An equilibrium existence result for games with incomplete information and indeterminate outcomes
type: Item
instance of: scholarly article
title: An equilibrium existence result for games with incomplete information and indeterminate outcomes
zbMATH Open document ID: 1106.91003
author: Erik J. Balder
published in: Journal of Mathematical Economics
publication date: 2004-08-19T00:00:00Z
review text: 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 i…
Mathematics Subject Classification ID: 91A10
Mathematics Subject Classification ID: 91A06
Mathematics Subject Classification ID: 91A28
zbMATH DE Number: 2092963
zbMATH Keywords: Bayesian-Nash equilibrium
reviewed by: Tadeusz Radzik
MaRDI profile type: Publication
full work available at URL: j.jmateco.2003.11.011
OpenAlex ID: W2005252991
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DOI: 10.1016/J.JMATECO.2003.11.011
description: scientific article; zbMATH DE number 2092963