  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.
