Preference for Randomization: Ambiguity Aversion and Inequality Aversion

In Anscombe and Aumann’s (1963) domain, there are two types of mixtures. One is an ex–ante mixture, or a lottery on acts. The other is an ex–post mixture, or a state–wise mixture of acts. These two mixtures have been assumed to be indifferent under the Reversal of Order axiom. However, we argue that the difference between these two mixtures is crucial in some important contexts. Under ambiguity aversion, an ex–ante mixture could provide only ex–ante hedging but not ex–post hedging. Under inequality aversion, an ex–ante mixture could provide only ex–ante equality but not ex–post equality. For each context, we develop a model that treats a preference for ex–ante mixtures separately from a preference for ex–post mixtures. One representation is an extension of Gilboa and Schmeidler’s (1989) Maxmin preferences. The other representation is an extension of Fehr and Schmidt’s (1999) Piecewise–linear preferences. In both representations, a single parameter characterizes a preference for ex–ante mixtures. For the both representations, instead of the Reversal of Order axiom, we propose a weaker axiom, the Indifference axiom, which is a criterion, suggested in Raiffa’s (1961) critique, for evaluating lotteries on acts. These models are consistent with much recent experimental evidence in each context.