PROFILING PROBLEMS WITH PARTIALLY IDENTIFIED STRUCTURE

This article studies two classes of profiling problems where the relevant structural inputs to the problems may be only partially identified: that where profilees are not 'strategic' in altering their characteristics and that where profilees strategically alter their characteristics at a cost. The analysis is unified by assuming there is a utilitarian planner that attempts to optimise the sum of welfare. We compare maximin solutions and minimax-regret solutions in two contexts. An unexpected feature of the minimax-regret solution is that a minimax-regret planner may end up learning unknown features of the population it faces rather quickly relative to a maximin planner. In many profiling problems, the available empirical evidence only partially identifies the structural inputs that a planner needs to choose an optimal policy. Consider a lender who must decide whether to approve loan applications from persons with specified observed attributes. To choose an optimal policy, the lender will want to know the default rate for such persons. If the lender has previously approved loans to similar persons, he will know their default rate empirically. However, if the lender has previously denied loans to such persons, then he will not know their default rate empirically. At most, the lender can try to predict the default rate by combining assumptions with evidence on the default rates of loans granted to other types of persons. The article examines how a lender or other planner faced with a profiling problem might reasonably behave when the relevant structural inputs are only partially identified. We consider two broad contexts, called 'bottom up' and 'top down' here. Both contexts are relevant to actual policy making. In a bottom up context, the planner feels able to make only minimal assumptions about the unknown inputs. Thus, he must cope with considerable ambiguity about the nature of the optimal profiling policy. In the lending problem, the decision maker is in a bottom up context if he feels unable to make a tight prediction of the default rates that would occur if he were in the future to grant loans to types of persons who were denied loans in the past. In a top down context, the decision maker initially imposes assumptions, based on economic theory or another source, that are strong enough to determine an optimal policy. Concerned that these assumptions may not be exactly correct, he chooses a policy that is robust to small perturbations of the assumptions. In the lending problem, the decision maker is in a top down context if he feels able to make a reasonably tight prediction of the counterfactual default rates, but not an exact prediction. The article explores the implications of two established decision-making principles that are applicable in both of the above broad contexts. The maximin principle * I thank Chuck Manski, Elie Tamer and a referee for very useful comments . I especially thank the referee for expositional assistance and Chuck Manski for his help during the writing of this article. For example, I used Chuck's version for the introduction. I thank the NSF and the Vilas Trust for essential financial support. None of the above are responsible for errors, shortcomings, or views expressed.