Decision making over necessities through the Choquet integral criterion

Since von Neuman and Morgenstern's (1944) contribution to game theory, a rational decision maker will rank risky prospects according to the celebrated Expected utility criterion. This method takes lotteries i.e. (simple) probability distributions to represent risky prospects. If the decision maker follows the vN-M axioms (e.g.Kreps (1988)) then there exists a utility function such that any probability can be resumed to a lottery having for support the best and the worst state, where the probability that he wins the bet is given by its expected utility. Probalities are precise objects to model risk, but the way they do it is incoherent (Dubois and Prade (1988)). A familiar object in fuzzy set theory is the one of necessity or its dual version a possibility. In which case the occurence of an event is given by an interval which expresses the imprecision. Nevertheless the description of risk is coherent. Our concern is to rank different necessity measures and rank them according to the Choquet Expectation criterion (Choquet (1953)). If the decision maker follows our set of axioms then there exists a fuzzy set (Zadeh (1978)) such that any necessity can be resumed to a bet on being perfectly informed of the state which occurs or being totally ignorant, where the degree of information he will get is given by its Choquet expectation.