Ricerc@Sapienza

We provide an axiomatic definition of conditional submodular capacity that allows conditioning on “null” events and is the basis for the notions of consistency and of consistent extension of a partial assessment. The same definition gives rise to an axiomatic definition of conditional submodular Choquet expected value, which is a conditional functional defined on conditional gambles, that can be expressed as the Choquet integral with respect to its restriction on conditional indicators.

In many decision problems under uncertainty, agents are only able to provide a possibly
incomplete preference relation on gambles, that can be “irrational” according to the
classical expected utility paradigm. Furthermore, agents can find it easier to express
their preference relation taking one or more specific scenarios as hypothesis. In order to
handle these situations two betting scheme rationality conditions are introduced, which
characterize those preference relations representable by a conditional Choquet expected

We manage decisions under “objective” ambiguity by considering generalized Anscombe-Aumann acts, mapping states of the world to generalized lotteries on a set of consequences. A generalized lottery is modeled through a belief function on consequences, interpreted as a partially specified randomizing device. Preference relations on these acts are given by a decision maker focusing on different scenarios (conditioning events).