Stable Matching with Uncertain Pairwise Preferences

We study a two-sided matching problem under preferences, where the agents have independent pairwise comparisons on their possible partners and these preferences may be uncertain. Preferences may be intransitive and agents may even have cycles in their preferences; e.g. an agent a may prefer b to c, c to d, and d to b, all with probability one. If an instance has such a cycle, then there may not exist any matching that is stable with positive probability. We focus on the computational problems of checking the existence of possibly and certainly stable matchings, i.e., matchings whose probability of being stable is positive or one, respectively. We show that finding possibly stable matchings is NP-hard, even if only one side can have cyclic preferences. On the other hand we show that the problem of finding a certainly stable matching is polynomial-time solvable if only one side can have cyclic preferences and the other side has transitive preferences, but that this problem becomes NP-hard when both sides can have cyclic preferences.