The $k$-flip Ising game

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 $φ=N^+/N$, 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.

💡 Research Summary

The paper introduces a “k‑flip Ising game,” a discrete‑time noisy binary‑choice model in which N agents occupy the vertices of a complete graph and, at each time step, a randomly selected subset of k agents (1 ≤ k ≤ N) simultaneously reconsider their strategies. Each agent i’s utility is given by

U_i(s|t) = H s + (J/N) ∑_{j≠i} s_j + ε_s(t),

where H is a uniform external field, J > 0 quantifies peer influence, and ε_s are independent Gumbel‑distributed random variables. The Gumbel noise yields a Boltzmann choice probability

p(s|t) = e^{β