Betting on Equilibrium: Monitoring Strategic Behavior in Multi-Agent Systems

TL;DR

This paper presents a real-time, statistically sound method for monitoring whether agents in multi-agent systems behave according to equilibrium, using a betting-based sequential testing framework that adapts to various equilibrium concepts.

Contribution

It introduces a novel sequential testing approach based on e-values for online detection of deviations from equilibrium in multi-agent systems, applicable to various equilibrium types and stochastic games.

Findings

Provides finite-time guarantees for equilibrium monitoring

Controls false discovery rate in large games

Extends to stochastic game settings

Abstract

In many multi-agent systems, agents interact repeatedly and are expected to settle into equilibrium behavior over time. Yet in practice, behavior often drifts, and detecting such deviations in real time remains an open challenge. We introduce a sequential testing framework that monitors whether observed play in repeated games is consistent with equilibrium, without assuming a fixed sample size. Our approach builds on the e-value framework for safe anytime-valid inference: by "betting" against equilibrium, we construct a test supermartingale that accumulates evidence whenever observed payoffs systematically violate equilibrium conditions. This yields a statistically sound, interpretable measure of departure from equilibrium that can be monitored online. We also leverage Benjamini-Hochberg-type procedures to increase detection power in large games while rigorously controlling the false discovery rate. Our framework unifies the treatment of Nash, correlated, and coarse correlated equilibria, offering finite-time guarantees and a detailed analysis of detection times. Moreover, we extend our method to stochastic games, broadening its applicability beyond repeated-play settings.