  We study $n$-agent Bayesian Games with $m$-dimensional vector types and
linear payoffs, also called Linear Multidimensional Bayesian Games. This class
of games is equivalent with $n$-agent, $m$-game Uniform Multigames. We
distinguish between games that have a discrete type space and those with a
continuous type space. More specifically, we are interested in the existence of
pure Bayesian Nash Equilibrium for such games and efficient algorithms to find
them. For continuous priors we suggest a methodology to perform Nash
Equilibrium search in simple cases. For discrete priors we present algorithms
that can handle two actions and two players games efficiently. We introduce the
core concept of threshold strategy and, under some mild conditions, we show
that these games have at least one pure Bayesian Nash Equilibrium. We
illustrate our results with several examples like Double Game Prisoner Dilemna
(DGPD), Chicken Game and Sustainable Adoption Decision Problem (SADP).
