On the behavior of separating equilibria of signaling games with a finite set of types as the set of types becomes dense in an interval
A signaling model consists of two players, player 1 and 2, the type space A (\(\subset [m,M]\subset {\mathfrak R})\) for player 1, and the action space \({\mathfrak R}\) (the real numbers) and a utility function \(u^ i\) for each player. A strategy of player 1 is a measurable function \(\tau: A\to {\mathfrak R}\), and a strategy of player 2 is a measurable function \(\zeta: {\mathfrak R}\to {\mathfrak R}\) from player 1's action space to player 2's action space. A separating Nash equilibrium is a strategy bundle \((\tau,\zeta)\) such that (1) \(\tau\) is 1-1, and such that (2) \(\tau(\alpha)\in_{y} u^ 1(\alpha,y,\zeta (y))\) for all \(\alpha\in A\), and \(\zeta(y)\in_{z} u^ 2(\tau^{-1}(y),y,z)\) for all \(y\in \tau (A)\). Denote by \(U(\alpha,{\hat \alpha},y)\) player 1's utility, when his type is \(\alpha\) and player 2 reacts to 1's action y believing that 1's type is \({\hat \alpha}\). Then a 1-1 strategy \(\tau\) is a separating Nash equilibrium strategy for player 1 if for all \(\alpha\in A\), \[ (*)\quad U(\alpha,\alpha,\tau (\alpha))\geq U(\alpha,\tau^{-1}(y),y)\quad for\quad all\quad y\in \tau (A). \] Consider a sequence of signaling models defined by a sequence of finite sets of types \(\{A_ n\}_ n\), \(A_ n\subset [m,M]\), \(A_ n\subset A_{n+1}\), and \(\overline{\cup_{n}A_ n}=[m,M]\). The following theorem implies that any limit of separating equilibrium path is a separating equilibrium path of the limit game: Let \(\{\tau_ n\}_ n\) be a sequence of 1-1 functions \(\tau_ n:\) \(A_ n\to {\mathfrak R}\) such that for each n, (*) is satisfied for all \(\alpha \in A_ n\). Suppose \(\cup_{n}\tau_ n(A_ n)\) is bounded. Then the sequence has a convergent subsequence. Such convergence is uniform, and the limit \(\tau\) has a unique extension to [m,M], which is continuous everywhere (except at a certain point), and which satisfies (*) for all \(\alpha\in \cup_{n}A_ n\).
scientific article; zbMATH DE number 4072753