Majorized correspondences and equilibrium existence in discontinuous games

Abstract

This paper is aimed at widening the scope of applications of majorized correspondences. A new class of majorized correspondences—domain ${\mathcal {U}}$-majorized correspondences—is introduced. For them, a maximal element existence theorem is established. Then, sufficient conditions for the existence of an equilibrium in qualitative games are provided. They are used to show the existence of a pure strategy Nash equilibrium in compact quasiconcave games that are either correspondence secure or correspondence transfer continuous.

Appendix

1.1 Proof of Lemma 2

Suppose that $F$ is locally ${\mathcal {U}}_{\theta }$-majorized. We have to show that it is ${\mathcal {U}}_{\theta }$-majorized. For every $x\in X$, there exists an open neighborhood $U(x)$ and a ${\mathcal {U}}_{\theta }$-majorant $F^{x}:U(x)\rightarrow Y$ of $F$ at $x$ such that $F(z)\subset F^{x}(z)$ and $\theta (z)\notin F^{x}(z)$ for every $z\in U(x)$. The open cover $\{U(x):x\in X\}$ has a finite subcover $\{U(x^{j}):j=1,\ldots ,J\}$. For each $j$, define $F_{j}^{\prime }:X\rightarrow Y$ by

$$\begin{aligned} F_{j}^{\prime }(z)=\left\{ \begin{array}{l@{\quad }l} F^{x^{j}}(z) &{} \text { if }z\in U(x^{j}), \\ Y&{}\text { if }z\in X\backslash U(x^{j}). \end{array} \right. \end{aligned}$$

Since each $F_{j}^{\prime }$ is upper hemicontinuous on $X$ and $Y$ is a compact subset of a Hausdorff topological vector space, the correspondence $\overline{F}:X\twoheadrightarrow$ $Y$ defined by $\overline{F}(z)=\cap _{j\in \{1,\ldots ,J\}}F_{j}^{\prime }(z)$ for every $z\in X$ is of class ${\mathcal {U}}_{\theta }$ (see, e.g., Aliprantis and Border 2006, Theorem 17.25). Clearly, $F(z)\subset \overline{F}(z)$ for every $z\in U(x)$.

1.2 Proof of Theorem 2

Assume, by contradiction, that $\varGamma$ has no equilibrium, i.e., $\mathrm { Dom}\varGamma =X$. Since $X$ is compact, the open cover $\{U(x):x\in X\}$ of $X$ contains a finite subcover $\{U(x^{j}):j\in J\}$, where $J$ is a finite set. Let $\{V_{j}:j\in J\}$ be an open refinement of $\{U(x^{j}):j\in J\}$ such that cl$V_{j}\subset U(x^{j})$ for every $j\in J$ (see, e.g., Urai 2010, Theorem 3.1.3). For each $j\in J$ and each $i\in I$, define a correspondence $\widehat{F}_{i}^{j}:X\twoheadrightarrow X_{i}$ by

We need to show that each $\widehat{F}_{i}^{j}$ is upper hemicontinuous on $X$. Fix some $i\in I$, $j\in J$, and $z\in X$. Assume that there exists a relatively open, proper subset of $X_{i}$, $W$, such that $\widehat{F} _{i}^{j}(z)\subset W$. Clearly, $i\in I(x^{j})$ and $z\in V_{j}$. Denote $J_{C}^{z}=\{s\in J:z\in \mathrm {cl}V_{s}\}$. Since, for each $s\in J_{C}^{z}$ , $D_{i}^{x^{s}}$ is either nonempty-valued and upper hemicontinuous at $z$ or empty-valued at $z$, there exists a neighborhood $U_{s}$ of $z$ such that $U_{s}\subset V_{j}\cap U(x^{s})$ and $D_{i}^{x^{s}}(U_{s})\subset W$. Then $\widehat{F}_{i}^{j}(\cap _{s\in J_{C}^{z}}U_{s}\cap (\cap _{s\in J\backslash J_{C}^{z}}(X\backslash \mathrm {cl}V_{s})))\subset W$.

For $i\in I$ and $j\in J$, define $F_{i}^{j}:X\twoheadrightarrow Y$ by $F_{i}^{j}(z)=\mathrm {co}\widehat{F}_{i}^{j}(z)$ for every $z\in X$. Then $F_{i}^{j}$ is compact-valued (see, e.g., Aliprantis and Border 2006, Lemma 5.29), and, moreover, $F_{i}^{j}$ is upper hemicontinuous on $X_{i}$ (see, e.g., Aliprantis and Border 2006, Theorem 17.35). Then, for each $i\in I$, the correspondence $F_{i}:X\twoheadrightarrow X_{i}$ defined by $F_{i}(z)=\cap _{j\in J}F_{i}^{j}(z)$ for every $z\in X$ is upper hemicontinuous on $X$. Therefore, the correspondence $\overline{F} :X\twoheadrightarrow X$ defined by $\overline{F}(z)=\Pi _{i\in I}F_{i}(z)$ is upper hemicontinuous on $X$.

Consider some $z\in X$. It lies in some $V_{j}$. Let us show that $z_{i^{\prime }}\notin F_{i^{\prime }}^{j}(z)$ for some $i^{\prime }\in I(x^{j})$. By (ii), there exists $i^{\prime }\in \cup _{s\in J_{C}^{z}}I(x^{s})$ such that $z_{i^{\prime }}\notin \mathrm {co}\{\cup _{s\in J_{C}^{z}}D_{i^{\prime }}^{x^{s}}(z)\}$. Since $\widehat{F} _{i^{\prime }}^{j}(z)=\cup _{s\in J_{C}^{z}}D_{i^{\prime }}^{x^{s}}(z)$, we have that $z_{i^{\prime }}\notin F_{i^{\prime }}^{j}(z)$. Therefore, $z_{i^{\prime }}\notin F_{i^{\prime }}(z)$, and, consequently, $z\notin \overline{F}(z)$.

Since $\overline{F}$ is of class ${\mathcal {U}}$ and $\mathrm {Dom}\varGamma = \mathrm {Dom}\overline{F}=X$, $\varGamma$ is ${\mathcal {U}}^{d}$-majorized. Then it has an equilibrium by Lemma 5, a contradiction.