On the Convergence of Swap Dynamics to Pareto-Optimal Matchings

We study whether Pareto-optimal stable matchings can be reached via pairwise swaps in one-to-one matching markets with initial assignments. We consider housing markets, marriage markets, and roommate markets as well as three different notions of swap rationality. Our main results are as follows. While it can be efficiently determined whether a Pareto-optimal stable matching can be reached when defining swaps via blocking pairs, checking whether this is the case for all such sequences is computationally intractable. When defining swaps such that all involved agents need to be better off, even deciding whether a Pareto-optimal stable matching can be reached via some sequence is intractable. This confirms and extends a conjecture made by Damamme et al. (2015), who have furthermore shown that convergence to a Pareto-optimal matching is guaranteed in housing markets with single-peaked preferences. We show that in marriage and roommate markets, single-peakedness is not sufficient for this to hold, but the stronger restriction of one-dimensional Euclidean preferences is.