The Impossibility of Compromise: Convexity and Uniqueness in Decision Making under Risk and Uncertainty

The main set of results of the paper shows that for expected utility preferences, agreement of two preferences on one (non-extreme) indifference class implies their equality. We show that, besides expected utility preferences under (objective) risk, this uniqueness property holds for subjective expected utility preferences under uncertainty, in both Anscombe-Aumann's (partially subjective) and Savage's (fully subjective) frameworks. For these two frameworks, we present an analogous result for beliefs, showing that if two decision makers agree on a likelihood indifference class, they must have identical probabilities. The second part of the paper shows two different sets of consequences of the uniqueness results. First, we study the preference aggregation results of the type pioneered by Harsanyi. We show that under the assumptions of those results, the existence of a weak form of agreement among the agents implies that all their preferences are identical. Then, we study the models which extend expected utility to incorporate ambiguity aversion, and show that for a class of these models a very weak condition is surprisingly equivalent to expected utility maximizing behavior.