The $k$-flip Ising game

TL;DR

This work analyzes how partially parallel, k-flip Ising dynamics on a complete graph governs transition times between metastable and stable states. By deriving an exact transition matrix Ω^{(k)} and computing the first two moments of the φ distribution, it uncovers a nontrivial, k-dependent competition between diffusion and restoring forces that can produce a minimum in mean transition time. The authors develop a Markov-chain framework to obtain mean and second moments of first hitting times, examine equilibrium distributions via an effective potential, and validate predictions with simulations showing robust, heavy-tailed transition times. The results illuminate how collective, multi-agent updates influence metastable decay rates in noisy binary-choice systems, with implications for understanding rapid or slow transitions under partial parallelism.

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.

Figures (15)

• Figure 1: Plots of the potential $V_{\varphi} (k, N, \beta, H) = - \ln \pi^{(k)}_\varphi (k, N, \beta, H)$ as a function of $\varphi$ for $\beta = 1.9, H = 0.8H^{*}, N = 150$ and $k = 1$ (solid line), $k = 10$ (dashed line), $k = 50$ (dotted line), $k = 100$ (dash-dotted line), $k = 150$ (long-dashed line). The right plots correspond to an enlarged scale displaying the metastable state.
• Figure 2: The structure of the effective potential $V_{\rm eff} (\varphi \vert \beta,H)$ (left) and the structure of solutions (\ref{['CWe']}) (right) for the low-temperature phase. The red solid arrow shows the transition from the metastable state $\varphi_1$ at the lower branch $\varphi^-(H)$ to the stable state $\varphi_2$ at the upper branch $\varphi^+(H)$. The blue dashed arrow shows the transition from unstable state $\varphi = 0.5$ to the stable state $\varphi_2$. The parameters are $\beta J = 1.9, \gamma=0.8$.
• Figure 3: Plots of mean first-hitting times as a function of $k$ for $\beta = 1.9, \gamma = 0.8$ and $N = 100$ (left), $N = 150$ (right). The red solid (blue dashed) line corresponds to the red solid (blue dashed) arrow in Fig. \ref{['pot_example']}.
• Figure 4: Plots of $r_\tau (\beta, \gamma, k, N)$ as a function of $k$ for $N =40$ (solid line), $N =80$ (dashed line), $N = 120$ (dotted line), $N = 150$ (dash-dotted line) with $\gamma = 0.8$ and $\beta = 1.9$ (left), $\beta = 2.5$ (right).
• Figure 5: Plots of $r_\tau (\beta, \gamma, \rho, N)$ as a function of $\rho$ with $N = 150$ (1) for $\beta = 1.5$ (solid line), $\beta = 1.7$ (dashed line), $\beta = 2$ (dotted line), $\beta = 2.5$ (dash-dotted line), $\beta = 3$ (long-dashed line) and $\gamma = 0.8$ (left figure); (2) for $\gamma = 0.75$ (solid line), $\gamma = 0.8$ (dashed line), $\gamma = 0.85$ (dotted line), $\gamma = 0.9$ (dash-dotted line), $\gamma = 0.95$ (long-dashed line) and $\beta = 2$ (right figure).