Ising Game on Graphs

Static equilibria and dynamic evolution in noisy binary choice (Ising) games on graphs are considered. Equations defining static quantal response equilibria (QRE) for Ising games on graphs with arbitrary topology and noise distribution are written. It is shown that in the special cases of complete graph and arbitrary noise distribution, and circular and star topology and logistic noise distribution the resulting equations can be cast in the form coinciding with that derived in the earlier literature. Explicit equations for non-directed random graphs in the annealed approximation are derived. It is shown that the resulting effect on the phase transition is the same as found in the literature on phase transition in the mean-field versions of the Ising model on graphs . Evolutionary Ising game having the earlier described QRE as its stationary equilibria in the mean field approximation is constructed using the formalism of master equation for the complete, star, circular and random annealed graphs and the formalism of population games for random annealed graphs.