Statistical mechanics of groups interacting in coevolutionary games
Systems in which the success of an individual depends on the actions of others are described by game theory. A dynamic learning mechanism has been introduced to this concept by evolutionary game theory, which is based on Darwinian selection. This learning mechanism increases the individual fitness and can be understood as an optimization procedure. As the fitness depends on interactions between individuals, the object that is optimized changes during the process in these systems. Darwinian selection is based on competition. However, cooperation between individuals is often observed in nature. A number of mechanisms have been proposed to explain the evolution of cooperation. The determination of the relevant mechanisms in biological and social systems is still subject of research. In this thesis, cooperation based on similarity is discussed. A minimal model for such a mechanism is derived. If agents only cooperate with others similar to themselves, segregation is observed in spatially extended systems. In social systems this is usually not desired. We discuss mechanisms to avoid this segregation. If the tolerance towards others is systematically increased, a cyclic dominance of the strategies emerges. This cyclic dominance leads to spiral waves in spatially extended systems. In well mixed systems described by the replicator dynamics, oscillations are observed. We derive a minimal model that allows analytical insight in the dynamics of such systems. Previous models in this direction have mainly been based on computer simulations. The cyclic nature of the abundances of the different strategies is similar to the one observed in cyclic games. In asymmetric games, these oscillations can be exploited by populations that adapt their selection rates to the current state of the system. In this way, an increased payoff is realized during the transient phase. This leads to a mechanism that increases the average payoff of one population under the influence of external noise. The mechanism works in cyclic games as well as in non-cyclic games. This is the first demonstration for an increased payoff in a game that arises from stochastic fluctuations.