Rationalizability in large games

Abstract

This paper characterizes both point-rationalizability and rationalizability in large games when societal responses are formulated as distributions or averages of individual actions. The sets of point-rationalizable and rationalizable societal responses are defined and shown to be convex, compact and equivalent to those outcomes that survive iterative elimination of never best responses, under point-beliefs and probabilistic beliefs, respectively. Given the introspection and mentalizing that rationalizability notions presuppose, one motivation behind the work is to examine their viability in situations where the terms rationality and full information can be given a more parsimonious, and thereby a more analytically viable, expression.

Appendices

Appendix A

The first two propositions are on correspondences, whose details can be found in Aliprantis and Border (2006, Chapters 17 and 18).

Let $S$ and $X$ be topological spaces and $F$ a correspondence from $S$ to $X$. $F: S\twoheadrightarrow X$ has closed values or is closed-valued if $F(s)$ is a closed set for each $s\in S$. The terms compact-valued, convex-valued and nonempty-valued are similarly defined. Let $F^{l}(E)\equiv \{s\in S: F(s)\bigcap E \ne \emptyset \}$ for any subset $E$ of $X$. If $F^l(C)$ is closed for each closed subset $C$ of $Y,\,F: S\twoheadrightarrow X$ is upper hemicontinuous. $F$ has closed graph is its graph $\{(s, x)\in S\times Y: x\in F(s) \}$ is a closed subset of $S\times X$.

Proposition 2

   $P1$: A correspondence with compact Hausdorff range space has closed graph if and only if it is upper hemicontinuous and closed-valued.

$P2$: The image of a compact set under a compact-valued upper hemicontinuous correspondence is compact.

Let $(S, \Sigma )$ be a measurable space and $X$ a topological space. A correspondence $F: S\twoheadrightarrow X$ is said to be measurable, if for each closed subset $C$ of $X,\,F^{l}(C)\in {\Sigma }$, then $F$ is said to be measurable; $F$ is said to be weakly measurable, if for each open subset $O$ of $X$, $F^{l}(O)\in {\Sigma }$. A function $f$ is said to be a selection of $F$ if $f(s) \in F(s)$ for all $s \in S$. With this notation, the next proposition is a collection of measurability results of correspondence and its selections.

Proposition 3

Let $(S, \Sigma )$ be a measurable space and $X$ a Polish space. Let $F,\,F_1,\,F_2$ be correspondences from $S$ to $X$.

$P1$::

If $F$ has nonempty compact values, then $F$ is measurable if and only if it is weakly measurable.

$P2$ :

(Kuratowski–Ryll–Nardzewski Selection Theorem): If $F$ is weakly measurable and has nonempty closed values, then it admits a measurable selection.

$P3$::

$F$ has closed graph if and only if it is upper hemicontinuous and closed-valued.

$P4$::

If $F_1$ and $F_2$ are closed-valued and measurable, then their intersection correspondence $G$, where $G$ is such that for all $s\in S,\,G(s)=F_1(s) \bigcap F_2(s)$, is measurable and closed-valued.

Let $X$ be a Polish space and $(\Omega ,\mathcal A ,P)$ an atomless probability space. A measurable function $f: \Omega \rightarrow X$ is a measurable selection of a correspondence $F: \Omega \twoheadrightarrow X$ if $f(\omega )\in F(\omega )$ for $P$-almost all $\omega \in \Omega$. The next proposition is culled from the corresponding results on the distribution of correspondences in Khan and Sun (1995) and Keisler and Sun (2009).

Proposition 4

Let $X$ be a compact metric space and $(\Omega ,\mathcal A ,P)$ an atomless probability space. Then the following results are valid if, in addition, (1) $X$ is a countable, or (2) $(\Omega ,\mathcal A ,P)$ is a saturated probability space.

$P1$::

Let $\{f_n\}$ be a sequence of measurable functions from $\Omega$ to X, such that $\tau _n =P {f_n}^{-1}$ converges weakly to $\tau \in \mathcal{M }(X)$ as $n\rightarrow \infty$. Let $D(\omega )=cl\text{-Lim }\,\{f_n(\omega )\}$. Then, $D(\omega )$ is nonempty for almost all $\omega$, and there exists a measurable selection $f$ of $D$, such that $P f^{-1} = \tau$.

$P2$::

For any correspondence $F$ from $(\Omega ,\mathcal A ,P)$ to $X,\,\mathcal D _F = \{Pf^{-1}:\,f$ is a measurable selection of $F \}$ is convex.

$P3$::

For any compact-valued correspondence $F$ from $(\Omega ,\mathcal A ,P)$ to $X,\,\mathcal D _F$ is compact.

The next two propositions are modified from Khan and Sun (1996), Sun (1997), Podczeck (2008) and Sun and Yannelis (2008). Proposition 5 is based on Bochner integral, whereas Proposition 6 is based on the Gel$^\prime$fand integral. See Yannelis (1991) and his references for an earlier treatment on the integration of correspondences over atomless probability spaces.

Let $(\Omega ,\mathcal A ,P)$ be a finite measure space and $X$ a Banach space. If $f: \Omega \rightarrow X$ is a Bochner integrable function, $\int _E f d P$ is the Bochner integral of $f$ over $E$ then for $E\in \mathcal{F }$. If $F: \Omega \twoheadrightarrow X$ is a correspondence, the Bochner integral of $F$ is, denoted by $\int _\Omega F d P$,

$$\begin{aligned} \displaystyle \int \limits _T F d P = \left\{ \int \limits _I f d P :f \text{ is } \text{ a } \text{ Bochner } \text{ integrable } \text{ selection } \text{ of } F\right\} . \end{aligned}$$

The next proposition deals with the Bochner integration of correspondences.

Proposition 5

Let $(\Omega ,\mathcal A ,P)$ be an atomless probability space and $X$ a separable Banach space. The following results are valid if, either (1) $X$ is a countable or (2) $(\Omega ,\mathcal A ,P)$ is a saturated probability space,

$P1$::

Let $\{f_n\}$ be a sequence of measurable functions from $\Omega$ to $X$, such that $\int {f_n} dP$ converges to $\iota$ as $n\rightarrow \infty$. Let $D(\omega )=w\text{- }cl\text{-Lim }\,\{f_n(\omega )\}$. Then, $D(\omega )$ is nonempty for almost all $\omega$, and there exists a measurable selection $f$ of $D$ such that $\int f d P = \iota$.

$P2$::

For any correspondence $F$ from $(\Omega ,\mathcal A ,P)$ into $X,\,\int _\Omega F d P$ is convex.

$P3$::

For any integrably bounded, weakly compact-valued correspondence $F$ from $(\Omega ,\mathcal A ,P)$ to $X,\,\int _\Omega F d P$ is weakly compact.

$P4$::

Let $Y$ be a metric space and $F$ a correspondence from $\Omega \times Y$ to $X$, such that for each fixed $y\in Y,\,F(\cdot , y)$ is a measurable, weakly compact-valued correspondence from $\Omega$ to $X$. If $F(\omega ,y)$ is upper hemicontinuous on $Y$ for each fixed $i$, then $\int _\Omega F(\omega ,y) d P$ is weakly upper hemicontinuous on $Y$.

Let $(\Omega ,\mathcal A ,P)$ be an atomless probability space and $X$ the dual of a separable Banach space. If $f: \Omega \rightarrow X$ is a Gel$^\prime$fand integrable function, $\int _E f d P$ is the Gel $^\prime$ fand integral of $f$ over $E$ then for $E\in \mathcal{F }$. If $F: \Omega \twoheadrightarrow X$ is a correspondence, the Gel $^\prime$ fand integral of $F$ is, denoted by $\int _\Omega F d P$,

$$\begin{aligned} \displaystyle \int \limits _T F d P = \left\{ \int \limits _I f d P :f \text{ is } \text{ a } \text{ Gel }^\prime \text{ fand } \text{ integrable } \text{ selection } \text{ of } F\right\} . \end{aligned}$$

The last proposition in this appendix is on the Gel$^\prime$fand integration of correspondences.

Proposition 6

Let $(\Omega ,\mathcal A ,P)$ be an atomless probability space and $X$ the dual of a separable Banach space. The following results are valid if, either (1) $X$ is a countable or (2) $(\Omega ,\mathcal A ,P)$ is a saturated probability space.

$P1$::

Let $\{f_n\}$ be a sequence of measurable functions from $\Omega$ to $X$, such that $\int {f_n} dP$ converges to $\iota$ as $n\rightarrow \infty$. Let $D(\omega )=w^*\text{- }cl\text{-Lim }\,\{f_n(\omega )\}$. Then, $D(\omega )$ is nonempty for almost all $\omega$, and there exists a measurable selection $f$ of $D$, such that $\int f d P = \iota$.

$P2$::

For any correspondence $F$ from $(\Omega ,\mathcal A ,P)$ into $X,\,\int _\Omega F d P$ is convex.

$P3$::

For any integrably bounded, weak* compact-valued correspondence $F$ from $(\Omega ,\mathcal A ,P)$ to $X,\,\int _\Omega F d P$ is weak* compact.

$P4$::

Let $Y$ be a metric space and $F$ a correspondence from $\Omega \times Y$ to $X$, such that for any fixed $y\in Y,\,F(\cdot , y)$ is a measurable, weak* compact-valued correspondence from $\Omega$ to $X$. If $F(\omega ,y)$ is upper hemicontinuous on $Y$ for each fixed $i$, then $\int _\Omega F(\omega ,y) d P$ is weakly upper hemicontinuous on $Y$.³⁵

Appendix B

One can now prove the following lemma which is essential in the proof of Lemma 1.

Lemma 3

Let $X,\,Y$ be two nonempty compact metric spaces. Let $F$ be a measurable correspondence with nonempty closed values from an atomless probability space $(I,\mathcal I ,\lambda )$ to $Y$, and for each $i\in I$ consider a upper hemicontinuous and closed-valued correspondence $M(i, \cdot ): Y \twoheadrightarrow X$. Let $G: I\twoheadrightarrow Y \times X$ be such that $G(i)$ is the graph of $M(i, \cdot )$ for all $i\in I$. If $G$ is measurable, then the correspondence $M(\cdot , F(\cdot )): I \twoheadrightarrow X$ is measurable and closed-valued.

Proof

Let $\phi : I\twoheadrightarrow Y\times X$ be a correspondence, such that $\phi (i)=F(i)\times X$ for all $i\in I$. For any open set $O$ of $Y\times X$, consider $\phi ^l(O)=\{i\in I: \phi (i) \bigcap O \ne \emptyset \}$. Let $P_X$ and $P_Y$ be the projection mappings from $Y\times X$ to $X$ and $Y$, respectively. If $P_X(O)\ne X,\,\phi ^l (O) = \emptyset \in \mathcal{I }$. If $P_X(O) = X,\,\phi ^l(O)=\{i\in I: F(i) \bigcap P_Y(O) \ne \emptyset \}= F^l(P_Y(O))\in \mathcal{I }$ too, because $F$ is measurable and $P_Y$ is continuous. Thus, $\phi$ is weakly measurable. By Proposition 3 (P1) in Appendix A, $\phi$ is also measurable because by construction $\phi$ has nonempty compact values. Let $\phi ^G$ be such that $\phi ^G (i)=\phi (i) \bigcap G(i)$ for all $i\in I$. For any fixed $i\in I$, since $M(i, \cdot )$ is upper hemicontinuous and closed-valued, $G(i)$ is closed by Proposition 3 (P3). Therefore, $\phi ^G$ is closed-valued and measurable. Furthermore, note that by construction of $\phi ^G,\,M(i, S(i))= P_X(\phi ^G(i))$ for all $i\in I$. Therefore, the continuity of $P_X$ implies that $M(\cdot , S(\cdot )): I \twoheadrightarrow X$ is measurable and closed-valued. $\square$

One can prove Lemma 1 now.

Proof of Lemma 1

For any fixed nonempty closed set $\mathcal D$ of $\mathcal{M }(A)$, let $\mathcal{D }(i)=\mathcal{D }$ for all $i\in I$. It is clear that $\mathcal D (\cdot )$ is a measurable and closed-valued correspondence. By Berge’s maximum theorem, for each $i\in I$, the joint continuity of $u_i$ on $A\times \mathcal M (A)$ implies that $B(i, \cdot )$ is upper hemicontinuous and has nonempty compact values on $\mathcal M (A)$. Thus, by Lemma 3, in order to show that $B(\cdot , \mathcal{D })$ is measurable and closed-valued, it suffices to show $G: I \twoheadrightarrow \mathcal M (A) \times A$ is measurable where $G(i)$ is the graph of $B(i, \cdot )$ for all $i\in I$.

Toward this end, for any given closed subset $C$ of $A \times \mathcal{M }(A)$, let $U_C=\{f\in \mathcal U _A: \text{ there } \text{ exists } (a, \tau )\in C, \text{ such } \text{ that, } f(a, \tau ) \ge f(a^{\prime }, \tau ), \text{ for } \text{ all } a^{\prime } \in A\}.$ It is clear that $\mathcal{G }^{-1}(U_C)=G^{l}(C)$ where $G^l(C)=\{i\in I: C \cap G(i) \ne \emptyset \}$. Thus, given $\mathcal{G }$ is measurable, it is sufficient to prove that $U_C$ is closed. Let $\{f^n\}$ be a sequence in $U_C$ that converges uniformly to $f^*$. As the uniform limit of a net of continuous real functions is still continuous (see, e.g., Aliprantis and Border (2006, Theorem 2.65)), $f^*\in \mathcal U _A$. So, for any $\varepsilon >0$, the continuity of $f^*$ implies that for any convergent sequence $\{(a^n, \tau ^n)\} \rightarrow (a^*, \tau ^*)$, there exists some $N_1 \in \mathbb N$ such that for any $n > N_1,\,|f^*(a^n, \tau ^n) - f^* (a^*, \tau ^*)| < \varepsilon /2$. The uniform convergence of $g^n$ implies that there exists some $N_2 \in \mathbb N$, such that for any $n > N_2$, and for all $(a, \tau ) \in A \times \mathcal U _A,\,|f^n(a, \tau ) - f^* (a, \tau )| < \varepsilon /2$. Therefore, for any $n > N$, where $N=\max \{N_1, N_2\}$,

$$\begin{aligned} |f^n(a^n, \tau ^n) - f^* (a^*, \tau ^*)|&\le |f^n(a^n, \tau ^n) - f^*(a^n, \tau ^n)|\\&+ |f^*(a^n, \tau ^n) - f^* (a^*, \tau ^*)| <\varepsilon . \end{aligned}$$

Thus, $f^n(a^n, \tau ^n) \rightarrow f^* (a^*, \tau ^*)$. I now show that there exists some element $(a, \tau )$ in $C$ such that $f^*(a, \tau ) \ge f^*(a^{\prime }, \tau ),$ for all $a^{\prime } \in A$. For each $n\in \mathbb N$, since $f^n\in U_C$, there exists $(a^n, \tau ^n)\in C$, such that

$$\begin{aligned} f^n(a^n, \tau ^n) \ge f^n(a^{\prime }, \tau ^n),\quad \text{ for } \text{ all } a^{\prime } \in A. \end{aligned}$$

Furthermore, the closeness of $C$ implies that there is a subsequence of $(a^n, \tau ^n)$ that converges to some $(\hat{a}, \hat{\tau })\in C$. Without loss of generality, let the subsequence be the sequence itself. It is now clear that $(\hat{a}, \hat{\tau })$ satisfies $f^*(\hat{a}, \hat{\tau }) \ge f^* (a^{\prime }, {\hat{\tau }})$ for any $a^{\prime } \in A$. Hence, $f^*\in U_C$. The proof is complete. $\square$

The next result is used in the proof of Theorem 3 to characterize expected payoffs under probabilistic beliefs. One can check that it holds, along the lines of the proof of Jara-Moroni (2012, Lemma 3.7).

Proposition 7

Let $Y$ and $X$ be compact metric spaces and $u$ a continuous real-valued function on $Y\times X$. Let $U: Y\times \mathcal{M }(X)$ be a function such that for any $(y, \mu )\in Y\times \mathcal{M }(X)$,

$$\begin{aligned} U(y, \mu )=\int \limits _{X} u(y, x) d\mu . \end{aligned}$$

$U$ is continuous on $Y\times \mathcal{M }(X)$.

Proofs of Lemma 2 and Theorem 4 are as follows.

Proof of Lemma 2

Let $\phi : \overline{con}\,(A) \twoheadrightarrow \overline{con}\,(A)$ be the Bochner integral correspondence such that for $\iota \in \overline{con}\,(A),\,\phi (\iota )= \int _I B(i, \iota ) d\lambda$. By Berge’s maximum theorem and Proposition 2 (P2) in Appendix A, $\phi$ has nonempty values. Then, together with the convexity and upper hemicontinuity results in Proposition 5 in Appendix A, one can appeal to the Fan-Glicksberg fixed point theorem to guarantee the existence of a Nash equilibrium. $\square$

Proof of Theorem 4

First, note that one can show that for any closed subset $X\subseteq \mathcal{A },\,B(\cdot , X)$ is measurable and has nonempty closed values by similar arguments in the proof of Lemma 1. Let $\{F^t\}$ be a sequence of correspondences, such that $F^t: I \twoheadrightarrow A, t \ge 0$ is given by

$$\begin{aligned} F^0(i)= A, \text{ for } \text{ all } i\in I\quad \text{ and }\quad F^t(i)= B\left( i, {Pr}^{t-1}(\mathcal{A })\right) , \text{ for } \text{ all } i\in I, \text{ if } t\ge 1, \end{aligned}$$

where ${Pr}^t(\mathcal{A }) = \int _I F^t d \lambda .$ Let $F: I \twoheadrightarrow A$ be a correspondence, such that for all $i\in I,\,F(i)= cl\text{-Lim }\,\{F^t(i)\}$. Then, one can now appeal to Proposition 5 (P1–P3) in Appendix A to complete the proof by arguments similar to proofs of Theorems 1, 2 and 3. $\square$