On random stable matchings: cyclic matchings with strict preferences and two-side matchings with partially ordered preferences

Consider a cyclically ordered collection of $r$ equinumerous agent sets with strict preferences of every agent over the agents from the next agent set. A weakly stable cyclic matching is a partition of the set of agents into disjoint union of $r$-long cycles, one agent from each set per cycle, such that there are no destabilizing $r$-long cycles, i.e. cycles in which every agent strictly prefers its successor to its successor in the matching. Assuming that the preferences are uniformly random and independent, we show that the expected number of stable matchings grows with $n$ (cardinality of each agent set) as $(n\log n)^{r-1}$. We also consider a bipartite stable matching problem where preference list of each agent forms a partially ordered set. Each partial order is an intersection of several, $k_i$ for side $i$, independent, uniformly random, strict orders. For $k_1+k_2>2$, the expected number of stable matchings is analyzed for three, progressively stronger, notions of stability. The expected number of weakly stable matchings is shown to grow super-exponentially fast. In contrast, for $\min(k_1,k_2)>1$, the fraction of instances with at least one strongly stable (super-stable) matching is super-exponentially small.