Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferences

TL;DR

This paper demonstrates that for a three-sided matching problem with cyclic incomplete preferences, minimal instances with no weakly stable matching exist for three agents, and none exist for fewer than three, extending previous knowledge.

Contribution

It provides the first example of a minimal three-agent instance with no weakly stable matching and proves such instances do not exist for fewer than three agents.

Findings

No stable matching exists for n=3 in the cyclic incomplete preferences model.

Such examples do not exist for n<3.

Constructed examples reduce the size of previous known examples for complete preferences.

Abstract

Given $n$ men, $n$ women, and $n$ dogs, each man has an incomplete preference list of women, each woman does an incomplete preference list of dogs, and each dog does an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that each of them enters in the preference list of the corresponding agent. We do a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog which do not live together currently but each of them would become "happier" if they do. Otherwise the matching is said to be stable (a weakly stable matching in 3-DSMI-CYC problem). We give an example of this problem for $n=3$ where no stable matching exists. Moreover, we prove the absence of such an example for $n<3$. Such an example was known earlier only for $n=6$ (Biro, McDermid, 2010). The constructed examples also allows one to decrease (in two times) the size of the recently constructed analogous example for complete preference lists (Lam, Plaxton, 2019).