The Learning Approach to Games
Melih İşeri, Erhan Bayraktar
TL;DR
Traditional game-theoretic analysis often treats agents as black-box decision-makers operating in external environments, neglecting the internal learning processes that shape behavior. The authors propose a player-centric framework that defines a player as $({\cal O}, {\mathfrak L}_{\varphi}, {\Upsilon})$ with consistency constraints, a recurrence-based notion of behavior, and a layered set of estimates, enabling dynamic, learning-driven strategies to influence decisions; they introduce an $(\varepsilon,r,\delta)$-uncertain equilibrium and connect it to correlated equilibrium and dynamic programming. The paper develops discrete and mean-field formulations, provides learning algorithms for estimating costs and opponent behavior, and illustrates the approach with a simple two-player toy and a CartPole RL example, highlighting how internal estimates drive evolving strategies beyond static equilibria. By explicitly incorporating internal representations, predictions, and learning dynamics, the framework offers a principled lens to analyze complex multi-agent systems in AI, economics, and policy design, with potential impact on how we design and regulate learning-enabled agents.
Abstract
This work introduces a unified framework for analyzing games in greater depth. In the existing literature, players' strategies are typically assigned scalar values, and equilibrium concepts are used to identify compatible choices. However, this approach neglects the internal structure of players, thereby failing to accurately model observed behaviors. To address this limitation, we propose an abstract definition of a player, consistent with constructions in reinforcement learning. Instead of defining games as external settings, our framework defines them in terms of the players themselves. This offers a language that enables a deeper connection between games and learning. To illustrate the need for this generality, we study a simple two-player game and show that even in basic settings, a sophisticated player may adopt dynamic strategies that cannot be captured by simpler models or compatibility analysis. For a general definition of a player, we discuss natural conditions on its components and define competition through their behavior. In the discrete setting, we consider players whose estimates largely follow the standard framework from the literature. We explore connections to correlated equilibrium and highlight that dynamic programming naturally applies to all estimates. In the mean-field setting, we exploit symmetry to construct explicit examples of equilibria. Finally, we conclude by examining relations to reinforcement learning.