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Credibility, efficiency, and stability: a theory of dynamic matching markets

Abstract

We introduce a dynamic framework to analyze two-sided matching interactions that occur repeatedly over time, such as teacher–student matching or hospital–intern markets in Britain. We propose a dynamic concept of credible group stability and show that implementing a men-optimal stable matching in each period is credibly group-stable. The result holds for a women-optimal stable matching. A credibly group-stable dynamic matching is individually rational and immune to any defensible group deviations with an appropriate definition of defensibility. We obtain several policy implications for market design. Moreover, a sufficient condition for Pareto efficiency is given for finitely repeated markets.

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Notes

  1. See http://www.cmcpgh.org. Tuition does not play a decisive role in matching, because the tuition is not differentiated by teachers or students. The web was accessed on October 22, 2015.

  2. The market was changed with the introduction of the UK Foundation Programme in 2005. See its official website (http://www.foundationprogramme.nhs.uk/ for the details (accessed on April 29, 2019). Also see Moore et al. (2009). Moreover, Péter Biró describes the transition for Scotland to the UK Foundation Progamme in terms of matching theory (http://econ.core.hu/kutatas/jatek_resz.html. The web was accessed on April 29, 2019.

  3. See recent exceptions: Damiano and Lam (2005), Doval (2018), Kadam and Kotowski (2018a, b) for a class of two-sided matching problems, and Abdulkadiroğlu and Loertscher (2007), Ünver (2010), Bloch and Cantala (2013), Kurino (2014), Kennes et al. (2014) for another class of indivisible goods resource allocation problems. The differences from our problem are explained in Sect. 1.1 of the related literature.

  4. The word “group” is used as a synonym of coalition that is a collection of agents. The use depends on which solution concept is used. Coalition is used for the characteristic function approach such as the core, while group is for the non-characteristic function approach such as group stability.

  5. Damiano and Lam (2005) calls an outcome path a matching plan.

  6. See Kadam and Kotowski (2018a, b): for non-time-separable utilities.

  7. This kind of approach has been taken in the matching literature, as we discussed in Sect. 1.1.

  8. Corbae et al. (2003) also consider this kind of contingent plan. Moreover, Doval (2018) has used our concept of dynamic matching in her dynamic matching model.

  9. Non-characteristic function approaches have been widely used in the many-to-one, many-to-many matching problems and network games, as we discussed in Sect. 1.1.

  10. A special case of pair deviations (the group consisting of one man and one woman) coincides with the one considered by Roth and Vande Vate (1990) where a new matching \({\hat{\mu }}\) is obtained from \(\mu \) by satisfying the blocking pair. The basic idea is also the same as Corbae et al. (2003). In addition, this notion is different from enforcement used to define a bargaining set in Klijn and Masso (2003).

  11. The term “group stability” used in many-to-one or many-to-many matching problems is different from ours, although we adopt the same approach of non-characteristic function. See Sect. 1.1.

  12. This example is from Damiano and Lam (2005).

  13. This is adapted from an example in footnote 5 in Corbae et al. (2003).

  14. We implicitly assume that \(u_{m_{2}}^{0}(w_{3})<u_{m_{2}}^{0}(w_{2})\) so that we have a unique stable matching.

  15. Even if we require the group deviation \((B,{\bar{\varphi }})\) to be history-independent, all of our results are not affected. In this case, since the set of the modified defensible group deviations is larger than that of the original one, the set of credibly group-stable dynamic matchings that use the modified defensibility is smaller than that of the original one.

  16. See Sect. 2.2 for the definition of weak stability. The idea of our credible group stability is similar to the bargaining set. The definition of our group deviation is different from that of enforcement which is used to define the Zhou ’s bargaining set as formalized by Klijn and Masso (2003) for a marriage model, and thus there is no obvious relationship between our credible group stability and the bargaining set. However, Klijn and Masso (2003) show that the set of weakly stable and weakly efficient matchings coincides with the bargaining set. Hence, the set of credible pairwise stable and weakly efficient matchings coincides with the bargaining set.

  17. This is a stronger statement than weak Pareto efficiency of stable matchings which is implied by Lemma 1-Parts (1) and (2).

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Acknowledgements

I am very grateful to Onur Kesten and Ted Temzelides, and especially to M. Utku Ünver for their guidance, support and discussions. I thank Masaki Aoyagi, Andreas Blume, Lars Ehlers, Isa E. Hafalir, Bettina Klaus, Hideo Konishi, Rene Saran, a referee and the guest editor of this journal, and seminar participants at Pittsburgh, SED 2008 5th Conference on Economic Design, and 9th International Meeting of the Society of Social Choice and Welfare for comments and discussions. I also thank Thomas Rawski for proofreading the paper. All remaining errors are my own responsibility.

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Correspondence to Morimitsu Kurino.

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This paper is based on Chapter 2 of my Ph.D. dissertation (Kurino 2009) submitted to the University of Pittsburgh.

A. Appendix: Proofs

A. Appendix: Proofs

A.1 Proposition 1

Consider a two-period dynamic market with \(M=\{m_{1},m_{2}\}\) and \(W=\{w_{1},w_{2},w_{3}\}\). The preferences are depicted in Fig. 9. In addition, the utility of being unmatched is 0 to each agent. Note that Fig. 9 just indicates the preferences for all agents, but does not show all matchings. There are 11 possible matchings. Denote \(\mu _{ij}\) by the matching in which \(m_{i}\) is matched with \(w_{j}\) and the other agents are unmatched. Denote \(\mu _{ij,k\ell }\) by the matching in which \(m_{i}\) (\(m_{k}\)) is matched with \(w_{j}\) (\(w_{\ell }\)) and the other agent is unmatched. Matching \(\mu _{U}\) is the one where all agents are unmatched. In total, we have \(121=11\times 11\) outcome paths. Out of them, we have 15 individual rational outcome paths: \((\mu _{11},\mu _{11})\), \((\mu _{11},\mu _{21})\), \((\mu _{13},\mu _{11})\), \((\mu _{13},\mu _{13})\), \((\mu _{13},\mu _{21})\), \((\mu _{13},\mu _{U})\), \((\mu _{21},\mu _{13})\), \((\mu _{21},\mu _{U})\), \((\mu _{U},\mu _{13})\), \((\mu _{U},\mu _{21})\), \((\mu _{U},\mu _{U})\); \((\mu _{21},\mu _{21})\), \((\mu _{21},\mu _{22})\), \((\mu _{22},\mu _{22})\),\((\mu _{11,22},\mu _{11,22})\). The first 11 outcome paths are blocked by the pair \((m_{2},w_{2})\) via \((\mu _{22},\mu _{22})\), and the last four are blocked by \((m_{1},w_{3})\) via \((\mu _{13},\mu _{13})\). \(\square \)

Fig. 9
figure 9

The preferences in the period market

A.2 Proposition 4

A.2.1 Part (1)

It is sufficient to show that if a matching \(\mu \) is individually rational, then [all blocking pairs are weak] \(\Leftrightarrow \) [for each pairwise deviation \((A,{\hat{\mu }})\) from \(\mu \), if \(u_{i}({\hat{\mu }})>u_{i}(\mu )\) for each \(i\in A\), then\((A,{\hat{\mu }})\) is not defensible]. Suppose that \(\mu \) is individually rational.

First, we show the direction \((\Rightarrow )\). Suppose that all blocking pairs for \(\mu \) are weak. Let \((A,{\hat{\mu }})\) be a pairwise deviation from \(\mu \) such that for each \(i\in A\), \(u_{i}({\hat{\mu }})>u_{i}(\mu )\) (condition *). Without loss of generality, take \(m\in A\cap M\). Since \(\mu \) is individually rational, \(u_{m}({\hat{\mu }})>u_{m}(\mu )\ge u_{m}(m)\). Thus, at \({\hat{\mu }}\), m is matched with some woman w. Thus, by the definition of group deviation, we have \(A=\{m,w\}\). Thus, by condition (*), the pair (mw) blocks \(\mu \). Since by our assumption all blocking pairs for \(\mu \) are weak, without loss of generality, there is \(w^{\prime }\in W\) such that

$$\begin{aligned}&u_{m}(w^{\prime })>u_{m}(w)\text { and} \end{aligned}$$
(2)
$$\begin{aligned}&(m,w^{\prime })\text { is a blocking pair for }\mu . \end{aligned}$$
(3)

Consider a group deviation \((B,{\bar{\mu }})\) from \({\hat{\mu }}\), where \(B:=\{m,w^{\prime }\}\) and \({\bar{\mu }}(m)=w^{\prime }\). Note that \(A\cap B=\{m\}\ne \emptyset \) and \(w^{\prime }\ne w\) by (2). We will show that for each \(i\in B\), \(u_{i}({\bar{\mu }})>u_{i}({\hat{\mu }})\), which means \((A,{\hat{\mu }})\) is not defensible. By the definition of pairwise deviation, under \({\hat{\mu }}\), \(w^{\prime }\) is matched with \(\mu (w)\). Then, it follows from (3) that \(u_{w^{\prime }}(m)=u_{w^{\prime }}({\bar{\mu }})>u_{w^{\prime }}(\mu )=u_{w}({\hat{\mu }})\). Moreover, it follows from (2) that \(u_{m}(w^{\prime })=u_{m}({\bar{\mu }})>u_{m}(w)=u_{m}({\hat{\mu }})\). Thus, the pairwise deviation \((A,{\hat{\mu }})\) is not defensible.

Next we show the other direction \((\Leftarrow )\). Suppose that the hypothesis is true. Let (mw) be a blocking pair for \(\mu \). Then consider the pairwise deviation \((\{m,w\},{\hat{\mu }})\) from \(\mu \) with \({\hat{\mu }}(m)=w\). Then \(u_{m}({\hat{\mu }})>u_{m}(\mu )\) and \(u_{w}({\hat{\mu }})>u_{w}(\mu )\). By our hypothesis, the pairwise deviation is not defensible. Thus there is a group deviation \((B,{\bar{\mu }})\) from \({\hat{\mu }}\) with \(\{m,w\}\cap B\ne \emptyset \) such that for each \(i\in B\), \(u_{i}({\bar{\mu }})>u_{i}({\hat{\mu }})\). Without loss of generality, take m in \(\{m,w\}\cap B\). Then,

$$\begin{aligned} u_{m}({\bar{\mu }})>u_{m}({\hat{\mu }})>u_{m}(\mu )\ge u_{m}(m). \end{aligned}$$
(4)

The last inequality follows from individual rationality of \(\mu \). By (4), m is matched with some woman \(w^{\prime }\in B\) at \({\bar{\mu }}\) with \(w^{\prime }\ne w\). Thus \(u_{m}({\bar{\mu }})=u_{m}(w^{\prime })>u_{m}(w)\ge u_{m}({\hat{\mu }})\). It remains to show that the pair \((m,w^{\prime })\) is a blocking pair for \(\mu \). First, by (4), \(u_{m}(w^{\prime })=u_{m}({\bar{\mu }})>u_{m}(\mu )\). Lastly we need to show that \(u_{w^{\prime }}(m)>u_{w^{\prime }}(\mu )\). By the definition of pairwise deviation \((\{m,w\},{\hat{\mu }})\), since \(w\ne w^{\prime }\), \(w^{\prime }\) is matched with \(\mu (w^{\prime })\) under \({\hat{\mu }}\). Then, since \(w^{\prime }\in B\), \(u_{w}(m)=u_{w^{\prime }}({\bar{\mu }})>u_{w^{\prime }}({\hat{\mu }})=u_{w^{\prime }}(\mu )\). Therefore, \((m,w^{\prime })\) blocks \(\mu \). \(\square \)

A.2.2 Part (3)

Consider a static market (Knuth 1996) with \(M=\{m_{1},m_{2},\)\(m_{3},m_{4}\}\), \(W=\{w_{1},w_{2},w_{3},w_{4}\}\) and the following preferences:

\(m_{1}\)\(m_{2}\)\(m_{3}\)\(m_{4}\)\(w_{1}\)\(w_{2}\)\(w_{3}\)\(w_{4}\)
\(\underline{w_{1}}\)\(w_{2}\)\(w_{3}\)\(\underline{w_{4}}\)\(m_{4}\)\(\underline{m_{3}}\)\(\underline{m_{2}}\)\(m_{1}\)
\(w_{2}\)\(w_{1}\)\(w_{4}\)\(w_{3}\)\(m_{3}\)\(m_{4}\)\(m_{1}\)\(m_{2}\)
\(w_{3}\)\(w_{4}\)\(w_{1}\)\(w_{2}\)\(m_{2}\)\(m_{1}\)\(m_{4}\)\(m_{3}\)
\(w_{4}\)\(\underline{w_{3}}\)\(\underline{w_{2}}\)\(w_{1}\)\(\underline{m_{1}}\)\(m_{2}\)\(m_{3}\)\(\underline{m_{4}}\)

where each column indicates the preference of an agent in the first row, all mates are acceptable in each column, and an upper mate is preferred to a lower one. Consider the matching

$$\begin{aligned} \mu =\left( \begin{array}{cccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4}\\ w_{1} &{} w_{3} &{} w_{2} &{} w_{4} \end{array}\right) , \end{aligned}$$

where each of underlined cells in the table indicates his or her partner from this matching. Matching \(\mu \) is not stable (for example, pair \((m_{2},w_{1})\) blocks it), but individually rational.

We show that \(\mu \) is credibly group-stable. Suppose for a contradiction that there is a defensible group deviation \((A,{\hat{\mu }})\) such that for each \(i\in A\), \(u_{i}({\hat{\mu }})>u_{i}(\mu )\). Note that A does not contain agents \(m_{1}\), \(m_{4}\), \(w_{2}\) nor \(w_{3}\), because each has the best mate in matching \(\mu \).

First, consider the case where A is a pair. Then, since A blocks \(\mu \), A is \((m_{2},w_{1})\), \((m_{3},w_{1})\), \((m_{2},w_{4})\), or \((m_{3},w_{4})\). Thus, we have the following four cases.

A\(\{m_{2},w_{1}\}\)\(\{m_{3},w_{1}\}\)
\({\hat{\mu }}\)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{3})\\ (m_{1}) &{} w_{1} &{} w_{2} &{} w_{4} &{} w_{3} \end{array}\right) \)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{2})\\ (m_{1}) &{} w_{3} &{} w_{1} &{} w_{4} &{} w_{2} \end{array}\right) \)
Blocking pair\((m_{3},w_{4})\)\((m_{3},w_{4})\)
A\(\{m_{2},w_{4}\}\)\(\{m_{3},w_{4}\}\)
\({\hat{\mu }}\)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{3})\\ w_{1} &{} w_{4} &{} w_{2} &{} (m_{4}) &{} w_{3} \end{array}\right) \)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{2})\\ w_{1} &{} w_{3} &{} w_{4} &{} (m_{4}) &{} w_{2} \end{array}\right) \)
Blocking pair\((m_{2},w_{1})\)\((m_{2},w_{4})\)

Hence, deviation \((A,{\hat{\mu }})\) is not defensible, which is a contradiction.

If A consists of three agents, it is not defensible since the deviation is similar to pairwise ones, which is a contradiction. Thus \(A=\{m_{2},m_{3},w_{1},w_{4}\}\). By defensibility, the restriction \({\hat{\mu }}|_{A}\) to A is stable in the restricted market consisting of A. Thus \({\hat{\mu }}|_{A}\) is either \(\{(m_{2},w_{1}),(m_{3},w_{4})\}\) or \(\{(m_{2},w_{4}),(m_{3},w_{1})\}\). In both cases, since \(m_{4}\) is unmatched at \({\hat{\mu }}\), \((m_{4},w_{1})\) blocks \({\hat{\mu }}\). A contradiction.

A.2.3 Part (4)

Consider a static market with \(M=\{m_{1},m_{2},m_{3},m_{4}\}\), \(W=\{w_{1},w_{2},w_{3},w_{4}\}\), and the following preferences:

\(m_{1}\)\(m_{2}\)\(m_{3}\)\(m_{4}\)\(w_{1}\)\(w_{2}\)\(w_{3}\)\(w_{4}\)
\(\underline{w_{1}}\)\(w_{1}\)\(w_{4}\)\(w_{1}\)\(m_{3}\)\(\underline{m_{2}}\)\(\underline{m_{3}}\)\(m_{2}\)
 \(w_{4}\)\(w_{1}\)\(\underline{w_{4}}\)\(m_{4}\)  \(m_{3}\)
 \(\underline{w_{2}}\)\(\underline{w_{3}}\) \(m_{2}\)  \(\underline{m_{4}}\)
    \(\underline{m_{1}}\)   

where each column indicates the preference of an agent in the first row, only acceptable mates are listed in each column, and an upper mate is preferred to a lower one. Consider the matching

$$\begin{aligned} \mu =\left( \begin{array}{cccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4}\\ w_{1} &{} w_{2} &{} w_{3} &{} w_{4} \end{array}\right) . \end{aligned}$$

Each of boldfaced cells in the table indicates his or her partner from this matching. Note that matching \(\mu \) is not stable but individually rational. All of blocking pairs are \((m_{2},w_{1})\), \((m_{3},w_{1})\), \((m_{4},w_{1})\), \((m_{2},w_{4})\), and \((m_{3},w_{4})\).

First, we show that \(\mu \) is credibly pairwise-stable. Let \((A,{\hat{\mu }})\) be a pairwise deviation from \(\mu \) such that for each \(i\in A\), \(u_{i}({\hat{\mu }})>u_{i}(\mu )\). We need to show that it is not defensible. Then, since A is a blocking pair, A is \(\{m_{2},w_{1}\}\), \(\{m_{3},w_{1}\}\), \(\{m_{4},w_{1}\}\), \(\{m_{2},w_{4}\}\), or \(\{m_{3},w_{4}\}\). Thus we have the following five cases.

A\(\{m_{2},w_{1}\}\)\(\{m_{3},w_{1}\}\)
\({\hat{\mu }}\)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{2})\\ (m_{1}) &{} w_{1} &{} w_{3} &{} w_{4} &{} w_{2} \end{array}\right) \)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{3})\\ (m_{1}) &{} w_{2} &{} w_{1} &{} w_{4} &{} w_{3} \end{array}\right) \)
Blocking pair\((m_{3},w_{4})\)\((m_{3},w_{4})\)
A\(\{m_{4},w_{1}\}\)\(\{m_{2},w_{4}\}\)
\({\hat{\mu }}\)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{4})\\ (m_{1}) &{} w_{2} &{} w_{3} &{} w_{1} &{} w_{4} \end{array}\right) \)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{2})\\ w_{1} &{} w_{4} &{} w_{3} &{} (m_{4}) &{} w_{2} \end{array}\right) \)
Blocking pair\((m_{3},w_{1})\)\((m_{2},w_{1})\)
A\(\{m_{3},w_{4}\}\) 
\({\hat{\mu }}\)\(\left( \begin{array}{ccccc} m_{1} &{} m_{2} &{} m_{3} &{} m_{4} &{} (w_{3})\\ w_{1} &{} w_{2} &{} w_{4} &{} (m_{4}) &{} w_{3} \end{array}\right) \) 
Blocking pair\((m_{2},w_{4})\) 

Hence, in any case, pair deviation \((A,{\hat{\mu }})\) is not defensible.

Next, we show that \(\mu \) is not credibly group-stable. Consider the group deviation \((A,{\hat{\mu }})\) from \(\mu \) where \(A=\{m_{2},m_{3},w_{1},w_{4}\}\), \({\hat{\mu }}(m_{2})=w_{4}\), and \({\hat{\mu }}(m_{3})=w_{1}\). Note that under \({\hat{\mu }}\), all women are matched with the best mates. Thus, there is no profitable deviation from \({\hat{\mu }}\). Thus \((A,{\hat{\mu }})\) is defensible. Moreover, each agent in A is better off in \({\hat{\mu }}\) than in \(\mu \). Hence \(\mu \) is not credibly group-stable.

A.3 Proposition 5

Fix a stable matching \(\mu \) and a defensible group deviation \((A,{\hat{\mu }})\) from \(\mu \). Let B be the set of all agents outside A whose partner is not in A under \(\mu \), and C be the set of all agents outside A whose partner is in A under \(\mu \). Then, by Definition 4, \({\hat{\mu }}|_{B}=\mu |_{B}\) and all agents in C are unmatched at \({\hat{\mu }}\).

We first show that if C is empty then \({\hat{\mu }}\) is stable. Let C be empty. Then \(M\cup W=A\cup B\). Suppose that some agent i blocks \({\hat{\mu }}\). If \(i\in A\), the blocking contradicts the defensibility of \((A,{\hat{\mu }})\). If \(i\in B\), since \({\hat{\mu }}|_{B}=\mu |_{B}\), i blocks \(\mu \), which contradicts the stability of \(\mu \). Thus no agent blocks \({\hat{\mu }}\). On the other hand, suppose that some pair (mw) blocks \({\hat{\mu }}\). If either \(m\in A\) or \(w\in A\), then the blocking contradicts the defensibility of \((A,{\hat{\mu }})\). If \(m\in B\) and \(w\in B\), then since \({\hat{\mu }}|_{B}=\mu |_{B}\), (mw) blocks \(\mu \). This contradicts the stability of \(\mu \). Hence no pair blocks \({\hat{\mu }}\). Therefore \({\hat{\mu }}\) is stable.

Thus, to prove that \({\hat{\mu }}\) is stable, we show that C is empty. Suppose for a contradiction that C is not empty. Without loss of generality, take a woman \(w_{0}\) in \(C\cap W\). We will recursively construct an infinite sequence \(\{(m_{k},w_{k})\}_{k=1}^{\infty }\) of distinct pairs in \(M\times W\) such that for each \(k\in \{1,2,\ldots \}\) , (a) \(\mu (m_{k})=w_{k-1}\), (b) \({\hat{\mu }}(m_{k})=w_{k}\), (c) \(m_{k},w_{k}\in A\), (d) \(u_{m_{k}}(\mu )<u_{m_{k}}({\hat{\mu }})\), (e) \(u_{w_{k}}(\mu )>u_{w_{k}}({\hat{\mu }})\). This contradicts the finiteness of M and W.

First, construct \(m_{1}\) and \(w_{1}\) that satisfy conditions (a) to (e). Since \(w_{0}\in C\), it follows from the definition of group deviation that \(w_{0}\) is matched with some man in A at \(\mu \). Denote this man by \(m_{1}\) so that \(m_{1}=\mu (w_{0})\) and thus (a) is satisfied. Since \(w_{0}\) is unmatched at \({\hat{\mu }}\) and \(\mu \) is individually rational, it follows from strict preferences that

$$\begin{aligned} u_{w_{0}}(m_{1})\equiv u_{w_{0}}(\mu )>u_{w_{0}}({\hat{\mu }})\equiv u_{w_{0}}(w_{0}). \end{aligned}$$
(5)

If \(m_{1}\) were unmatched at \({\hat{\mu }}\), \(u_{m_{1}}(w_{0})\equiv u_{m_{1}}(\mu )>u_{m_{1}}({\hat{\mu }})\equiv u_{m_{1}}(m_{1})\) by strict preferences and individual rationality of \(\mu \). Then pair \((m_{1},w_{0})\) would block \({\hat{\mu }}\), violating defensibility of \((A,{\hat{\mu }})\) as \(m_{1}\in A\). Thus, it follows from the definition of group deviation that \(m_{1}\) is matched with some woman in A at \({\hat{\mu }}\). Denote this woman by \(w_{1}\). Now, \(m_{1},w_{1}\in A\) and \(w_{1}={\hat{\mu }}(m_{1})\) so that (b) and (c) are satisfied. Note \(w_{0}\ne w_{1}\). Since \(w_{0}\ne w_{1}\), it follows from strict preferences that

$$\begin{aligned}&\text {either }\quad u_{m_{1}}(w_{0})\equiv u_{m_{1}}(\mu )>u_{m_{1}}(w_{1})\equiv u_{m_{1}}({\hat{\mu }}), \end{aligned}$$
(6)
$$\begin{aligned}&\text {or }\quad u_{m_{1}}(w_{0})\equiv u_{m_{1}}(\mu )<u_{m_{1}}(w_{1})\equiv u_{m_{1}}({\hat{\mu }}). \end{aligned}$$
(7)

If inequality (6) were true, then with (5), pair \((m_{1},w_{0})\) would block \({\hat{\mu }}\), violating the defensibility of \((A,{\hat{\mu }})\) as \(m_{1}\in A\). Thus inequality (7) is true so that (d) is satisfied. Now, \(\mu (w_{1})\ne {\hat{\mu }}(w_{1})\equiv m_{1}\), otherwise \(w_{0}=w_{1}\), a contradiction. Since \(\mu \) is stable, by inequality (7) and strict preferences, \(u_{w_{1}}(\mu )>u_{w_{1}}({\hat{\mu }})\) so that (e) is satisfied. Now, \(\{m_{1},w_{1},w_{0}\}\) satisfies conditions (a)–(e).

Suppose that we are given \(w_{0}\) and \(\{(m_{k},w_{k})\}_{k=1}^{K-1}\) which satisfy conditions (a) to (e) and all of whom are distinct. We construct \(m_{K}\) and \(w_{K}\) that satisfy the conditions. First, by our hypothesis,

$$\begin{aligned} u_{w_{K-1}}(\mu )>u_{w_{K-1}}({\hat{\mu }}). \end{aligned}$$
(8)

If \(w_{K-1}\) were unmatched at \(\mu \), then \(w_{K-1}\) would block \({\hat{\mu }}\) by inequality (8), violating the defensibility of \((A,{\hat{\mu }})\) as \(w_{K-1}\in A\) by our hypothesis. Thus \(w_{K-1}\) is matched with some man at \(\mu \). Denote this man by \(m_{K}\) so that \(m_{K}=\mu (w_{K-1})\) and so (a) is satisfied. Since by our hypothesis \(w_{K-1}\) is different from \(w_{1},\ldots ,w_{K-2}\) and \(m_{k}=\mu (w_{k-1})\) for each \(k\in \{1,\ldots ,K-1\}\), \(m_{K}=\mu (w_{K-1})\) implies that \(m_{K}\ne m_{1},\ldots ,m_{K-1}\) and thus \(m_{1},\ldots ,m_{K}\) are distinct. If \(m_{K}\not \in A\), then \(m_{K}\) would be unmatched at \({\hat{\mu }}\) by the definition of group deviation. Then, since \(\mu \) is individually rational, \(u_{m_{K}}(w_{K-1})\equiv u_{m_{K}}(\mu )>u_{m_{K}}({\hat{\mu }})\equiv u_{m_{K}}(m_{K})\) from strict preferences. Thus, with inequality (8), pair \((m_{K},w_{K-1})\) would block \({\hat{\mu }}\), violating the defensibility as \(w_{K-1}\in A\) by our hypothesis. Thus \(m_{K}\in A\). If \(m_{K}\) were unmatched at \({\hat{\mu }}\), then we would violate the defensibility like before. So, by the definition of group deviation, \(m_{K}\) is matched with some woman in A at \({\hat{\mu }}\). Denote this woman by \(w_{K}\) so that \(w_{K}={\hat{\mu }}(m_{K})\) and \(w_{K}\in A\), and now (b) and (c) are satisfied. Since \(m_{1},\cdots ,m_{K}\) are distinct and \(w_{k}={\hat{\mu }}(m_{k})\) for each \(k\in \{1,\ldots ,K\}\), we have \(w_{K}\ne w_{1},\ldots ,w_{K-1}\) and thus \(w_{1},\ldots ,w_{K}\) are distinct. Now, because \(w_{K-1}\ne w_{K}\), strict preferences imply that

$$\begin{aligned}&\text {either }\quad u_{m_{K}}(w_{K-1})\equiv u_{m_{K}}(\mu )>u_{m_{K}}(w_{K})\equiv u_{m_{K}}({\hat{\mu }}), \end{aligned}$$
(9)
$$\begin{aligned}&\text {or }\quad u_{m_{K}}(w_{K-1})\equiv u_{m_{K}}(\mu )<u_{m_{K}}(w_{K})\equiv u_{m_{K}}({\hat{\mu }}). \end{aligned}$$
(10)

If inequality (9) were true, then with (8), pair \((m_{K},w_{K-1})\) would block \({\hat{\mu }}\), violating the defensibility as \(m_{K}\) and \(w_{K-1}\) are in A. Thus inequality (10) holds so that (d) is satisfied. Finally, \(\mu (w_{K})\ne m_{K}\) as \(m_{K}\) is matched with \(w_{K-1}\ne w_{K}\) at \(\mu \). This implies from stability of \(\mu \) and inequality (10) that \(u_{w_{K}}(\mu )>u_{w_{K}}({\hat{\mu }})\equiv u_{w_{K}}(m_{K})\) so that (e) is satisfied. Now, we have the desired sequence. \(\square \)

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Kurino, M. Credibility, efficiency, and stability: a theory of dynamic matching markets. JER 71, 135–165 (2020). https://doi.org/10.1007/s42973-019-00004-z

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  • DOI: https://doi.org/10.1007/s42973-019-00004-z

Keywords

  • Dynamic matching market
  • Credibility
  • Efficiency
  • Group stability

JEL Classification

  • C71
  • C78
  • D47