Decision theory and game theory are both interdisciplinary domains that focus on modeling and studying decision-making processes. On the one hand, decision theory aims to account for the possible behaviors of an agent with respect to an uncertain situation. It thus provides several frameworks to describe the decision-making processes in this context, including that of belief functions. On the other hand, game theory focuses on multi-agent decisions, typically with probabilistic uncertainty (if any), hence the so-called class of Bayesian games. In this paper, we extend Bayesian games to the theory of belief functions. We obtain a more expressive class of games we refer to as Bel games; it make it possible to better capture human behaviors with respect to lack of information. Next, we prove an extended version of the so-called Howson-Rosenthal's theorem, showing that Bel games can be turned into games of complete information, i.e., without any uncertainty. Doing so, we embed this class of games into classical game theory and thereby enable the use of existing algorithms. Using the Coq proof assistant, we formalize a theory of belief functions, and formally verify three different proofs of Howson-Rosenthal's theorem.