Likelihood Equilibria in the Ising Game

TL;DR

The paper addresses static equilibria in noisy binary choice Ising games on networks by identifying configurations that maximize the likelihood ${\cal P}(s_1,\dots,s_N|\Theta)$. It develops a general likelihood framework for agent choices with noisy utilities parameterized by $H$, $J$, and a noise function $g$, derives the corresponding equilibrium conditions, and shows that these likelihood equilibria are equivalent to Bayes-Nash quantal response equilibria when agent expectations are consistent; it also demonstrates that the same equilibria can be obtained from a partition function analysis. The analysis is carried out for both complete graphs and annealed random graphs via the configuration model, yielding self-consistency equations for the order parameters $m$ and $m_k$ that reproduce known QRE results in the consistent-expectations limit. Overall, the work links probabilistic game-theoretic equilibria with statistical-mechanical descriptions of multiagent systems, offering a unified view of equilibrium structure in noisy Ising-like interactions with potential applications to social and economic networks.

Abstract

A description of static equilibria in the noisy binary choice (Ising) game on complete and random graphs resulting from maximisation of the likelihood of system configurations is presented. An equivalence of such likelihood equilibria to the competitive Bayes-Nash quantal response expectation equilibria in the special case of consistent agents expectations is established. It is shown that the same likelihood equilibria are obtained by considering the system's partition function.