Prior Independent Equilibria and Linear Multi-dimensional Bayesian Games
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We show that a Bayesian strategy map profile is a Bayesian Nash Equilibrium independent of any prior if and only if the Bayesian strategy map profile, evaluated at any type profile, is the Nash equilibrium of the so-called local deterministic game corresponding to that type profile. We call such a Bayesian game type-regular. We then show that an m-dimensional n-agent Bayesian game whose utilities are linearly dependent on the types of the agents is equivalent, following a normalisation of the type space of each agent into the (m-1)-simplex, to a simultaneous competition in nm so-called basic n-agent games. If the game is own-type-linear, i.e., the utility of each player only depends linearly on its own type, then the Bayesian game is equivalent to a simultaneous competition in m basic n-agent games, called a multi-game. We then prove that an own-type-linear Bayesian game is type-regular if it is type-regular on the vertices of the (m-1)-simplex, a result which provides a large class of type-regular Bayesian maps. The class of m-dimensional own-type-linear Bayesian games can model, via their equivalence with multi-games, simultaneous decision-making in m different environments. We show that a two dimensional own-type-linear Bayesian game can be used to give a new model of the Prisoner's Dilemma (PD) in which the prosocial tendencies of the agents are considered as their types and the two agents play simultaneously in the PD as well as in a prosocial game. This Bayesian game addresses the materialistic and the prosocial tendencies of the agents. Similarly, we present a new two dimensional Bayesian model of the Trust game in which the type of the two agents reflect their prosocial tendency or trustfulness, which leads to more reasonable Nash equilibria. We finally consider an example of such multi-environment decision making in production by several companies in multi-markets.
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