Asymmetric games on networks: towards an Ising-model representation

TL;DR

This paper models the Battle of the Sexes game on small networks using a generalized Ising model, revealing how network structure influences equilibrium correlations and suggesting potential for analytical solutions.

Contribution

It introduces a novel Ising-model framework to analyze asymmetric games on networks, extending from two to three players and exploring network effects on equilibria.

Findings

Correlations between players depend on initial conditions and network structure.

Local network features significantly influence equilibrium states.

Certain parameters of equilibrium are network-independent.

Abstract

We here study the Battle of the Sexes game, a textbook case of asymmetric games, on small networks. Due to the conflicting preferences of the players, analytical approaches are scarce and most often update strategies are employed in numerical simulations of repeated games on networks until convergence is reached. 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 the players can bring the network into a steady configuration, starting from an out-of-equilibrium one. We obtain these configurations using game-theoretical tools, and describe the results using Ising parameters. We exhaust the two-player case, giving a detailed account of all the equilibrium possibilities. Going to three players, we generalize the Ising model and compare the equilibrium solutions of three representative types of network. We find that players that are not directly linked retain a degree of correlation that is proportional to their initial correlation. We also find that the local network structure is the most relevant for small values of the magnetic field and the interaction strength of the Ising model. Finally, we conclude that certain parameters of the equilibrium states are network independent, which opens up the possibility of an analytical description of asymmetric games played on networks.