The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games

We prove that the Burkill-Cesari integral is a value on a subspace of and then discuss its continuity with respect to both the and the Lipschitz norm. We provide an example of value on a subspace of strictly containing as well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley’s one, on a subspace of . 1. Introduction Since the seminal Aumann and Shapley's book [1], it is widely recognized that the theory of value of nonatomic games is strictly linked with different concepts of derivatives. A few papers, up to the recent literature, have investigated these relations (see, e.g., [2–4]). In [1] Aumann and Shapley proved the existence and uniqueness of a value on the space , namely, the space spanned by the powers of nonatomic measures (which, under suitable hypotheses, contains, for instance, games of interest in mathematical economics such as transferable utility economies with finite types). Moreover, in [1, Theorem H], the authors provided an explicit formula for the value of games in in terms of a derivative of their “ideal” set function . To the best of our knowledge, the more general contribution so far on the link between derivatives of set functions and value theory is Mertens [3]; his results led to the proof of the existence of a value on spaces larger than . A more recent contribution on the same subject is due to Montrucchio and Semeraro [4]. The problem of the existence of a value on the whole space of absolutely continuous games (which contains ) is instead still unsolved and challenging. Therefore, proofs of the existence of a value on other subspaces of , beyond , can represent a step forward, and investigations of this kind appear to be in order. In Epstein and Marinacci [2] the question of the relation between their refinement derivative and the value was posed and a possible direction sketched; in Montrucchio and Semeraro [4], the authors applied their more general (i.e., without the nonatomicity restriction) notion of refinement derivative to the study of the value on certain spaces of games by extending the potential approach of Hart and Mas-Colell [5] to infinite games. In a previous paper [6] we had pointed out that, in a nonatomic context, the refinement derivative is connected with the classical Burkill-Cesari (BC) integral of set functions and, for BC integrable functions, the BC integral coincides with the refinement derivative at the empty set. Though less general, the BC integral is analytically more treatable. Motivated by all these facts, in [6] we have started the study of the BC