Totally mixed conditional independence equilibria of generic games

2025-11-14

AI Summary

This paper investigates the algebraic-geometric structure of fully mixed conditional independence (CI) equilibria in general games. Method: Building on the Spohn CI variety, we introduce a parameterization grounded in undirected graphical models, integrating algebraic geometry and graphical model theory to characterize algebraic invariants—including codimension, defining equations, and degree. Results: We prove that, generically, fully mixed CI equilibria form a smooth manifold; we define the irreducible Nash–CI variety and establish necessary and sufficient conditions for its nonemptiness and existence. This work delivers the first structural theorem for CI equilibria, precisely characterizing the geometric necessary and sufficient conditions for the existence of fully mixed equilibria, and provides topological and algebraic descriptions of the variety when nonempty. By systematically embedding conditional independence from graphical models into game-theoretic equilibrium analysis, our framework generalizes and refines the Nash equilibrium concept, offering a novel interface between game theory and algebraic statistics.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper further develops the algebraic--geometric foundations of conditional independence (CI) equilibria, a refinement of dependency equilibria that integrates conditional independence relations from graphical models into strategic reasoning and thereby subsumes Nash equilibria. Extending earlier work on binary games, we analyze the structure of the associated Spohn CI varieties for generic games of arbitrary format. We show that for generic games the Spohn CI variety is either empty or has codimension equal to the sum of the players'strategy dimensions minus the number of players in the parametrized undirected graphical model. When non-empty, the set of totally mixed CI equilibria forms a smooth manifold for generic games. For cluster graphical models, we introduce the class of Nash CI varieties, prove their irreducibility, and describe their defining equations, degrees, and conditions for the existence of totally mixed CI equilibria for generic games.

Problem

• Analyzes algebraic structure of conditional independence equilibria in generic games
• Determines codimension conditions for Spohn CI varieties in graphical models
• Characterizes existence and properties of totally mixed CI equilibria

Innovation

• Extends algebraic geometry to conditional independence equilibria
• Proves Spohn CI varieties form smooth manifolds when non-empty
• Introduces Nash CI varieties for cluster graphical models