rewiring on its own. While most models only consider a single type of reaction to adverse neighborhoods, the coexistence of several such reactions may actually be an optimal setting for the evolution of cooperation.Comment: 12 pages, 5 figures; accepted for publication in PLoS ON

Selection of noise level in strategy adoption for spatial social dilemmas

We studied spatial Prisoner's Dilemma and Stag Hunt games where both the strategy distribution and the players' individual noise level could evolve to reach higher individual payoff. Players are located on the sites of different two-dimensional lattices and gain their payoff from games with their neighbors by choosing unconditional cooperation or defection. The way of strategy adoption can be characterized by a single $K$ (temperature-like) parameter describing how strongly adoptions depend on the payoff-difference. If we start the system from a random strategy distribution with many different player specific $K$ parameters, the simultaneous evolution of strategies and $K$ parameters drives the system to a final stationary state where only one $K$ value remains. In the coexistence phase of cooperator and defector strategies the surviving $K$ parameter is in good agreement with the noise level that ensures the highest cooperation level if uniform $K$ is supposed for all players. In this paper we give a thorough overview about the properties of this evolutionary process.Comment: 10 two-column pages, 10 figures; accepted for publication in Physical Review

Coevolutionary games - a mini review

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.Comment: 24 two-column pages, 10 figures; accepted for publication in BioSystem