Preference for Flexibility and the Opportunities of Choice

A decision-maker exhibits preference for flexibility if he always prefers any set of alternatives to its subsets, even when two of them contain the same best element. Desire for flexibility can be explained as the consequence of the agent's uncertainty along a two-stage process, where he must first preselect a subset of alternatives from which to make a final choice later on. We investigate conditions on the rankings of subsets that are compatible with the following assumptions: (1) the agent is endowed with a VN-M utility function of alternatives, (2) the agent attaches a subjective probability to the survival of each subset of alternatives, and (3) the agent will make a best choice out of any set which becomes available, and ranks sets ex-ante in terms of the expected utility of the best choices within them. We first prove that any total ordering respecting set inclusion is rationalizable in these terms. This result is essentially the same obtained by Kreps (1979) under an alternative interpretation. We also show that we cannot learn anything about the underlying utilities of agents unless we impose further restrictions on the admissible distributions of survival probabilities. Then we investigate the additional consequences of assuming that the survival probabilities of individual alternatives are independently distributed. We prove that this reduces significantly the class of set rankings which can be rationalized and that then one can infer some of the characteristics of the agent's preferences. We offer a full characterization for the case of three alternatives. We also provide necessary conditions for rationalizability in the general case.