Models for decision-making under uncertainty use probability distributions to represent variables whose values are unknown when the decisions are to be made. Often the distributions are estimated with observed data. Sometimes these variables depend on the decisions but the dependence is ignored in the decision maker's model, that is, the decision maker models these variables as having an exogenous probability distribution independent of the decisions, whereas the probability distribution of the variables actually depend on the decisions. It has been shown in the context of revenue management problems that such modeling error can lead to systematic deterioration of decisions as the decision maker attempts to refine the estimates with observed data. Many questions remain to be addressed. Motivated by the revenue management, newsvendor, and a number of other problems, we consider a setting in which the optimal decision for the decision maker's model is given by a particular quantile of the estimated distribution, and the empirical distribution is used as estimator. We give conditions under which the estimation and control process converges, and show that although in the limit the decision maker's model appears to be consistent with the observed data, the modeling error can cause the limit decisions to be arbitrarily bad. To remedy the problem, we propose data-driven learning algorithms to allow the decision maker to optimize the system directly using the resulting revenue data, thus by-passing the need to build forecasts and therefore avoiding the potentially huge losses that can result from modeling error.
Groupe d’études et de recherche en analyse des décisions