The paper is an addition to the intensionalist approach to decision theory, with emphasis on game theoretic modelling. Extensionality in games is an a priori requirement that players exhibit the same behavior in all algebraically equivalent games on pain of irrationality. Intensionalism denies that it is always irrational to play differently in differently represented (described, understood) but algebraically equivalent versions of a game. I offer a framework to integrate game non-extensionality with the more familiar idea of linguistic non-extensionality from philosophy of language, followed by applications of it based on toy examples of well-known game models. I argue that the notion of what I call “Intensional Nash Equilibrium” is, in effect, very useful in understanding human decision-making. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.