The $k$-flip Ising game

TL;DR

This paper analyzes a stochastic binary choice game on complete graphs, deriving transition probabilities, moments of state distributions, and examining how the number of players allowed to change strategies affects metastable state decay.

Contribution

It provides analytical expressions for the transition matrix and moments of the distribution in the $k$-flip Ising game, highlighting the impact of $k$ on metastable state decay.

Findings

Transition matrix and moments derived analytically.

Dependence of metastable decay on $k$ with minima at certain $k^*$.

Competition between diffusion and restoring forces influences dynamics.

Abstract

A partially parallel dynamical noisy binary choice (Ising) game in discrete time of $N$ players on complete graphs with $k$ players having a possibility of changing their strategies at each time moment called $k$-flip Ising game is considered. Analytical calculation of the transition matrix of game as well as the first two moments of the distribution of $\varphi=N^+/N$, where $N^+$ is a number of players adhering to one of the two strategies, is presented. 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 $k^*$ is attributed to a competition between $k$-dependent diffusion and restoring forces.