A theorem of the maximin and applications to Bayesian zero-sum games (Q548068)

The authors consider a two-player parametrized zero-sum game in which a parameter \(p \in P\) affects the players' feasible strategy sets and their payoffs. Let \(X_i\) denote the set of \(i\)-th player potential strategies \((i = 1,2)\) and, for a given \(p \in P,\) let \(\varphi_i(p) \subset X_i\) be the \(i\)-th player's nonempty set of feasible strategies. Let \(u: X_1 \times X_2 \times P \rightarrow \mathbb{R}\) be the first player's payoff function. The value function \(v_1: P \rightarrow \mathbb{R}\) and the solution multimap \(\psi_1: P \multimap X_1\) for the first player are defined as \[ v_1(p) = \sup_{x_1 \in \varphi_1(p)}\,\, \inf_{x_2 \in \varphi_2(p)} u(x_1,x_2,p), \] \[ \psi_1(p) = \text{arg\,max}_{x_1 \in \varphi_1(p)}\,\, \inf_{x_2 \in \varphi_2(p)} u(x_1,x_2,p). \] The function \(v_2\) and the multimap \(\psi_2\) are defined analogously. The authors study conditions under which the functions \(v_i\) are continuous and the multimaps \(\psi_i\) are upper hemicontinuous. An application to Bayesian zero-sum games in which each player's information is viewed as a parameter, is considered.