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O'Neill's Theorem for Games

Srihari Govindan, Rida Laraki, Lucas Pahl

Abstract

We present an analog of O'Neill's Theorem (Theorem 5.2 in [17]) for finite games, which reveals some of the structure of equilibria under payoff perturbations in finite games.

O'Neill's Theorem for Games

Abstract

We present an analog of O'Neill's Theorem (Theorem 5.2 in [17]) for finite games, which reveals some of the structure of equilibria under payoff perturbations in finite games.
Paper Structure (7 sections, 13 theorems, 13 equations, 35 figures, 1 table)

This paper contains 7 sections, 13 theorems, 13 equations, 35 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary Definitions and Statement of the Theorem
  3. Fixed Point Index
  4. Statement of the Theorem
  5. A Guide to the Proof of Theorem \ref{['main thm']}
  6. Proof of Theorem \ref{['main thm']}
  7. Appendix: Proof of Lemma \ref{['lem triangulation T tilde']}

Key Result

Theorem 2.6

For each $\varepsilon>0$, there exist pairwise disjoint sets $V_1, \ldots, V_k \subseteq \Sigma$, $V_i$ a neighborhood of $C_i$ for each $i$, such that for any choice of finitely many distinct points $\{\sigma^{ij}\}_{ij}, \sigma^{ij} \in V_i \setminus \partial V_i$ and numbers $r_{ij} \in \{-1,1\}$

Figures (35)

  • Figure 1: Triangulation of a square
  • Figure 2: The closed star of vertex $f$
  • Figure 3: The simplicial neighborhood of the closed star of $f$
  • Figure 4: Face complex and subcomplex
  • Figure 5: Carrier of point in $x$ in a triangulation
  • ...and 30 more figures

Theorems & Definitions (41)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Example 3.1
  • Example 3.2
  • Lemma 4.1
  • ...and 31 more