Continuity of equilibria for two-person zero-sum games with noncompact action sets and unbounded payoffs

Abstract

This paper extends Berge’s maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.

Appendix: Properties of \(\texttt {A}\)-lower semi-continuous multifunctions

This appendix describes some properties of $\texttt {A}$-lower semi-continuous multifunctions. Definition 4 and the definition of lower semi-continuous multifunctions imply that an $\texttt {A}$-lower semi-continuous multifunction is lower semi-continuous. The following example demonstrates that a lower semi-continuous multifunction may not be $\texttt {A}$-lower semi-continuous.

Example 5

Let $\texttt {X}=\texttt {A}=[0,1],$ $\texttt {B}=\mathbb {R},$ $\mathtt {\Phi }_\texttt {A}(x)=\{x\}\cup \{\frac{1}{x}\}$ for $x\in (0,1],$ $\mathtt {\Phi }_\texttt {A}(0)=\{0\},$ and $\mathtt {\Phi }_\texttt {B}(x,a)=\{a\}$ for all $(x,a)\in \mathrm{Gr}(\mathtt {\Phi }_\texttt {A}).$ Since each set $\mathtt {\Phi }_\texttt {B}(x,a)$ is a singleton, where $(x,a)\in \mathrm{Gr}(\mathtt {\Phi }_\texttt {A}),$ and the graph of the multifunction $\mathtt {\Phi }_\texttt {B}$ is closed, the multifunction $\mathtt {\Phi }_\texttt {B}$ is lower semi-continuous. Let us consider the sequence $\{x_n\}_{n=1,2,\ldots }=\{\frac{1}{n}\}_{n=1,2,\ldots }$ converging to $x=0.$ Then $b:=0\in \mathtt {\Phi }_\texttt {B}(0,0)$ and $a^{(n)}=n\in \mathtt {\Phi }_\texttt {A}(x^{(n)}),$ $n=1,2,\ldots \ .$ However, the sequence $\{b_n\}_{n=1,2,\ldots }:=\{n\}_{n=1,2,\ldots }$ does not have a limit point. Thus, the multifunction $\mathtt {\Phi }_\texttt {B}$ is not $\texttt {A}$-lower semi-continuous.

Let us provide sufficient conditions for $\texttt {A}$-lower semi-continuity.

Lemma 7

Let $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ be a lower semi-continuous set-valued mapping. Then the following statements hold:

• (a) if $\mathtt {\Phi }_{\texttt {A}}:\texttt {X}\mapsto S(\texttt {A})$ is upper semi-continuous and compact-valued at each $x\in \texttt {X},$ then $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous;

• (b) if $\mathtt {\Phi }_{\texttt {B}}(x,a)$ does not depend on $a\in \mathtt {\Phi }_\texttt {A}(x)$ for each $x\in \texttt {X},$ that is, $\mathtt {\Phi }_{\texttt {B}}(x,a_*)=\mathtt {\Phi }_{\texttt {B}}(x,a^*)$ for each $(x,a_*),(x,a^*)\in \mathrm{Gr}(\mathtt {\Phi }_\texttt {A}),$ then $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous.

Remark 26

Let $\mathtt {\Phi }:\texttt {X}\mapsto S(\texttt {B}),$ where $\mathtt {\Phi }(x)$ can be interpreted as the set of actions for Player II, when this set does not depend on the actions of Player I, as this takes place for games with simultaneous moves. Then we can define the sets

$$\begin{aligned} \mathtt {\Phi }_\texttt {B}(x,a):= \mathtt {\Phi }(x),\qquad \qquad (x,a)\in \mathrm{Gr}(\mathtt {\Phi }_\texttt {A}). \end{aligned}$$

(52)

The definition of a lower semi-continuous multifunction implies that, if the multifunction $\mathtt {\Phi }:\texttt {X}\mapsto S(\texttt {B})$ is lower semi-continuous, then the multifunction $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is lower semi-continuous too. Lemma 7 implies that the lower semi-continuity of $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is equivalent to its $\texttt {A}$-lower semi-continuity in the following two cases: (a) for two-person zero-sum games with perfect information, when the decision sets $\{\mathtt {\Phi }_{\texttt {A}}(x)\}_{x\in \texttt {X}}$ for the first player are compact and the dependence of $\mathtt {\Phi }_{\texttt {A}}(x)$ by the state variable x is upper semi-continuous, and (b) for two-person zero-sum games with simultaneous moves.

Proof of Lemma 7

(a) Let $\{x^{(n)}\}_{n=1,2,\ldots }$ be a sequence with values in $\texttt {X}$ that converges and its limit x belongs to $\texttt {X}.$ Let also $a^{(n)}\in \mathtt {\Phi }_\texttt {A}(x^{(n)}),$ for each $n=1,2,\ldots ,$ and $b\in \mathtt {\Phi }_\texttt {B}(x,a)$ for some $a\in \mathtt {\Phi }_\texttt {A}(x).$ Let us prove that b is a limit point for a sequence $\{b^{(n)}\}_{n=1,2,\ldots }$ with $b^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)})$ for each $n=1,2,\ldots \ .$ Indeed, Lemma 4, being applied to $\mathbb {X}:=\texttt {X},$ $\mathbb {Y}:=\texttt {A},$ and $\varPhi :=\mathtt {\Phi }_\texttt {A},$ implies that the sequence $\{a^{(n)}\}_{n=1,2,\ldots }$ has a limit point $a\in \mathtt {\Phi }_\texttt {A}(x).$ Therefore, b is a limit point of a sequence $\{b^{(n)}\}_{n=1,2,\ldots }$ with $b^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)})$ for each $n=1,2,\ldots ,$ since $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is a lower semi-continuous set-valued mapping.

(b) Since $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is a lower semi-continuous set-valued mapping and $\mathtt {\Phi }(x,a)$ does not depend on $a\in \mathtt {\Phi }_\texttt {A}(x)$ for each $x\in \texttt {X},$ the following statement holds: if a sequence $\{x^{(n)}\}_{n=1,2,\ldots }$ with values in $\texttt {X}$ converges and its limit x belongs to $\texttt {X},$ $a^{(n)}\in \mathtt {\Phi }_\texttt {A}(x^{(n)})$ for each $n=1,2,\ldots ,$ and $b\in \mathtt {\Phi }_\texttt {B}(x,a)$ for some $a\in \mathtt {\Phi }_\texttt {A}(x),$ then b is a limit point of a sequence $\{b^{(n)}\}_{n=1,2,\ldots }$ with $b^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)})$ for each $n=1,2,\ldots ,$ that is, $\mathtt {\Phi }_{\texttt {B}}:\mathrm{Gr}(\mathtt {\Phi }_{\texttt {A}})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous set-valued mapping. $\square$

The following two statements, which are not used in this paper, provide additional properties of $\texttt {A}$-lower semi-continuous set-valued mappings for the case, when $\texttt {B}$ is a vector space. Let $\texttt {B}$ be a vector space and $\mathtt {\Phi }_\texttt {B},\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ be set-valued mappings. Let us define for each $(x,a)\in \mathrm{Gr}(\mathtt {\Phi }_\texttt {A})$

$$\begin{aligned} \mathtt {\Phi }_\texttt {B}(x,a)+\mathtt {\Psi }_\texttt {B}(x,a):=\{b_0+b_1\,:\,b_1\in \mathtt {\Phi }_\texttt {B}(x,a),\, b_2\in \mathtt {\Psi }_\texttt {B}(x,a) \}. \end{aligned}$$

Lemma 8

Let $\texttt {B}$ be a vector space and $\mathtt {\Phi }_\texttt {B},\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ be $\texttt {A}$-lower semi-continuous set-valued mappings. Then the set-valued mapping $\mathtt {\Phi }_\texttt {B}+\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous.

Proof of Lemma 8

Let $\{x^{(n)}\}_{n=1,2,\ldots }$ be a sequence with values in $\texttt {X}$ that converges and its limit x belongs to $\texttt {X}.$ Assume that $a^{(n)}\in \mathtt {\Phi }_\texttt {A}(x^{(n)})$ for each $n=1,2,\ldots ,$ and $b\in \mathtt {\Phi }_\texttt {B}(x,a)$ for some $a\in \mathtt {\Phi }_\texttt {A}(x).$ Let us prove that b is a limit point of a sequence $\{b^{(n)}\}_{n=1,2,\ldots }$ with $b^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)}),$ $n=1,2,\ldots .$ Indeed, since $\mathtt {\Phi }_\texttt {B}(x,a)=\mathtt {\Phi }_\texttt {B}(x,a)+\mathtt {\Psi }_\texttt {B}(x,a),$ there exist $b_1\in \mathtt {\Phi }_\texttt {B}(x,a)$ and $b_2\in \mathtt {\Psi }_\texttt {B}(x,a)$ such that $b=b_0+b_1.$ The $\texttt {A}$-lower semi-continuity of $\mathtt {\Phi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ and $\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ imply that $b_i,$ $i=0,1,$ is a limit point of a sequence $\{b_i^{(n)}\}_{n=1,2,\ldots }$ with $b_1^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)})$ and $b_2^{(n)}\in \mathtt {\Psi }_\texttt {B}(x^{(n)},a^{(n)})$ Therefore, $b=b_0+b_1$ is a limit point of a sequence $\{b^{(n)}\}_{n=1,2,\ldots }$ with $b^{(n)}:=b_0^{(n)}+b_1^{(n)}\in \mathtt {\Phi }_\texttt {B}(x^{(n)},a^{(n)})+\mathtt {\Psi }_\texttt {B}(x^{(n)},a^{(n)}),$ $n=1,2,\ldots \ .$ Thus, the set-valued mapping $\mathtt {\Phi }_\texttt {B}^0+\mathtt {\Phi }_\texttt {B}^1:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous. $\square$

Corollary 8

Let $\texttt {B}$ be a vector space, $\mathtt {\Phi }:\texttt {X}\mapsto S(\texttt {B})$ be a lower semi-continuous set-valued mapping, $\mathtt {\Phi }_{\texttt {A}}:\texttt {X}\mapsto \mathbb {K}(\texttt {A})$ be an upper semi-continuous set-valued mapping, and $\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ be a lower semi-continuous set-valued mapping. Let us consider the set-valued mapping $\mathtt {\Phi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ defined in (52). Then the set-valued mapping $\mathtt {\Phi }_\texttt {B}+\mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ is $\texttt {A}$-lower semi-continuous.

Proof

According to Lemma 7, the set-valued mappings $\mathtt {\Phi }_\texttt {B}, \mathtt {\Psi }_\texttt {B}:\mathrm{Gr}(\mathtt {\Phi }_\texttt {A})\subset \texttt {X}\times \texttt {A}\mapsto S(\texttt {B})$ are $\texttt {A}$-lower semi-continuous. Therefore, Lemma 8 implies that their sum is $\texttt {A}$-lower semi-continuous. $\square$