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Index and Robustness of Mixed Equilibria: An Algebraic Approach

Lucas Pahl

Abstract

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be $0$, $+1$ or $-1$ only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.

Index and Robustness of Mixed Equilibria: An Algebraic Approach

Abstract

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be , or only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.
Paper Structure (8 sections, 11 theorems, 28 equations, 2 tables)

This paper contains 8 sections, 11 theorems, 28 equations, 2 tables.

Table of Contents

  1. Introduction
  2. A motivating example
  3. Preliminaries
  4. Algebraic Preliminaries
  5. Topological Preliminaries
  6. Index and Robustness
  7. Extensions
  8. Final Remarks

Key Result

Proposition 2.1

Suppose $f: (\mathbb{R}^n,0) \to (\mathbb{R}^m, 0)$ is a polynomial map with an isolated root at $0$. Suppose $I = \langle f_1,...,f_m \rangle$ is an ideal of $\mathbb{R}[[X_1,....,X_n]]$ and $\mathbb{R}[[X_1,....,X_n]] / I$ has finite dimension. Then where $I_{max}$ is a maximal square-zero ideal of $\mathbb{R}[[X_1,....,X_n]] / I$.

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof : Proof of the Lemma
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof : Proof of the Lemma
  • Definition 3.3
  • ...and 19 more