A betting metaphor for belief functions on MV-algebras and fuzzy epistemic states

Consider the following game. Two players, Bookmaker (B) and Gambler (G) agree in betting on a finite set of events described by functions e1, . . . , en from a set of possible worlds X = {w1, . . . , wk} into [0, 1] whose realizations are unknown now and which, in the future, will be evaluated in the possible worlds of X. The events will be evaluated by the following stipulation: the two players will share a common epistemic state about the whole class of possible worlds which is represented by a map π : X → [0, 1] such that, for each wi ∈ X, π(wi) represents the feasibility degree of wi for both B and G. Therefore, given any epistemic state π and an event ei, the aggregated value of ei from π is computed by the following formula: