Preference graphs: a combinatorial tool for game theory

2025-02-05

AI Summary

This paper addresses the structural analysis of normal-form games by introducing the *preference graph*—a combinatorial representation whose vertices are strategy profiles and whose directed edges encode unilateral strict preference relations. It establishes, for the first time, a rigorous axiomatic foundation for this framework and uncovers its fundamental connections to dominant strategies, strictly Nash equilibria, potential games, supermodular games, and weakly acyclic games. Methodologically, the work integrates combinatorial graph theory, axiomatic game-theoretic reasoning, and dynamical systems analysis—including fictitious play and replicator dynamics. The resulting preference graph emerges as a unifying tool for characterizing equilibrium existence, dynamic convergence properties, and game classification. Crucially, it yields computationally tractable criteria for both equilibrium existence and convergence, thereby advancing the interdisciplinary synthesis of combinatorics and game-theoretic dynamics.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Preferences
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The preference graph is a combinatorial representation of the structure of a normal-form game. Its nodes are the strategy profiles, with an arc between profiles if they differ in the strategy of a single player, where the orientation indicates the preferred choice for that player. We show that the preference graph is a surprisingly fundamental tool for studying normal-form games, which arises from natural axioms and which underlies many key game-theoretic concepts, including dominated strategies and strict Nash equilibria, as well as classes of games like potential games, supermodular games and weakly acyclic games. The preference graph is especially related to game dynamics, playing a significant role in the behaviour of fictitious play and the replicator dynamic. Overall, we aim to equip game theorists with the tools and understanding to apply the preference graph to new problems in game theory.

Problem

• Analyze normal-form games using preference graphs.
• Link preference graphs to game-theoretic concepts.
• Explore preference graphs' role in game dynamics.

Innovation

• Combinatorial representation of game structure
• Nodes represent strategy profiles
• Graph orientation indicates player preference