Dynamic Matching Markets and Voting Paths

We consider a matching market, in which the aim is to maintain a popular matching between a set of applicants and a set of posts, where each applicant has a preference list ranking a subset of posts in some order of preference. A matching M is popular if there is no other matching M′ such that the number of applicants who prefer their partners in M′ to M exceeds the number of applicants who prefer their partners in M to M′. Popular matchings M are stable in the sense that no coalition of applicants can force a switch to another matching M′ by requesting a pairwise election between M and M′ with one vote per applicant (i.e. an up-or-down vote). The setting here is dynamic: applicants and posts can enter and leave the market, and applicants can also change their preferences arbitrarily. After any change, the matching we have in place may no longer be popular, in which case we are required to update it. However, we cannot simply recompute a popular matching from scratch after every such change. This is because there are instances in which no popular matching is directly more popular than the existing non-popular matching, and hence there would be no consensus for the applicants to agree to the switch. The aim then is to find a voting path, which is a sequence of matchings, each more popular than its predecessor, that ends in a popular matching. In this paper, we show that, as long as some popular matching exists, there exists a 2-step voting path from any given matching to some popular matching. Furthermore, given any popular matching, we show how to find a shortest-length such voting path in linear time.