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 intransitive even cycles in preferences; e.g. an agent prefer b to c, c d, d b, all with probability one. If instance has such cycle, then there not exist any that is stable positive probability. focus computational problems of checking existence possibly certainly matchings, i.e., matchings whose being or one, respectively. show finding NP-hard, if only one side can cyclic preferences. On other hand we polynomial-time solvable transitive but this becomes NP-hard when both sides