Uncertainty and binary stochastic choice

Experimental evidence suggests that decision-making has a stochastic element and is better described through choice probabilities than preference relations. Binary choice probabilities admit a strong utility representation if there exists a utility function u such that the probability of choosing a over b is a strictly increasing function of the utility difference $$u(a) -u(b) $$ u ( a ) - u ( b ) . Debreu (Econometrica 26(3):440–444, 1958) obtained a simple set of sufficient conditions for the existence of a strong utility representation when alternatives are drawn from a suitably rich domain. Dagsvik (Math Soc Sci 55:341–370, 2008) specialised Debreu’s result to the domain of lotteries (risky prospects) and provided axiomatic foundations for a strong utility representation in which the underlying utility function conforms to expected utility. This paper considers general mixture set domains. These include the domain of lotteries, but also the domain of Anscombe–Aumann acts: uncertain prospects in the form of state-contingent lotteries. For the risky domain, we show that one of Dagsvik’s axioms can be weakened. For the uncertain domain, we provide axiomatic foundations for a strong utility representation in which the utility function represents invariant biseparable preferences (Ghirardato et al. in J Econ Theory 118:133–173, 2004). The latter is a wide class that includes subjective expected utility, Choquet expected utility and maxmin expected utility preferences. We prove a specialised strong utility representation theorem for each of these special cases.