Two-sided matching with partial information

Abstract

The traditional model of two-sided matching assumes that all agents fully know their own preferences. As markets grow large, however, it becomes impractical for agents to precisely assess their rankings over all agents on the other side of the market. We propose a novel model of two-sided matching in which agents are endowed with known partially ordered preferences and unknown true preferences drawn from known distributions consistent with the partial order. The true preferences are learned through interviews, revealing the pairwise rankings among all interviewed agents, performed according to a centralized interview policy, i.e., an algorithm that adaptively schedules interviews. Our goal is for the policy to guarantee both stability and optimality for a given side of the market, with respect to the underlying true preferences of the agents. As interviews are costly, we seek a policy that minimizes the number of interviews. We introduce three minimization objectives: (very weak) dominance, which minimizes the number of interviews for any underlying true preference profile; Pareto optimality, which guarantees that no other policy dominates the given policy; and optimality in expectation with respect to the preference distribution. We formulate our problem as a Markov decision process, implying an algorithm for computing an optimal-in-expectation policy in time polynomial in the number of possible preference orderings (and thus exponential in the size of the input). We then derive structural properties of dominant policies which we call optimality certificates. We show that computing a minimum optimality certificate is NP-hard, suggesting that optimal-in-expectation and/or Pareto optimal policies could be NP-hard to compute. Finally, we restrict attention to a setting in which agents on one side of the market have the same partially ordered preferences (but potentially distinct underlying true preferences), and in which agents must interview before matching. In this restricted setting, we show how to leverage the idea of minimum optimality certificates to design a computationally efficient interview-minimizing policy. This policy works without knowledge of the distributions and is dominant (and so is also Pareto optimal and optimal-in-expectation).