Asymmetric games on networks: Towards an Ising-model representation

• Published in: Physica A 2022-05, Vol.593, p.126972, Article 126972
• Main Authors: Correia, A.D., Leestmaker, L.L., Stoof, H.T.C., Broere, J.J.
• Format: Article
• Language: English
• Subjects: Asymmetric games; Correlated games; Games on networks; Ising model
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2022.126972

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. First, we show that these correlations emerge in simulations and can have an Ising-like form. Then, 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-theoretic 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 networks. 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. •Correlations emerge from update strategy in networks of asymmetric games.•A simple Ising model explains the behavior partially, but not completely.•We use an extended Ising model that translates the emergent correlated games.•Players can change the game from an out-of-equilibrium to an equilibrium correlation.•We find convergent behavior for different types of small networks in equilibrium.