Random Utility Model with Endogenously Assigned Menus

There is a largely overlooked assumption underlying stochastic choice theory: menus are assigned exogenously, as if by a hidden randomized controlled trial. This assumption is not innocuous, because in many real-world settings menus are assigned endogenously according to the decision makers' preferences. This paper studies Random Utility Model under menu endogeneity and shows that any seemingly anomalous choice behavior can be generated by a population of rational decision makers with heterogeneous preferences facing endogenously assigned menus. To address this problem, I propose a new causal estimand: the probability that an alternative would be chosen from a given menu if that menu were presented to the entire population. I then characterize sharp bounds on this estimand using observed menu and choice frequencies. In addition, I show that when choices are observed across multiple markets with different preference distributions and a common menu-assignment rule, the causal estimands from those markets and the menu assignment rule are uniquely identified.