Abstract
We reexamine the existence of stable solutions in a class of three-sided matching problems previously studied by Zhang and Zhong (J Comb Optim 42:928–245, 2021). The sets of participants are U, V, and W. Agents in U have strict preferences defined on V, agents in V have strict preferences defined on W, and agents in W have strict preferences defined on \(U\times V\). In this framework, we show that a weakly stable matching may not exist.
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Notes
Under alternative assumptions on preferences, there is a vast literature that focuses on the existence of stable outcomes in three-sided matching problems (see, for instance, Danilov 2003; Boros et al. 2004; Eriksson et al. 2006; Biró and McDermid 2010; Huang 2010; Zhang et al. 2019; Zhong and Bai 2019; Pashkovich and Poirrier 2020; Lam and Plaxton 2019, 2021).
In our example, \((u_1,v_1)\notin M\vert _{U\times V}\) and \((v_1,w_1)\notin M\vert _{V\times W}\). Moreover, \((u_1,v_4,w_4)\in M\) and agent \(u_1\) never proposes to \(v_1\). In the proof of Theorem 3 in Zhang and Zhong (2021, page 933) it is argued that these properties imply that \(v_4 \succ _{u_1} v_1\) (see the “case (3)" in the proof of this result). Unfortunately, this is not necessarily true, as in Example 1 we have that \(v_1 \succ _{u_1} v_4\). Notice that the reason why agent \(u_1\) never proposes to \(v_1\) is that \(v_1\) was inactive when \(u_1\) wanted to propose to her.
The 3MC-CYC instance with no weakly stable matchings (implicitly) constructed by Lam and Plaxton (2019, 2021) has 90 agents on each market side. Recently, Lerner (2022) provides an example of a 3MC-CYC instance with no weakly stable matchings and 20 agents on each market side (cf., Lerner and Lerner (2022)).
Zhang and Zhong (2021, Theorem 2) shown that any problem \([U,V,W, (\succ _h)_{h\in U\cup V}, (\succ _{U,w}, \succ _{V,w})_{w\in W}]\) has a stable matching.
Let \([U,V,W, (\succ _h)_{h\in U\cup V}, (\succ _{U,w}, \succ _{V,w})_{w\in W}]\) be a three-sided matching problem with separable preferences and \([U,V,W, (\succ _h)_{h\in H}]\) be the associated three-sided problem with mixed preferences, in which every linear order \(\succ _w\) is strongly \({\mathrm{UV}}\)-lexicographic and represented by \((\succ _{U,w},\succ _{V,w})\). Given a matching \(\overline{M}\) that is stable in the problem \([U,V,W, (\succ _h)_{h\in U\cup V}, (\succ _{U,w}, \succ _{V,w})_{w\in W}]\) but blocked by a triplet \((\overline{u},\overline{v},\overline{w})\) in \([U,V,W, (\succ _h)_{h\in H}]\), Zhang and Zhong (2021) claim that \(\overline{v} \succ _{V,\overline{w}} \overline{M}_V(\underline{w})\) (see the item (2) in the “necessity" part of the proof of Theorem 4 in Zhang and Zhong (2021, page 935)). Unfortunately, this is not necessarily true. Indeed, by choosing \(\overline{M}=M\) and \((\overline{u},\overline{v},\overline{w})=(u_1,v_1,w_1)\) in Example 2, we have that \(\overline{M}_V(\underline{w}) \succ _{V,\overline{w}} \overline{v}\) because of \(v_2 \succ _{V,w_1} v_1\). Essentially, since \(\succ _{\overline{w}}\) is \({\mathrm{UV}}\)-lexicographic, \((\overline{u},\overline{v}) \succ _{\underline{w}} \overline{M}(\overline{w})\) does not necessarily imply that \(\overline{v} \succ _{V,\overline{w}} \overline{M}_V(\underline{w})\).
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Arenas, J., Torres-Martínez, J.P. Reconsidering the existence of stable solutions in three-sided matching problems with mixed preferences. J Comb Optim 45, 62 (2023). https://doi.org/10.1007/s10878-023-00990-2
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DOI: https://doi.org/10.1007/s10878-023-00990-2