Totally mixed conditional independence equilibria of generic games

TL;DR

This paper explores the algebraic and geometric structure of conditional independence equilibria in generic strategic games, revealing their manifold structure and algebraic properties, and extending previous binary game analyses.

Contribution

It generalizes the study of CI equilibria to arbitrary game formats, introduces Nash CI varieties, and characterizes their algebraic properties and conditions for equilibria existence.

Findings

Spohn CI variety is either empty or has predictable codimension.

Totally mixed CI equilibria form a smooth manifold in generic games.

Nash CI varieties are irreducible with explicitly described algebraic properties.

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.