Regret, equilibrium, and learning in games: A guided tour

2026-08-10

AI Summary

This study addresses how intelligent agents can achieve rational decision-making in unknown, non-stationary, and even adversarial environments, and investigates the emergence and stability of equilibria in multi-agent interactions. The work proposes a unified analytical framework that integrates regularized learning strategies, adversarial multi-armed bandit models, and fictitious play dynamics, accommodating both oracle information and bandit feedback settings. Key contributions include deriving optimal regret bounds for single-agent learning in adversarial environments, establishing ergodic convergence to Nash equilibria in zero-sum games, and formulating a folk-theorem-like correspondence between attractors of regularized learning dynamics and Nash equilibria, thereby revealing a fundamental alignment between strategic stability and adaptive learning behavior.

Abstract

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.

Problem

• regret
• equilibrium
• learning in games
• multi-agent systems
• adversarial bandits

Innovation

• regularized learning
• regret bounds
• Nash equilibrium
• multi-agent learning
• bandit feedback