The Study Of Evolutionary Games On Dynamic Networks

Cooperation is commonly observed throughout biological systems and humansocieties. But from a Darwinian viewpoint, cooperators are at a disadvantage in nat-ural selection, which help others at the cost of their own chance of survival and re-production. Understanding cooperative behavior is a fundamental problem in biology,sociology and economics. Evolutionary game theory provides a powerfully theoreticalframework to study the emergence of cooperation. Since Nowak and May introducedspatial dimensions, there has been much work in evolutionary games on networks. Onthe static network, it has been found that the network structure can greatly in?uencethe cooperation level. However, the network structure is often determined by the dy-namics of each node. To gain a further insight of the emergence of cooperation, it isnecessary to investigate evolutionary games on dynamic networks, which are drivenby the dynamics of each agent.As an important characteristic, migration of individual is often observed in realworld. But the effects of mobility is often neglected in evolutionary games. To under-stand the role of mobility in the emergence of cooperation, we investigate the evolutionof cooperation in dynamic networks, which result from the movement of agents, andstudy how to promote cooperation when plays move. We first investigate how mo-bility in?uence cooperation by separating the movement of agents from the evolutionof strategies. Then we study how to promote cooperation by reputation and adap-tive migration respectively. Our work may be helpful for the design of multi-agentcoordination system. The main contributions can be summarized as follows:We study the evolution of cooperation in the formation of ?ocks, when interactionneighborhoods are determined by sorting distance. When agents move, they try toalign their directions with that of neighbors, which are defined as k nearest agents to the focal one. And interactions among agents are modeled by the prisoner's dilemmagame. Under the deterministic update rule, we find that cooperators can live withsimple strategies. Compared with the static case, the cooperation level can even beenhanced for low velocities. But such effects rely on the size of neighborhood. Forsmall values of velocity and temptation, the cooperation level reaches the maximum atan intermediate size of neighborhood. Besides, the cooperation level is also in?uencedby the initial density of agents. We find that the cooperation level decreases with theinitial density.We study the evolution of cooperation in the formation of ?ocks, when interactionneighborhoods are determined by a given distance. We define neighbors as agents inthe spherical neighborhood of a given radius centered on the focal one. And we takethe prisoner's dilemma game and the snowdrift game as metaphors. Compared withfixed networks appearing at the initial moment and the equilibrium, updating strategiesand positions simultaneously is beneficial for cooperation in the formation of ?ocks.Whether the movement of players promotes cooperation is dependent on the valuesof payoff parameters and intermediate radii. Similar to the so called evolutionarycoherence resonance, the maximum level of cooperation can be reached at an optimalvalue of the velocity. Besides, the interaction radius R and the initial densityρhavesimilar in?uence on the evolution of cooperation. We find that intermediate values ofR orρare most favorable for cooperation. In the snowdrift game, the system can reachan absorbing state of full cooperation as the increase of R orρfor small cost-to-benefitratios.Cooperation is often inhibited by large velocities and strategy mutations, whenplays move randomly in the plane. To promote cooperation, we incorporate reputationmechanisms into the process of strategy update. We determine the player's reputationaccording to its strategy and reputation before. During the update of strategy, individ-uals with higher reputation have more chances to be imitated by others. We find thatwhen compared with the case that neighbors are chosen at random, the cooperator fre-quency can be enhanced even for high velocities and mutation noise. The cooperationlevel reaches the maximum when reputation in only determined by the latest strategyof players, while incorporating more previous records of reputation may inhibit co-operation. But reputation mechanisms only work for intermediate radii. In addition, reputation mechanisms make the state of full defection unstable to mutation noise, andafter a long time, one can observe the outbreak of prevailing cooperation.To characterize contingent responses of individuals to the environment, we com-bine the evolution of strategies with the migration of individuals, proposing an adap-tive migration rule based on average payoffs. According to average payoffs of theirneighbors, each individual can judge whether its current site good or bad. If there aresome empty sites in the migration neighborhood, which have higher payoffs than thatof the current location, the individual will migrate to the site with the highest payoff.Compared with the static case, adaptive migration can effectively promote cooper-ation, making cooperation stable in the presence of strategy mutations and randomrelocations. But migration based on average payoffs can only promote cooperation forintermediate densities. For a fixed noise level, there is an optimal value of density,which induces the maximum cooperation level. And for small densities, enlarging thesize of migration neighborhood will promote cooperation. Besides, migration basedon average payoffs makes the state of full defection unstable to strategy mutations.resulting in the outbreak of cooperation. The emergence of cooperation is in?uencedby the level of mutation noise and the density of agents.