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.
Highlights
IntroductionWe study linear multidimensional Bayesian games in which the type of each agent is a finite dimensional real vector and the utility of each agent only depends linearly on its own type
In this paper, we study linear multidimensional Bayesian games in which the type of each agent is a finite dimensional real vector and the utility of each agent only depends linearly on its own type.Multidimensional Bayesian games have been studied by Krishna and Perry in the context of multiple object auctions whose utilities are piecewise affine maps of the types of the agents [1]
The multigame representation of a linear multidimensional Bayesian game enables us, as described above, to develop an efficient algorithm to decide if a standard uniform multigame has a compatible ex post Nash equilibrium (NE) in which case the ex post NE is computed in constant time
Summary
We study linear multidimensional Bayesian games in which the type of each agent is a finite dimensional real vector and the utility of each agent only depends linearly on its own type. Games 2018, 9, 85 a simultaneous competition in m basic games, called a uniform multigame, as introduced in [6,7], in which the utility of each agent only depends linearly on its own type and each agent plays the same strategy in all basic games. While this equivalent representation provides no reduction in the size of input parameters, i.e., it is not more concise, it does establish a new and, we argue, conceptually useful representation of the multidimensional Bayesian game. We present an outline of the main sections of the paper after deriving some simple properties of ex post NE in Section 2 which gives the motivation for our key results
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