Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Combinatorics of Correlated Equilibria

Marie-Charlotte Brandenburg, Benjamin Hollering, Irem Portakal

TL;DR

The paper addresses the combinatorial structure of correlated equilibrium polytopes arising from finite games by formulating them via the correlated equilibrium cone $C_G$ and a parametric matrix $A(Y)$. It shows that the region where the polytope has maximal dimension is semialgebraic and can be stratified using oriented matroids, with a concrete description of the algebraic boundary for $(2×n)$-games in terms of coordinate hyperplanes and binomial minors. For generic $(2×3)$-games there are exactly three combinatorial types of $P_G$, including a unique full-dimensional type with a specific $f$-vector, and a conjecture that non-full-dimensional cases stem from smaller games. The results blend convex geometry with real algebraic geometry to yield a structured, computable framework for understanding correlated equilibria in low-dimensional games, aided by explicit computations and openly available MathRepo code.

Abstract

We study the correlated equilibrium polytope $P_G$ of a game $G$ from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of $P_G$, and show that for $(2 \times n)$-games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic $(2 \times 3)$-games.

Combinatorics of Correlated Equilibria

TL;DR

The paper addresses the combinatorial structure of correlated equilibrium polytopes arising from finite games by formulating them via the correlated equilibrium cone and a parametric matrix . It shows that the region where the polytope has maximal dimension is semialgebraic and can be stratified using oriented matroids, with a concrete description of the algebraic boundary for -games in terms of coordinate hyperplanes and binomial minors. For generic -games there are exactly three combinatorial types of , including a unique full-dimensional type with a specific -vector, and a conjecture that non-full-dimensional cases stem from smaller games. The results blend convex geometry with real algebraic geometry to yield a structured, computable framework for understanding correlated equilibria in low-dimensional games, aided by explicit computations and openly available MathRepo code.

Abstract

We study the correlated equilibrium polytope of a game from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of , and show that for -games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic -games.
Paper Structure (5 sections, 6 theorems, 39 equations, 4 figures, 1 table)

This paper contains 5 sections, 6 theorems, 39 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Correlated Equilibrium Polytope
  3. The Correlated Equilibrium Cone
  4. The Region of Full-Dimensionality
  5. Combinatorial Types of Correlated Equilibrium Polytopes

Key Result

Proposition 2.4

Any affine linear transformation of the payoff tensors $X^{(i)}$ with positive scalars leaves the polytope $P_G$ invariant. More precisely, let $G = (n, S, X)$ be a game. For each $i \in [n]$, fix $t_i \in \mathbb R, \ \lambda_i \in \mathbb R_{>0}$ and let $\tilde{X}^{(i)}_{j_1 \dots j_n} = \lambda_

Figures (4)

  • Figure 1: The correlated equilibrium polytope of the Traffic Lights example is a bipyramid where three of its vertices are Nash equilibria. Its $f$-vector is $(1,5,9,6,1)$ and its face lattice can be seen on the right.
  • Figure 2: The vertices of the correlated equilibrium polytope for the Traffic Lights example (\ref{['ex:traffic-lights-2']})
  • Figure 3: A $3$-dimensional correlated equilibrium polytope (green) inside the probability simplex $\Delta_3$ (yellow) for a $(2\times 2)$-game. Its Nash equilibria (black) are the intersection with the Segre variety (red). This picture applies to the Traffic Lights example (\ref{['ex:traffic-lights-1', 'ex:traffic-lights-2', 'ex:traffic-lights-3']}) as well as the Hawk-Dove game (\ref{['ex: correlated polytope']}).
  • Figure 4: The graph of the combinatorially unique $5$-dimensional polytope that arises as the correlated equilibrium polytope of a $(2\times 3)$-game, as described in \ref{['thm:2x3typesFromStrata']}.

Theorems & Definitions (30)

  • Example 2.1: Traffic Lights
  • Definition 2.2: Correlated Equilibrium
  • Example 2.3: Traffic Lights
  • Proposition 2.4
  • proof
  • Definition 2.5: Nash Equilibrium
  • Example 2.6: Traffic Lights
  • Proposition 2.7: nau_geometrynashequilibria
  • Example 3.1: $(2 \times 2)$-games
  • Definition 3.2
  • ...and 20 more