An equilibrium existence result for games with incomplete information and indeterminate outcomes
The author starts with considering an \(n\)-person noncooperative incomplete information game \(\Gamma\) in the sense of Harsanyi, with the following structure: The set of the players is \(I = \{1,\dots, n\}\). Each player \(i\in I\) has a nonempty separable metric action space \(A_i\) and an abstract measurable space \((T_i, \mathcal{T}_i)\). Next, a probability measure \(\tau\) on the associated product \((T,\mathcal{T})\) of the spaces \((T_i, \mathcal{T}_i)\) is given. Each player \(i\) has a payoff integrand \(v_i : T \times A \rightarrow \mathbb{R}\) where \(T := (T_1,\dots,T_n)\) and \(A := (A_1,\ldots,A_n)\). According to \(\tau\) a vector \(t = (t_1, \dots, t_n) \in T\) is chosen and player \(i\) is informed only on the value of the component \(t_i\) of \(t\), \(i\in I\). A (mixed) private information-based strategy \(\delta_i = \delta_i(t_i,\cdot)\) for player \(i\) is any transition probability from \(T_i\) into \(A_i\). Under the strategy profile \(\delta = (\delta_1,\dots, \delta_n)\), the expected payoff for player \(i\) is defined as \( J_i(\delta) := \int_T \{ \int_A v_i(t,a)\widetilde{\delta}(t) (da)\}\tau (dt), \) where \(t = (t_1, \dots, t_n)\), \(a = (a_1, \dots, a_n)\) and \(\widetilde{\delta}\) is the product probability measure on \(A\) determined by \(\widetilde{\delta}(B_1\times \dots \times B_n) := \prod_{i\in I}\delta_i(t_i,B_i)\) for all \((B_1,\dots, B_n) \in A\). For such incomplete information game \(\Gamma\), a Bayesian-Nash equilibrium strategy profile is defined to be a vector \(\delta^* = (\delta_1^*,\dots, \delta_n^*)\) such that for every \(i\in I\), \[ J(\delta^*)=\max_{\delta_i}J((\delta_1^*,\dots,\delta_{i-1}^*,\delta_i,\delta_{i+1}^*, \dots, \delta_n^*). \] The first result obtained about the existence of a Bayesian-Nash equilibrium strategy profile in the game \(\Gamma\) when the spaces \(A_i\) are compact, \(\tau\) is absolutely continuous with respect to \(\times_{i\in I}\tau_i\) of the \(T_i\)-marginals of \(\tau\), and if the functions \(v_i(t,\cdot)\) are continuous. It generalizes an earlier result of Milgrom and Weber. Next the author extends the model of the game \(\Gamma\) to a game with incomplete information and indeterminate outcomes and communication by adding an outcome correspondence \(\Theta : A \rightarrow 2^{\Omega}\) and modifying player \(i\)'s payoff integrand to the form \(u_i : T\times A \times \Omega \rightarrow \mathbb{R}\). For such extended game model, it is proved (under some additional ``continuity'' assumptions on \(\Theta\) an \(u_i\)) that there exists an ``optimal'' strategy profile \(\eta^* = (\eta_1^*,\ldots, \eta_n^*)\) called an announcement-action devise profile with a special property (making truthful type announcement). The profile \(\eta^*\) is defined as a counterpart of a Bayesian-Nash equilibrium \(\delta^*\) for the previous model. This second theorem is the main result of the paper and extends a recent equilibrium existence result of \textit{M. O. Jackson}, \textit{L. K. Simon, J. M. Swinkels} and \textit{W. R. Zame} [Econometrica 70, No. 5, 1711--1740 (2002; Zbl 1106.91348)] to a situation with more abstract type spaces.
scientific article; zbMATH DE number 2092963
scientific article