We here study the Battle of the Sexes game on small networks, extending the response strategy analysis to asymmetric gamesand three players. We model the different steady states, that are obtained in numerical simulations from the various update strategies, as different response strategy equilibria. These are then mapped to a generalized Ising model , describing the correlations between the players ' outcomes in a way that is agnostic to the existence of a link, but that coincides with the correlation device if there is one. Going to three players, we look at and compare the equilibrium solutions for three representative types of networks . We find that players that are not directly connected 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. This research paves the way to a statistical physics description of games on networks that is scale sensitive and has the potential to give an analytical description of both long- and short-range behavior observed in numerical simulations .
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 .
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on small networks, extending the response strategy analysis to asymmetric gamesand three players. We model the different steady states, that are obtained
, 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
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from the various update strategies, as different response strategy equilibria. These are then mapped to
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
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, describing the correlations between the players ' outcomes in a way that is agnostic to the existence of a link, but that coincides with the correlation device if there is one.
. 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.
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This research paves the way to a statistical physics description of games on networks that is scale sensitive and has the potential to give
Finally, we conclude that certain parameters of the equilibrium states are network independent, which opens up the possibility of
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both long- and short-range behavior observed in numerical simulations
asymmetric games played on networks