Essays on Random Choice

Abstract

Chapter 1 introduces and axiomatizes a new class of representations for incomplete preferences called confidence models, which describe decision makers who behave as if they have probabilistic uncertainty over their true preferences, and are only willing to express a binary preference if it is sufficiently likely to hold. Confidence models provide a natural way to connect incomplete preferences with stochastic choice; this connection is characterized by a simple condition that serves to identify the behavioral content of incomplete preferences. Chapter 2 studies random choice rules over finite sets that obey regularity but potentially fail to satisfy all of the Block-Marschak inequalities. Such random choice rules can be represented by capacities on the space of preferences. The higher-order Block-Marschak inequalities are shown to be related to the degree of monotonicity that can be achieved by a capacity representation. Finally, Chapter 3 shows that failures of uniqueness for random utility representations are widespread. Uniqueness can be restored by introducing a finite state space and considering random choice over Savage acts. A representation is characterized in which acts are chosen according to the probability that they are optimal in every state.