Ising game on graphs

Static and dynamic equilibria in noisy binary choice (Ising) games on complete and random graphs in the annealed approximation are analysed. Two versions, an Ising game with interaction term defined in accordance with the Ising model in statistical physics and a reduced Ising game with a customary definition of interaction term in game theory on graphs, are considered. A detailed analysis of hysteresis phenomenon shaping the pattern of static equilibria based on consideration of elasticity with respect to external influence is conducted. Fokker-Planck equations describing dynamic versions of the games under consideration are written and their asymptotic stationary solutions derived. It is shown that domains of parameters corresponding to the maxima of these probability distributions are identical with the corresponding hysteresis ranges for static equilibria. Same result holds for domains defining local stability of solutions of the evolution equations for the moments. In all the cases considered it is shown that the results for the reduced Ising game coincide with those obtained for the Ising game on complete graphs. It is shown that for s special case of logistic noise the results obtained for static equilibria for the Ising game reproduce those in the Ising model on graphs in statistical physics.