Evolutionary games on minimally structured populations

Conformity effect on the evolution of cooperation in signed networks

Human behaviors are often subject to conformity, but little research attention has been paid to social dilemmas in which players are assumed to only pursue the maximization of their payoffs. The present study proposed a generalized prisoner dilemma model in a signed network considering conformity. Simulation shows that conformity helps promote the imitation of cooperative behavior when positive edges dominate the network, while negative edges may impede conformity from fostering cooperation. The logic of homophily and xenophobia allows for the coexistence of cooperators and defectors and guides the evolution toward the equality of the two strategies. We also find that cooperation prevails when individuals have a higher probability of adjusting their relation signs, but conformity may mediate the effect of network adaptation. From a population-wide view, network adaptation and conformity are capable of forming the structures of attractors or repellers.

The evolution of cooperation in signed networks under the impact of structural balance

Structural balance plays an important role in the dynamics of signed networks. Based on structural balance, we generalize the evolution of cooperation in signed networks. Here we develop a new simulation model to study the impact of structural balance on the evolution of cooperation in signed networks. The simulation shows that cooperation prevails when an individual has a higher probability of adjusting the signs of its relations. We also find that structural balance forces the coexistence of cooperators and defectors, while the initial attributes of networks have little impact on the evolution of cooperation in the presence of structural balance, although they have a strong effect on the evolution of structural balance.

Network reciprocity created in prisoner's dilemma games by coupling two mechanisms

We found that a nontrivial enhancement of network reciprocity for 2 × 2 prisoner's dilemma games can be achieved by coupling two mechanisms. The first mechanism presumes a larger strategy update neighborhood than the conventional first neighborhood on the underlying network. The second is the strategy-shifting rule. At the initial time step, the averaged cooperation extent is assumed to be 0.5. In the case of strategy shifting, an agent adopts a continuous strategy definition during the initial period of a simulation episode (when the global cooperation fraction decreases from its initial value of 0.5; that is, the enduring period). The agent then switches to a discrete strategy definition in the time period afterwards (when the global cooperation fraction begins to increase again; that is, the expanding period). We explored why this particular enhancement comes about, and to summarize, the continuous strategy during the initial period relaxes the conditions for the survival of relatively cooperative clusters, and the large strategy adaptation neighborhood allows those cooperative clusters to expand easily.

Cyclic dominance in evolutionary games: a review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

Evolutionary prisoner's dilemma game on graphs and social networks with external constraint

A game-theoretical model is constructed to capture the effect of external constraint on the evolution of cooperation. External constraint describes the case where individuals are forced to cooperate with a given probability in a society. Mathematical analyses are conducted via pair approximation and diffusion approximation methods. The results show that the condition for cooperation to be favored on graphs with constraint is b¯/c¯>k/A¯ (A¯=1+kp/(1-p)), where b¯ and c¯ represent the altruistic benefit and cost, respectively, k is the average degree of the graph and p is the probability of compulsory cooperation by external enforcement. Moreover, numerical simulations are also performed on a repeated game with three strategies, always defect (ALLD), tit-for-tat (TFT) and always cooperate (ALLC). These simulations demonstrate that a slight enforcement of ALLC can only promote cooperation when there is weak network reciprocity, while the catalyst effect of TFT on cooperation is verified. In addition, the interesting phenomenon of stable coexistence of the three strategies can be observed. Our model can represent evolutionary dynamics on a network structure which is disturbed by a specified external constraint.

Characterization of spiraling patterns in spatial rock-paper-scissors games

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

Dynamically generated cyclic dominance in spatial prisoner's dilemma games

We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a steplike function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.

Phase diagrams for three-strategy evolutionary prisoner's dilemma games on regular graphs

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.

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.