On the Integration of Shapley-Scarf Housing Markets

We study the welfare consequences of merging disjoint Shapley-Scarf housing markets. We obtain tight bounds on the number of agents harmed by integration and on the size of their losses. We show that, in the worst-case scenario, market integration may harm the vast majority of agents, and that the average rank of an agent's house can decrease (asymptotically) by 50% of the length of their preference list. We also obtain average-case results. We exactly compute the expected gains from integration in random markets, where each of the preference profiles is chosen uniformly at random. We show that, on average, market integration benefits all agents, particularly those in smaller markets. Using the expected number of cycles in the top trading cycles algorithm, we bound the expected number of agents harmed by integration. In particular, the expected fraction of agents harmed by integration is less than 50% if each market has the same size and this is below 26 (independent of the number of markets that merge). We conclude by providing a preference domain that ensures that those harmed by market integration are a minority