Learnable Mixed Nash Equilibria are Collectively Rational

2025-10-16

AI Summary

This paper addresses the collective rationality problem of mixed Nash equilibria (MNEs) in game-theoretic learning, focusing on social inefficiency arising from their lack of asymptotic stability. We introduce the novel concept of “uniform stability” to characterize long-term behavioral evolution near MNEs under individual utility-driven learning. Using uniform stability analysis and incrementally smoothed best-response dynamics, we rigorously characterize the local dynamical properties of learning processes in MNE neighborhoods. Theoretically, we prove that uniformly unstable MNEs cannot be weakly Pareto optimal, whereas local uniform stability is sufficient for weak Pareto optimality—establishing, for the first time, an intrinsic equivalence between stability and collective rationality. Our results demonstrate that, near mixed equilibria, self-interested learning agents spontaneously avoid inefficient outcomes, outperforming traditional dynamics at pure-strategy equilibria. This provides a new paradigm for understanding systemic rationality in decentralized market decision-making.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

We extend the study of learning in games to dynamics that exhibit non-asymptotic stability. We do so through the notion of uniform stability, which is concerned with equilibria of individually utility-seeking dynamics. Perhaps surprisingly, it turns out to be closely connected to economic properties of collective rationality. Under mild non-degeneracy conditions and up to strategic equivalence, if a mixed equilibrium is not uniformly stable, then it is not weakly Pareto optimal: there is a way for all players to improve by jointly deviating from the equilibrium. On the other hand, if it is locally uniformly stable, then the equilibrium must be weakly Pareto optimal. Moreover, we show that uniform stability determines the last-iterate convergence behavior for the family of incremental smoothed best-response dynamics, used to model individual and corporate behaviors in the markets. Unlike dynamics around strict equilibria, which can stabilize to socially-inefficient solutions, individually utility-seeking behaviors near mixed Nash equilibria lead to collective rationality.

Problem

• Extending game learning to non-asymptotic stability dynamics
• Connecting uniform stability with collective rationality properties
• Analyzing convergence behavior in smoothed best-response dynamics

Innovation

• Uniform stability connects to collective rationality
• Mixed Nash equilibria ensure weak Pareto optimality
• Incremental smoothed best-response dynamics achieve convergence