Robust and Approximately Stable Marriages Under Partial Information

Abstract

We study the stable marriage problem in the partial information setting where the agents, although they have an underlying true strict linear order, are allowed to specify partial orders either because their true orders are unknown to them or they are unwilling to completely disclose the same. Specifically, we focus on the case where the agents are allowed to submit strict weak orders and we try to address the following questions from the perspective of a market-designer: (i) How can a designer generate matchings that are robust—in the sense that they are “good” with respect to the underlying unknown true orders? (ii) What is the trade-off between the amount of missing information and the “quality” of solution one can get? With the goal of resolving these questions through a simple and prior-free approach, we suggest looking at matchings that minimize the maximum number of blocking pairs with respect to all the possible underlying true orders as a measure of “goodness” or “quality”, and subsequently provide results on finding such matchings. In particular, we first restrict our attention to matchings that have to be stable with respect to at least one of the completions (i.e., weakly-stable matchings) and show that in this case arbitrarily filling-in the missing information and computing the resulting stable matching can give a non-trivial approximation factor for our problem in certain cases. We complement this result by showing that, even under severe restrictions on the preferences of the agents, the factor obtained is asymptotically tight in many cases. We then investigate a special case, where only agents on one side provide strict weak orders and all the missing information is at the bottom of their preference orders, and show that in this special case the negative result mentioned above can be circumvented in order to get a much better approximation factor; this result, too, is tight in many cases. Finally, we move away from the restriction on weakly-stable matchings and show a general hardness of approximation result and also discuss one possible approach that can lead us to a near-tight approximation bound.