Totally mixed conditional independence equilibria of generic games
Matthieu Bouyer, Irem Portakal, Javier Sendra-Arranz
TL;DR
This work develops an algebro-geometric theory of CI equilibria by integrating graphical-model conditional independencies into finite games and analyzing the resulting Spohn CI varieties. It establishes that, for generic games, the Spohn CI variety within the parametrized graphical model either vanishes or attains codimension $d_1+\cdots+d_n-n$, with totally mixed CI equilibria forming a smooth manifold when nonempty. The paper then specializes to cluster graphs, proving irreducibility and providing defining equations and degree formulas (via the Chow ring and Porteous’ formula) for Nash CI varieties, along with a precise nonemptiness criterion. Together, these results bridge algebraic statistics and game theory, showing CI equilibria can exist and be structurally richer than Nash equilibria, and enabling explicit computations of dimensions, degrees, and existence conditions for nonbinary games. The methods, including monomial parametrizations, saturation arguments, and Bertini-type dimension results, offer a tractable framework for analyzing CI equilibria in broader nonbinary settings and graph structures.
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.