Signaling games with endogenous types

Abstract

This paper examines a signaling game with endogenous types in which the sender can influence the property of his private information. We propose an equilibrium selection criterion, called criterion D3, which builds on criterion D1 from signaling games without endogenous types (Cho and Kreps 1987 and Cho and Sobel 1990). We examine the selected equilibrium properties and compare them to the equilibria selected under the existing selection rules.

Appendix A. Proof of Proposition 3.4

We only prove the “only if” part. Let $\epsilon _n=(\epsilon _n(\pi ,m),\epsilon _n(a))$ be the profile of minimum probabilities for $(\pi ,m,a)\in \Pi \times {{\mathcal {M}}}\times A$. Let $G^{\epsilon _n}$ be a perturbed game of G, where $\epsilon _n\rightarrow 0$ is a profile of the minimum probabilities for each pure strategy in G. We need to construct a Nash equilibrium of $G^{\epsilon _n}$ whose outcome is close to $[\sigma ,r]$.

• (1) By U4, $\forall k$, $(\pi ,m)$ with $\pi <{{\overline{\pi }}}$ is never a weak best response. We consider a game obtained by eliminating these “never weak best responses.” Given G, define ${{\overline{G}}}$ as the signaling game in which the set of endogenous types is $\Pi =\{{{\overline{\pi }}}\}$ instead of $[0,{{\overline{\pi }}}]$. ${{\overline{G}}}$ is a monotonic signaling game where criterion D1 is generically equivalent to strategic stability (Cho and Sobel 1990).

• (2) We perturb each player’s payoffs in ${{\overline{G}}}$ according to $\epsilon _n(\pi ,m)$. Define

  $$\begin{aligned} {{\overline{u}}}_n(k,{{\overline{\pi }}},m,a)=\sum _{\pi<{{\overline{\pi }}}}\epsilon _n(\pi ,m) u(k,\pi ,m,a) +\left( 1-\sum _{\pi <{{\overline{\pi }}}}\epsilon _n(\pi ,m)\right) u(k,{{\overline{\pi }}},m,a) \end{aligned}$$

  and

  $$\begin{aligned} {{\overline{v}}}_n(k,{{\overline{\pi }}},m,a)=\sum _{\pi<{{\overline{\pi }}}}\epsilon _n(\pi ,m) v(k,\pi ,m,a) +\left( 1-\sum _{\pi <{{\overline{\pi }}}}\epsilon _n(\pi ,m)\right) v(k,{{\overline{\pi }}},m,a). \end{aligned}$$

  ${{\overline{u}}}_n$ is the expected payoff of choosing ${{\overline{\pi }}}$ subject to the constraint that all other $\pi <{{\overline{\pi }}}$ is used with minimum probabilities $\{\epsilon _n(\pi ,m)\}$. Similarly, ${{\overline{v}}}_n$ is the receiver’s expected payoff when the receiver chooses $a\in A$, while the sender chooses $\pi <{{\overline{\pi }}}$ with the minimum probabilities. Define ${{\overline{G}}}_n$ as the signaling game obtained by replacing (u, v) by $({{\overline{u}}}_n,{{\overline{v}}}_n)$. Note that ${{\overline{G}}}_n$ remains a monotonic signaling game.

• (3) Define $\epsilon _n(m\mid k)=\sum _\pi \epsilon _n(\pi ,m\mid k)$ as the minimum probability to choose m in ${{\overline{G}}}_n$ by type k sender. Let $\epsilon _n(m)=(\epsilon _n(m\mid k))_{k\in {{\mathcal {K}}}}$. Define ${{\overline{G}}}^{\epsilon _n(m),\epsilon _n(a)}_n$ as the perturbed game of ${{\overline{G}}}_n$ by forcing each player to use each action with minimum probabilities $\{\epsilon _n(m),\epsilon _n(a)\}$.

Fix a sequential equilibrium outcome $[\sigma ,r]$ that survives criterion D3. We know that $[\sigma ,r]$ survives criterion D1 in ${{\overline{G}}}$. Since ${{\overline{G}}}$ is a monotonic signaling game, $[\sigma ,r]$ is a stable outcome generically. In a generic game, a D1 outcome changes continuously with respect to the payoffs of the game. For a small $(\epsilon _n(m),\epsilon _n(a))$, there exists a sequential equilibrium surviving D1 in ${{\overline{G}}}_n$, $(\sigma _n,r_n)$, so that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert [\sigma _n,r_n]-[\sigma ,r] \Vert =0. \end{aligned}$$

Since ${{\overline{G}}}_n$ is a monotonic signaling game, $[\sigma _n,r_n]$ is a stable outcome generically. Therefore, for any small $(\epsilon _n(m),\epsilon _n(a))$, there is a Nash equilibrium $({{\overline{\sigma }}}_n,{{\overline{r}}}_n)$ in ${{\overline{G}}}_n$ so that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert [{{\overline{\sigma }}}_n,{{\overline{r}}}_n ] - [\sigma _n,r_n ] \Vert =0. \end{aligned}$$

We now move from ${{\overline{G}}}_n$ to $G^{\epsilon _n}$ by adding $(\pi ,m)$ with $\pi <{{\overline{\pi }}}$ back to the sender’s feasible actions.

We claim that $[{{\overline{\sigma }}}_n,{{\overline{r}}}_n]$ induces Nash equilibrium outcome $({{\tilde{\sigma }}}_n,{{\tilde{r}}}_n)$ in $G^{\epsilon _n}$ so that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert [{{\overline{\sigma }}}_n,{{\overline{r}}}_n ] - [{{\tilde{\sigma }}}_n,{{\tilde{r}}}_n ] \Vert =0. \end{aligned}$$

from which the conclusion of the proof follows.

By the construction of ${{\overline{G}}}_n$, the expected payoff of each player is equal to the equilibrium payoff from $[{{\overline{\sigma }}}_n,{{\overline{r}}}_n]$ in $G^{\epsilon _n}$, if $\forall \pi <{{\overline{\pi }}}$ is assigned with the minimum probability. By U4, $(\pi ,m)$ with $\pi <{{\overline{\pi }}}$ is never a best response. Thus, it is optimal for the sender to use $\pi <{{\overline{\pi }}}$ only with the minimum probability in any Nash equilibrium of $G^{\epsilon _n}$. By V1, the receiver’s response remains close to the best response for the belief in ${{\overline{G}}}_n$. Define ${{\tilde{r}}}_n(m)={{\overline{r}}}_n(m)$ and a mixed strategy ${{\tilde{\sigma }}}_n(\cdot |k)$ as

$$\begin{aligned} {{\tilde{\sigma }}}_n(\pi ,m| k)={\left\{ \begin{array}{ll} \epsilon _n(\pi ,m) &{} \text {if } \ \pi<{{\overline{\pi }}}\\ (1-\sum _{\pi <{{\overline{\pi }}}}\epsilon _n(\pi ,m)) {{\overline{\sigma }}}_n(\pi ,m|k) &{} \text {if } \ \pi ={{\overline{\pi }}}. \end{array}\right. } \end{aligned}$$

By the construction of ${{\overline{G}}}_n$, $({{\tilde{\sigma }}}_n,{{\tilde{r}}}_n)$ constitutes a Nash equilibrium in $G^{\epsilon _n}$.