Statistical Equilibrium of Optimistic Beliefs

2025-02-13

AI Summary

This paper addresses the limitations of traditional equilibria in the mixed extension of finite games—specifically, their inadequate modeling of risk sensitivity and belief optimism. To this end, we propose the Statistical Optimistic Belief equilibrium (SE-OB), a novel equilibrium concept integrating optimistic Bayesian beliefs, stochastic payoff perturbations, and a risk-preference function. Methodologically, we pioneer the coupling of optimistic beliefs with multi-marginal optimal transport to formulate a risk-driven equilibrium paradigm, and design the first zeroth-order convergent algorithm for general structured Quantal Response Equilibria (QRE). We establish the existence of SE-OB and prove the convergence of our algorithm. Theoretical analysis reveals that heightened risk sensitivity improves equilibrium stability, while optimistic beliefs mitigate strategic over-aggressiveness. This work unifies and extends both Nash equilibrium and QRE, offering a new behavioral game-theoretic framework grounded in empirically plausible assumptions about bounded rationality, risk perception, and belief formation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We introduce the Statistical Equilibrium of Optimistic Beliefs (SE-OB) for the mixed extension of finite normal-form games, drawing insights from discrete choice theory. Departing from the conventional best responders of Nash equilibrium and the better responders of quantal response equilibrium, we reconceptualize player behavior as that of optimistic better responders. In this setting, the players assume that their expected payoffs are subject to random perturbations, and form optimistic beliefs by selecting the distribution of perturbations that maximizes their highest anticipated payoffs among belief sets. In doing so, SE-OB subsumes and extends the existing equilibria concepts. The player's view of the existence of perturbations in their payoffs reflects an inherent risk sensitivity, and thus, each player is equipped with a risk-preference function for every action. We demonstrate that every Nash equilibrium of a game, where expected payoffs are regularized with the risk-preference functions of the players, corresponds to an SE-OB in the original game, provided that the belief sets coincide with the feasible set of a multi-marginal optimal transport problem with marginals determined by risk-preference functions. Building on this connection, we propose an algorithm for repeated games among risk-sensitive players under optimistic beliefs when only zeroth-order feedback is available. We prove that, under appropriate conditions, the algorithm converges to an SE-OB. Our convergence analysis offers key insights into the strategic behaviors for equilibrium attainment: a player's risk sensitivity enhances equilibrium stability, while forming optimistic beliefs in the face of ambiguity helps to mitigate overly aggressive strategies over time. As a byproduct, our approach delivers the first generic convergent algorithm for general-form structural QRE beyond the classical logit-QRE.

Problem

• Introduces Statistical Equilibrium of Optimistic Beliefs (SE-OB).
• Extends Nash equilibrium with risk-preference functions.
• Proposes algorithm for games with zeroth-order feedback.

Innovation

• Optimistic better responders concept
• Risk-preference function integration
• Convergent algorithm for SE-OB