We study the problem of eliciting the preferences of a decision-maker through a moderate number of pairwise comparison queries to make them a high quality recommendation for a specific problem. We are motivated by applications in high stakes domains, such as when choosing a policy for allocating scarce resources to satisfy basic needs (e.g., kidneys for transplantation or housing for those experiencing homelessness) where a consequential recommendation needs to be made from the (partially) elicited preferences. We model uncertainty in the preferences as being set based and} investigate two settings: a) an offline elicitation setting, where all queries are made at once, and b) an online elicitation setting, where queries are selected sequentially over time in an adaptive fashion. We propose robust optimization formulations of these problems which integrate the preference elicitation and recommendation phases with aim to either maximize worst-case utility or minimize worst-case regret, and study their complexity. For the offline case, where active preference elicitation takes the form of a two and half stage robust optimization problem with decision-dependent information discovery, we provide an equivalent reformulation in the form of a mixed-binary linear program which we solve via column-and-constraint generation. For the online setting, where active preference learning takes the form of a multi-stage robust optimization problem with decision-dependent information discovery, we propose a conservative solution approach. Numerical studies on synthetic data demonstrate that our methods outperform state-of-the art approaches from the literature in terms of worst-case rank, regret, and utility. We showcase how our methodology can be used to assist a homeless services agency in choosing a policy for allocating scarce housing resources of different types to people experiencing homelessness.
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