The term “belief function” is generally used to refer to a class of capacities that can be viewed as representing ambiguity averse preferences. This paper introduces a notion of a strategic product integral for belief functions, and presents game-theoretic applications. The integral suitably captures independence of strategies in a game-theoretic setting, but is shown to be different from the well-known Choquet integral of an appropriate product capacity. The resulting equilibrium definitions incorporate stronger consistency conditions than those required by previous notions of equilibria under ambiguity defined by profiles of general capacities.