Decisions under Risk and Partial Knowledge Modelling Uncertainty and Risk Aversion

We deal with decisions under risk starting from a partial preference relation on a finite set of generalized convex lotteries, that are random quantities equipped with a convex capacity. A necessary and sufficient condition (Choquet rationality) is provided for its representability as a Choquet expected utility of a strictly increasing utility function. The restriction to concave utility functions is discussed. Moreover, we show that this condition, with or without the constraint of concavity for the utility function, assures the extension of the preference relation and it actually guides the decision maker in the extension process.