We introduce the notion of a multi-game to model the interaction of a finite number of agents playing a finite number of basic games simultaneously. Each agent allocates to each game a specific weight representing the fraction of its investment in that game. An agent’s weights for the basic games are considered as private information or types with a joint probability distribution describing the agents’ beliefs about their opponents’ weights. Multi-games thus form a class of Bayesian games which can model decision making in multi-environments in a variety of circumstances. Given a set of pure Nash equilibria (NE), one for each basic game in a multi-game, we construct a pure Bayesian NE for the multi-game. We then look at the class of so-called uniform multi-games in which each agent is constrained to play the same strategy in all basic games. A notion of regularity for uniform multi-games is developed and it is shown that a regular multi-game has a pure Bayesian NE that is computed in constant time. We then develop an algorithm, linear in the number of types of the agents, which tests if a multi-game is regular in which case it returns a pure Bayesian NE for the multi-game.