Stable Matching with Uncertain Linear Preferences

TL;DR

This paper explores the computational complexity of stable matchings under various models of preference uncertainty, revealing diverse difficulty levels and highlighting the impact of uncertainty representation.

Contribution

It introduces three models of preference uncertainty in stable matching and analyzes their computational complexities, providing new insights into stability probability calculations.

Findings

Complexity varies significantly across models.

Deciding the existence of a certainly stable matching is computationally challenging.

The form of preference uncertainty greatly influences problem difficulty.

Abstract

We consider the two-sided stable matching setting in which there may be uncertainty about the agents' preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model --- in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model --- for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model --- there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.