Dependency equilibria: Boundary cases and their real algebraic geometry

2024-05-29

arXiv.org

AI Summary

Dependency equilibria—generalizations of Nash equilibria proposed by Spohn—lack a rigorous algebraic-geometric foundation, limiting structural analysis and computational tractability. Method: Within real algebraic geometry, we introduce two equivalent, algebraically amenable reformulations of Spohn’s original definition, yielding a unified algebraic model—the Spohn variety—that captures both pure and mixed dependency equilibria. Contribution/Results: We prove that all Nash equilibria lie on the Spohn variety, and its real points are Zariski-dense in generic games. For (2×2) bimatrix games, we provide a complete geometric classification of dependency equilibria and derive a sufficient condition for the existence of pure dependency equilibria. This work establishes the first complete algebraic-geometric characterization of dependency equilibria, rigorously demonstrating their strict containment of Nash equilibria and opening a new paradigm for interdisciplinary research at the interface of game theory and algebraic geometry.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for $(2 imes2)$-games.

Problem

• Study dependency equilibria using real algebraic geometry
• Propose new definitions for comprehensive algebro-geometric analysis
• Analyze geometric structure of equilibria in 2x2 games

Innovation

• Alternative definitions for dependency equilibria study
• Sufficient condition for pure dependency equilibrium
• Geometric analysis of dependency equilibria structure