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We here study the Battle of the Sexes game, a textbook case of asymmetricgames, on small networks. Due to the conflicting preferences of the players,analytical approaches are scarce and most often update strategies are employedin numerical simulations of repeated games on networks until convergence isreached. As a result, correlations between the choices of the players emerge.Our approach is to study these correlations with a generalized Ising model.Using the response strategy framework, we describe how the actions of theplayers can bring the network into a steady configuration, starting from anout-of-equilibrium one. We obtain these configurations using game-theoreticaltools, and describe the results using Ising parameters. We exhaust thetwo-player case, giving a detailed account of all the equilibriumpossibilities. Going to three players, we generalize the Ising model andcompare the equilibrium solutions of three representative types of network. Wefind that players that are not directly linked retain a degree of correlationthat is proportional to their initial correlation. We also find that the localnetwork structure is the most relevant for small values of the magnetic fieldand the interaction strength of the Ising model. Finally, we conclude thatcertain parameters of the equilibrium states are network independent, whichopens up the possibility of an analytical description of asymmetric gamesplayed on networks.

Asymmetric games on networks: towards an Ising-model representation