Dutch book rationality conditions for conditional preferences under ambiguity

We study preference relations on conditional gambles of a decision maker acting under ambiguity. Dutch book rationality conditions are provided under a linear utility scale, encoding either an optimistic or a pessimistic attitude towards uncertainty. These conditions characterize possibly incomplete preferences representable by totally alternating or monotone conditional functionals. In general, the uniqueness of the representation is not guaranteed, but it can be obtained by adding the hypothesis of existence of a conditional fair price for every conditional gamble. The given rationality conditions have a betting scheme interpretation relying on “penalty fees” for betting on strict preference comparisons.