If it is surely better, do it more? Implications for preferences under ambiguity

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe-Aumann) mixtures of two acts f and g, \left\{ \alpha f+(1-\alpha)g:\alpha\in[0,1]\right\}. We propose a preference condition, Monotonicity in Optimal Mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between Monotonicity in Optimal Mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg (1961) paradoxes. We find that Monotonicity in Optimal Mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class of Ghirardato and Marinacci, 2001). This implies, for example, that maxmin expected utility (MEU) preferences (Gilboa and Schmeidler, 1989) are consistent with Monotonicity in Optimal Mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between Monotonicity in Optimal Mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences (Klibanoff, Marinacci and Mukerji, 2005) can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which Monotonicity in Optimal Mixtures places upper bounds on the intensity of ambiguity averse behavior.