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Games 2018, 9(4), 85; https://doi.org/10.3390/g9040085

Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments

1
Department of Computing, Imperial College London, London SW7 2RH, UK
2
School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Lavasani Av., P.O. Box 19395-5746, Tehran, Iran
Current address: Department of Computer Engineering, University of Science and Culture, Bahar Ave., P.O. Box 14619-681, Tehran, Iran.
*
Author to whom correspondence should be addressed.
Received: 2 August 2018 / Revised: 11 October 2018 / Accepted: 19 October 2018 / Published: 24 October 2018
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Abstract

We show that a Bayesian game where the type space of each agent is a bounded set of m-dimensional vectors with non-negative components and the utility of each agent depends linearly on its own type only is equivalent to a simultaneous competition in m basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the ( m 1 ) -dimensional simplex. This class of m-dimensional Bayesian games, via their equivalence with uniform multigames, can model decision making in multi-environments in a variety of circumstances, including decision making in multi-markets and decision making when there are both material and social utilities for agents as in the Prisoner’s Dilemma and the Trust Game. We show that, if a uniform multigame in which the action set of each agent consists of one Nash equilibrium inducing action per basic game has a pure ex post Nash equilibrium on the boundary of its type profile space, then it has a pure ex post Nash equilibrium on the whole type profile space. We then develop an algorithm, linear in the number of types of the agents in such a multigame, which tests if a pure ex post Nash equilibrium on the vertices of the type profile space can be extended to a pure ex post Nash equilibrium on the boundary of its type profile space in which case we obtain a pure ex post Nash equilibrium for the multigame. View Full-Text
Keywords: Multidimensional Bayesian game; multigame; type space partition; Prisoner’s Dilemma; Trust Game Multidimensional Bayesian game; multigame; type space partition; Prisoner’s Dilemma; Trust Game
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Edalat, A.; Hossein Ghorban, S.; Ghoroghi, A. Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments. Games 2018, 9, 85.

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