The $k$-flip Ising game

Abstract

The game-theoretic view on the classical Ising model - the noisy binary choice Ising game, - is a well-known conceptual framework that, on the one hand, highlights similar patterns between social or artificial agent-based systems and ensembles of interacting physical spins and, on the other hand, helps to stress important differences between the ones. The study considers one of such differences - the possibility of simultaneous decision making of several agents that is typical for game theoretic interaction. To do this we analyse a partially parallel discrete time dynamics of an Ising-type system of $N$ interacting binary units on a complete graph in which at each time steps $k$ arbitrarily chosen units can change their states is analysed. The particular problem under study is a $k$ - dependence of the decay of a metastable configuration into a stable one. The analysis is based on an explicit analytical calculation, for arbitrary noise, of the transition matrix characterising the $k$ - flip evolution as well as the first two moments of the distribution of the fraction of players choosing one of the two available strategies. First two moments of the first hitting time distribution for sample trajectories corresponding to transition from a metastable and unstable states to a stable one are considered. A nontrivial dependence of these moments on $k$ for the decay of a metastable state is discussed. A presence of the minima at certain $k_{\rm min}$ is attributed to a competition between $k$-dependent diffusion and restoring forces.