On random stable partitions

Abstract

It is well known that the one-sided stable matching problem (“stable roommates problem”) does not necessarily have a solution. We had found that, for the independent, uniformly random preference lists, the expected number of solutions converges to $$e^{1/2}$$ as n, the number of members, grows, and with Rob Irving we proved that the limiting probability of solvability is below $$e^{1/2}/2$$ , at most. Stephan Mertens’s extensive numerics compelled him to conjecture that this probability is of order $$n^{-1/4}$$ . Jimmy Tan introduced a notion of a stable cyclic partition, and proved existence of such a partition for every system of members’ preferences, discovering that presence of odd cycles in a stable partition is equivalent to absence of a stable matching. In this paper we show that the expected number of stable partitions with odd cycles grows as $$n^{1/4}$$ . However the standard deviation of that number is of order $$n^{3/8}\gg n^{1/4}$$ , i.e. too large to conclude that the odd cycles exist with probability $$1-o(1)$$ . Still, as a byproduct, we show that with probability $$1-o(1)$$ the fraction of members with more than one stable “predecessor” is of order $$n^{-1/2+o(1)}$$ . Furthermore, with probability $$1-o(1)$$ the average rank of a predecessor in every stable partition is of order $$n^{1/2}$$ . The likely size of the largest stable matching is $$n/2-O(n^{1/4+o(1)})$$ , and the likely number of pairs of unmatched members blocking the optimal complete matching is $$O(n^{3/4+o(1)})$$ .