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Game theory of undirected graphical models

Irem Portakal, Javier Sendra-Arranz

TL;DR

This work develops an algebraic-geometry framework for analyzing dependency equilibria in $n$-player normal-form games by modeling them as undirected graphical models and studying Spohn conditional independence varieties. It proves that for generic binary games, the Spohn CI variety $ u_{X,oldsymbol{C}}$ has codimension $n$ in the CI model $oldsymbol{ ext{M}}_{oldsymbol{C}}$, and that the totally mixed CI equilibria form a smooth semialgebraic manifold within the open simplex. The paper also proves a dimension formula: the dimension of $ u_{X,oldsymbol{C}}$ equals the number of nonempty cliques of size at least two in the clique complex, and establishes affine universality results for Nash CI varieties in decomposable graphs, showing that universal real affine varieties appear as open subsets of Nash CI spaces. Additionally, Nash CI varieties are shown to be complete intersections with computable degrees, and smooth Nash CI surfaces are of general type, providing deep connections between algebraic statistics, game theory, and complex geometry. These results offer a unified geometric lens on Nash, dependency, and CI equilibria with potential implications for learning, optimization, and theoretical understanding of equilibrium spaces.

Abstract

An $n$-player game $X$ in normal form can be modeled via undirected discrete graphical models where the discrete random variables represent the players and their state spaces are the set of pure strategies. There exists an edge between the vertices of the graphical model whenever there is a dependency between the associated players. We study the Spohn conditional independence (CI) variety $\mathcal{V}_{X,\mathcal{C}}$, which is the intersection of the independence model $\mathcal{M}_{\mathcal{C}}$ with the Spohn variety of the game $X$. We prove a conjecture by the first author and Sturmfels that $\mathcal{V}_{X,\mathcal{C}}$ is of codimension $n$ in $\mathcal{M}_{\mathcal{C}}$ for a generic game $X$ with binary choices. We show that the set of totally mixed CI equilibria i.e., the restriction of the Spohn CI variety to the open probability simplex is a smooth semialgebraic manifold for a generic game $X$ with binary choices. If the undirected graph is a disjoint union of cliques, we analyze certain algebro-geometric features of Spohn CI varieties and prove affine universality theorems.

Game theory of undirected graphical models

TL;DR

This work develops an algebraic-geometry framework for analyzing dependency equilibria in -player normal-form games by modeling them as undirected graphical models and studying Spohn conditional independence varieties. It proves that for generic binary games, the Spohn CI variety has codimension in the CI model , and that the totally mixed CI equilibria form a smooth semialgebraic manifold within the open simplex. The paper also proves a dimension formula: the dimension of equals the number of nonempty cliques of size at least two in the clique complex, and establishes affine universality results for Nash CI varieties in decomposable graphs, showing that universal real affine varieties appear as open subsets of Nash CI spaces. Additionally, Nash CI varieties are shown to be complete intersections with computable degrees, and smooth Nash CI surfaces are of general type, providing deep connections between algebraic statistics, game theory, and complex geometry. These results offer a unified geometric lens on Nash, dependency, and CI equilibria with potential implications for learning, optimization, and theoretical understanding of equilibrium spaces.

Abstract

An -player game in normal form can be modeled via undirected discrete graphical models where the discrete random variables represent the players and their state spaces are the set of pure strategies. There exists an edge between the vertices of the graphical model whenever there is a dependency between the associated players. We study the Spohn conditional independence (CI) variety , which is the intersection of the independence model with the Spohn variety of the game . We prove a conjecture by the first author and Sturmfels that is of codimension in for a generic game with binary choices. We show that the set of totally mixed CI equilibria i.e., the restriction of the Spohn CI variety to the open probability simplex is a smooth semialgebraic manifold for a generic game with binary choices. If the undirected graph is a disjoint union of cliques, we analyze certain algebro-geometric features of Spohn CI varieties and prove affine universality theorems.
Paper Structure (11 sections, 23 theorems, 98 equations, 4 figures)

This paper contains 11 sections, 23 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Algebraic game theory preliminaries
  3. Spohn variety and dependency equilibria
  4. Graphical models and conditional independence equilibria
  5. Dimension of Spohn CI varieties
  6. Filtration of Spohn CI varieties
  7. Nash conditional independence varieties
  8. Equations of Nash CI varieties
  9. Algebro-geometric properties
  10. Universality of Nash CI varieties
  11. Acknowledgements

Key Result

Proposition 2

The parametrized discrete undirected graphical model associated to $G$ consists of all joint probability distributions $P \in \Delta_{d_1 \cdots d_n -1}$ given by the following monomial parametrization where $\sigma = (\sigma^{(C)})_{C \in \mathcal{C}(G)}$ is the vector of parameters and $Z(\sigma)$ is the normalizing constant. Moreover, the positive part of the parametrized model is the hierarch

Figures (4)

  • Figure 1: Line graph and cycle on four vertices
  • Figure 2: The decomposable graph $G$ has $4$ maximal cliques. Two of them have $3$ vertices and the other two have $4$ vertices.
  • Figure 3: The poset structure of the set of subgraphs of the complete graph on $3$ vertices with respect to the inclusion.
  • Figure 4: Poset of subgraphs of the $4$ vertex graph $G_4$, their CI equilibria and payoff regions.

Theorems & Definitions (48)

  • Example 1
  • Proposition 2: Sul
  • Proposition 3
  • Example 4
  • Conjecture 5: BI
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 38 more