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A vector bundle approach to Nash equilibria

Hirotachi Abo, Irem Portakal, Luca Sodomaco

TL;DR

The paper develops an algebro-geometric framework to study totally mixed Nash equilibria of n-player normal-form games by modeling the equilibrium locus as the zero set of a globally generated vector bundle on a product of projective spaces. It defines the Nash equilibrium scheme ${oldsymbol{ m Z}}_X$ and introduces the Nash discriminant and Nash resultant varieties to capture nongeneric and boundary-format phenomena, respectively. The authors derive a suite of structural results: in balanced formats the real part of the Nash discriminant has codimension one (with special behavior in the $(d,d)$ binary case and boundary-format nuances), and the Nash resultant is irreducible with explicit codimension and degree formulas; for three-player binary and boundary-format cases they provide complete classifications and compute specific degrees using Chern class, discriminant, and GKZ methods. This framework links Nash equilibria with vector bundle theory, leading to a robust, computable description of nongeneric equilibria and highlighting deep connections between algebraic geometry and game theory with potential for broad applications in understanding generic and nongeneric equilibrium structures.

Abstract

We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.

A vector bundle approach to Nash equilibria

TL;DR

The paper develops an algebro-geometric framework to study totally mixed Nash equilibria of n-player normal-form games by modeling the equilibrium locus as the zero set of a globally generated vector bundle on a product of projective spaces. It defines the Nash equilibrium scheme and introduces the Nash discriminant and Nash resultant varieties to capture nongeneric and boundary-format phenomena, respectively. The authors derive a suite of structural results: in balanced formats the real part of the Nash discriminant has codimension one (with special behavior in the binary case and boundary-format nuances), and the Nash resultant is irreducible with explicit codimension and degree formulas; for three-player binary and boundary-format cases they provide complete classifications and compute specific degrees using Chern class, discriminant, and GKZ methods. This framework links Nash equilibria with vector bundle theory, leading to a robust, computable description of nongeneric equilibria and highlighting deep connections between algebraic geometry and game theory with potential for broad applications in understanding generic and nongeneric equilibrium structures.

Abstract

We use vector bundles to study the locus of totally mixed Nash equilibria of an -player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.

Paper Structure

This paper contains 11 sections, 21 theorems, 121 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Games, tensors, and Nash
  3. General notations
  4. Preliminaries
  5. Vector bundles
  6. Intermezzo: totally mixed Nash equilibria vs singular vector tuples
  7. Nash discriminants and Nash resultants
  8. The Nash discriminant variety
  9. The Nash discriminant variety of a three-player binary game
  10. The Nash discriminant variety of games of boundary format
  11. The Nash resultant variety

Key Result

Theorem 1

If $X$ is a generic game, then the following three conditions are equivalent:

Figures (1)

  • Figure 1: $3$-player game of format $(3,3,3)$

Theorems & Definitions (64)

  • Theorem : Theorem \ref{['thm: number tmNe generic game']}
  • Theorem : Proposition \ref{['prop: Nash resultant variety two players']}, Theorem \ref{['thm: codim degree Nash resultant variety']}
  • Theorem : Proposition \ref{['prop: discriminant_two_players']}, Theorem \ref{['thm: codimension real part discriminant of E is 1']}
  • Theorem : Theorem \ref{['thm: degree Nash discriminant variety 2x2x2']}, Proposition \ref{['prop:curve_component']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 54 more