Derivation of potential for group interactions in evolutionary games

For spatial evolutionary games, a potential can be derived if the interaction is described by two-player, two-strategy, symmetric games that yield thermodynamical behavior for a suitable dynamical rule. Here, we show the existence of a potential for two-strategy group interactions in which the players with identical strategies receive equal payoffs. This type of group interaction includes some extended versions of the public goods game. Some peculiar consequences of these group interactions are illustrated in a simple model in which the players are located at the sites of a square lattice with periodic boundary conditions. This five-player group interaction supports the formation of striplike arrangements of the two strategies. Monte Carlo simulations, however, indicate the existence of many other ground states and the relevance of frustration in shaping the macroscopic behavior when the effect of noise is taken into consideration.