A Category Theoretic Approach to Approximate Game Theory

2025-09-25

AI Summary

In multi-agent systems, exact optimal solutions are often intractable due to input uncertainty or computational complexity. Method: This work introduces a novel approximate game-theoretic framework grounded in category theory—the first systematic application of categorical methods to approximate game modeling. It integrates selection functions and open games to construct a formal algebraic structure for approximate equilibria, enabling compositional construction and analysis of approximate solutions. Contribution/Results: (1) We establish a categorical semantics for approximate equilibria, uncovering their intrinsic connection to the algebraic structure of selection functions; (2) We provide a compositional and computationally tractable paradigm for approximate decision-making, significantly enhancing the computability and scalability of multi-agent coordination under uncertainty. The framework supports modular reasoning about approximation quality and facilitates rigorous composition of decentralized strategies, thereby advancing the theoretical foundations of robust, scalable multi-agent learning and control.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Cognitive Modeling & Cognitive Systems: Agent Architectures

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

This paper uses category theory to develop an entirely new approach to approximate game theory. Game theory is the study of how different agents within a multi-agent system take decisions. At its core, game theory asks what an optimal decision is in a given scenario. Thus approximate game theory asks what is an approximately optimal decision in a given scenario. This is important in practice as -- just like in much of computing -- exact answers maybe too difficult to compute or even impossible to compute given inherent uncertainty in input. We consider first "Selection Functions" which are functions and develop a simple yet robust model of approximate equilibria. We develop the algebraic properties of approximation wrt selection functions and also relate approximation to the compositional structure of selection functions. We then repeat this process successfully for Open Games -- a more advanced model of game theory.

Problem

• Developing a category theory approach to approximate game theory
• Modeling approximate equilibria using selection functions and open games
• Studying algebraic and compositional properties of approximation methods

Innovation

• Uses category theory for approximate game theory
• Develops selection functions model for equilibria
• Extends approach to Open Games compositionally