Random Popular Matchings with Incomplete Preference Lists

For a set A of n people and a set B of m items, with each person having a preference list that ranks some items in order of preference, we consider the problem of matching every person with a unique item. A matching M is called popular if for any other matching M ′, the number of people who prefer M to M ′ is not less than the number of those who prefer M ′ to M . For given n and m, consider the probability of existence of a popular matching when each person’s preference list is independently and uniformly generated at random. Previously, Mahdian showed that in the case that people’s preference lists are strict (containing no ties) and complete (containing all items in B), if α = m/n > α∗, where α∗ ≈ 1.42 is the root of equation x = e, then a popular matching exists with high probability; and if α < α∗, then a popular matching exists with low probability. The point α∗ can be regarded as a transition point, at which the probability of existence of a popular matching rises from asymptotically zero to asymptotically one. In this paper, we investigate transition points in more general cases when people’s preference lists are not complete. In particular, we show that in the case that each person has a preference list of length k, if α > αk, where αk ≥ 1 is the root of equation xe−1/2x = 1 − (1 − e−1/x)k−1, then a popular matching exists with high probability; and if α < αk, then a popular matching exists with low probability.