Reducible equilibrium properties: comments on recent existence results

Abstract

We show that each of the properties of equilibrium problems considered in several recent Nash equilibrium existence results satisfy the following condition: All the existence problems that satisfy the property on a compact subset $K$ of a fixed strategy space have, on $K$, an equilibrium set that contains the equilibrium set of an existence problem that is well behaved on $K$. The properties that satisfy this condition are called reducible equilibrium properties. This notion is used to clarify recent existence results and the relationship of some of the properties considered in those results.

Appendix

In this section, and for completeness, we state and prove Barelli and Soza’s (2009) gluing lemma (Lemma 5.2 of that paper).

Lemma 3

Let $X$ be a nonempty, compact and convex subset of a Hausdorff locally convex topological vector space, $K$ be a compact subset of $X$, $\lambda :K\rightrightarrows X$ be convex-valued and, for each $x\in K$, $U^x$ be an open neighborhood of $x$ and $\varphi ^x:U^x\rightrightarrows X$ be a closed correspondence with nonempty, compact and convex values such that $\varphi ^x(x')\subseteq \lambda (x')$ for all $x'\in U^x$. Then there exists a closed correspondence $\Psi :K\rightrightarrows X$ with nonempty, compact and convex values such that $\Psi (x)\subseteq \lambda (x)$ for all $x\in K$.

Proof

For each $x\in K$, define $\psi ^x:K\rightrightarrows X$ by

$$\begin{aligned} \psi ^x(x')=\left\{ \begin{array}{l@{\quad }l} \varphi ^x(x') &{}\quad \text {if }x'\in U^x,\\ X &{}\quad \text {otherwise} \end{array} \right. \end{aligned}$$

for each $x'\in K$ and note that $\psi ^x$ is a closed correspondence with nonempty, compact and convex values. We have that $\{U^x,\varphi ^x\}_{x\in K}$ is such that $\{U^x\}_{x\in K}$ is an open cover of $K$. Since $K$ is compact, there exists a finite open cover $\{U^{x_j}\}_{j=1}^m$ of $K$ and, by Aliprantis and Border (2006, Lemma 2.92, p. 67), there exists a partition of unity $\{\beta _j\}_{j=1}^m$ subordinate to $\{U^{x_j}\}_{j=1}^m$. Define $\Psi :K\rightrightarrows X$ by $\Psi (x)=\sum \nolimits _{j=1}^m\beta _j(x)\psi ^{x_j}(x)$. It is easy to see that $\Psi$ is a closed correspondence with nonempty, compact and convex values.

Let $x\in K$ and define $J(x)=\{j\in \{1,\ldots ,m\}:x\in U^{x_j}\}$. Since $\beta _j(x)=0$ for all $j\not \in J(x)$, $\psi ^{x_j}(x)=\varphi ^{x_j}(x)\subseteq \lambda (x)$ if $j\in J(x)$ and $\lambda (x)$ is convex, it follows that $\Psi (x)=\sum \nolimits _{j\in J(x)}\beta _j(x)\varphi ^{x_j}(x)\subseteq \lambda (x)$. This completes the proof.$\square$