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Sequential versus simultaneous assignment systems and two applications

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Abstract

We study assignment systems where objects are assigned to agents sequentially. Student placement to exam and mainstream schools in the USA and centralized teacher appointment in Turkey are two of many examples. Despite their prevalence in practice, research on sequential systems has been rather limited. We analyze the properties of the systems in use in these places and show that they do not satisfy desirable fairness, welfare, and incentive criteria. It turns out such shortcomings are inherent in more general sequential assignment systems as well. We then analyze preference revelation games associated with various sequential systems including those comprising of combinations of well-known mechanisms.

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Notes

  1. There are different types of regular schools.

  2. There is by now a growing series of papers, pioneered by the seminal work of Abdulkadiroğlu et al. (2003) that exclusively study this second round of student assignments to public schools. As far as we are aware, the school choice literature has abstracted away from the two-round nature of the problem. In contrast, we study both rounds as distinct but related parts of a sequential decision and assignment process.

  3. Every year, the TMoE offers a standardized test to those university graduates who wish to serve in state-sponsored jobs. Although this test is taken mostly by new university graduates, many who have graduated in the past are also eligible to take it if they wish to do so. Some of them may be currently employed as a teacher.

  4. Due to proximity, a teacher may prefer a contractual position over a tenured position.

  5. In this first round, existing contractual teachers who are seeking a new position are also restricted to rank-list only tenured positions. Existing teachers employed in tenured positions are not allowed to participate in this assignment procedure. Nor can any contractual position currently filled be rank-listed by any applicants in the first round.

  6. A sequential assignment system may also offer more flexibility in implementation as it does not need to be fully centralized. It may in fact be composed of a decentralized assignment round followed by a centralized assignment round. For example, in Turkey each private high school makes its own admission decisions individually before a centralized assignment process for public high school admissions takes place. We do not study this type of hybrid systems in this paper.

  7. The historical separation of the two type of high schools in New York is attributed to the Calandra–Hecht Act which took effect in 1972. The original bill had been introduced into the state legislature as “a measure to protect the current status and quality of specialized academic high schools in New York City.” We thank Parag Pathak for helping us find this information.

  8. Westkamp (2013) makes a similar observation for the college admissions system in Germany where a sequential assignment system is in place to facilitate the accommodation of affirmative action considerations.

  9. We prove Theorems 2 and 3 by using axioms to cover larger classes of mechanisms.

  10. As far as we are aware, this transition took place without the involvement of any economists in the decision process.

  11. In this context, stability is characterized by the combination of individual rationality, fairness, and non-wastefulness. See, for example, Guillen and Kesten (2012).

  12. A major reason behind the current two-round German college admissions system is to accommodate affirmative action considerations.

  13. Westkamp conjectures that the incentive issues observed in the current German college admissions system may not be solved by adopting another sequential system.

  14. To be consistent with the sequential assignment applications, in our analysis we assume that if \(h(i)\in S\) then \(h(i)P_{i}s\) implies \(\emptyset P_{i}s\). In particular, i would not prefer to report any objects ranked below h(i) as acceptable and this should not be considered as a manipulation.

  15. Our definition differs slightly from the standard non-wastefulness notion (see Balinski and Sönmez 1999). Here, we also add the condition that i\({\overline{\succ }}_{s} \emptyset \). In a standard student placement or school choice problem, all agents are acceptable for all objects. In our case, if an agent is unacceptable for an object which he desires, then the unfilled seats for that object are not considered wasted.

  16. The notation \(\succ |(S^{2},I^{2})\) denotes the restriction of \(\succ \) to the set of objects in \(S^{2}\) and set of agents in \(I^{2}\).

  17. The notations \(P|S^{1}\) and \(P|(S^{2},I^{2})\) denote the restrictions of (true) preference profile P to the set of objects in \(S^{1}\cup \{\emptyset \}\) and to the set of objects in \(S^{2}\cup \{\emptyset \}\) and agents in \(I^{2}\), respectively.

  18. We assume that \(c_{i}\ne c_{j}\) for any \(i,j\in I\). In practice, any possible tie is broken via exogenous tie-breaking rule.

  19. Since only tenured schools are considered in this round, the priority order for all available schools is the same and follows the order of test scores.

  20. In practice, contractual teachers are not allowed to participate. Instead, they keep their current positions. Therefore, only the positions of assigned contractual teachers become available for unassigned new graduates of the first round. For consistency with our model, we describe an equivalent procedure here.

  21. In the problems we use for the Proof of Proposition 1, all the seats of available schools in Round 1 are allocated to the agents. Hence, this result holds for any system which allows unused seats to be available in the second round.

  22. Although the actual timing of the two reports may be different, since we assume that no information is revealed between the two rounds, it is without loss of generality to model the game in this way. Indeed, if we were to allow for full information revelation between the two rounds and use a more demanding solution concept such a SPNE, then the results are more troubling than what we have for simultaneous game. In fact, we show in Example 7 in Appendix A.12 that a system featuring any combination of the known mechanisms may lead to an unfair and wasteful SPNE outcome under true preferences.

  23. Ergin and Sönmez (2006), Pathak and Sönmez (2008), Haeringer and Klijn (2009), and Bando (2014) also focus on equilibrium analysis under complete information.

  24. Once again, we do not describe this mechanism for brevity.

  25. On the contrary, suppose that there exists an NE outcome in which \(i_{1}\) is assigned to \(s_{3}\). This is possible if \(i_{1}\) is unassigned in Round 1. Then, \(i_{2}\) should be assigned to \(s_{1}\) in that NE outcome. However, as explained in Example 1, at that NE outcome \(i_{3}\) gets \(s_{3}\). In any NE strategy profile in which \(i_{1}\) is assigned to \(s_{2},\) he needs to rank \( s_{2}\) as top choice and \(i_{3}\) best responds to this strategy by ranking \( s_{1}\) as top choice. Therefore, we get the same NE outcome given in Example 1. If there exists an NE outcome in which \(i_{1}\) is unassigned, no agent will be assigned to \(s_{2}\). This is a contradiction.

  26. In the Turkish context, for instance, although an otherwise identical tenured position is preferable to a contractual position, one commonly observes strong preference for contractual positions in major metropolitan cities such as Istanbul over tenured positions in smaller cities or rural areas.

  27. In the recently adopted new system in Turkey, all teachers are assigned via a serial dictatorship. The practice of hiring contractual teachers has been discontinued in recent years.

  28. Favoring higher ranks was introduced by Kojima and Ünver (2014). In contrast to them, we do not require all the seats of school s to be filled.

  29. In the sequel we provide descriptions of the (agent-proposing) DA and the serial dictatorship mechanisms. We refer the reader to the extant literature for the descriptions of the remaining mechanisms.

  30. Note that \(\succ _{s_{1}}\) is consistent with \(h(i_{1})=s_{1}\).

  31. Note that \(\succ _{s_{1}}\) is consistent with \(h(i_{1})=s_{1}\).

  32. The definitions of the axioms used in the proof can be found in Sect. 2 and Appendix A.1.

  33. These schools are Boston Latin Academy, Boston Latin School, and the John D. O’Bryant School of Mathematics and Science.

  34. In 2012–2013 school year, 836 of 3795 seventh-grade students enrolled in exam schools.

  35. Sixth graders can also apply to be transferred to another regular school after mid-March.

  36. These schools are Bronx High School of Science, Brooklyn Latin School, Brooklyn Technical High School, High School for Math, Science and Engineering at City College, High School of American Studies at Lehman College, Queens High School for the Sciences at York College, Staten Island Technical High School, and Stuyvesant High School.

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Correspondence to Onur Kesten.

Additional information

We thank Battal Doğan, Thayer Morrill, Tayfun Sönmez, Utku Ünver, Alex Westkamp, and seminar participants at University of Texas, Boston College, GAMES 2012, Montreal Matching Workshop, Rochester, Social Choice and Welfare conference, and Koç University Winter Workshop for helpful discussions and comments.

Appendix

Appendix

1.1 Axioms used in the proofs

In Sect. 2, we provide definitions of non-wastefulness (NW), individual rationality (IR), population monotonicity (PM), and weak population monotonicity (wPM). Here, we provide the rest of the axioms used in the proofs. We define the axioms for a given assignment problem \(({\overline{I}}, {\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }})\).

A matching \(\mu \) is mutually fair (MF) if there does not exist an agent–object pair (is) such that (1) \(i\in {\overline{I}}\) ranks \(s\in {\overline{S}}\) at the top of \({\overline{P}}_{i}\), (2) \(\mu (i)\ne s\), and (3) there exists an agent \(i^{\prime }\in \mu ^{-1}(i^{\prime })\) such that \(i {\overline{\succ }}_{s}i^{\prime }\).

A matching \(\mu \) is mutually best (MB) if there does not exist an agent–object pair (is) such that (1) \(i\in {\overline{I}}\) ranks \(s\in {\overline{S}}\) at the top of \({\overline{P}}_{i}\), (2) \(\mu (i)\ne s\), and (3) i is one of the top \({\overline{q}}_{s}\) agent in \({\overline{\succ }}_{s}\).

Let \(r({\overline{P}}_{i},s)\) be the rank of \(s\in {\overline{S}}\) in the preference list \({\overline{P}}_{i}\). A matching \(\mu \) favors higher ranks (FHR) if \(i\in {\overline{I}}\) is assigned to a worse object than \(s\in {\overline{S}}\), then all agents assigned to s rank s at least as high as i. Formally, \(\mu \) favors higher ranks if whenever there exists an agent–object pair (is) such that \(s{\overline{P}}_{i}\mu (i)\) then \(r({\overline{P}}_{i},s)\ge r({\overline{P}}_{j},s)\) for all \(j\in \mu ^{-1}(s)\).Footnote 28

We say a mechanism \(\varphi \) is non-bossy if for any \((\overline{ I},{\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }})\)\(\varphi _{i}( {\overline{I}},{\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }} )=\varphi _{i}({\overline{I}},{\overline{S}},(P_{i}^{\prime },{\overline{P}}_{-i}), {\overline{q}},{\overline{\succ }})\) implies \(\varphi ({\overline{I}},{\overline{S}}, {\overline{P}},{\overline{q}},{\overline{\succ }})=\varphi ({\overline{I}},\overline{ S},(P_{i}^{\prime },{\overline{P}}_{-i}),{\overline{q}},{\overline{\succ }})\). Similarly, a mechanism \(\varphi \) is weakly non-bossy if for any \(( {\overline{I}},{\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }})\)\(\ \varphi _{i}({\overline{I}},{\overline{S}},{\overline{P}},{\overline{q}},\overline{ \succ })=\varphi _{i}({\overline{I}},{\overline{S}},(P_{i}^{\prime },{\overline{P}} _{-i}),{\overline{q}},{\overline{\succ }})=\emptyset \) implies \(\varphi ( {\overline{I}},{\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }} )=\varphi ({\overline{I}},{\overline{S}},(P_{i}^{\prime },{\overline{P}}_{-i}), {\overline{q}},{\overline{\succ }})\).

A mechanism \(\varphi \) is independent of irrelevant agents (IIA) if for every \(({\overline{I}},{\overline{S}},{\overline{P}},{\overline{q}},\overline{ \succ })\)\(\ \varphi _{i}({\overline{I}}\diagdown \{k\},{\overline{S}},( {\overline{P}}_{j})_{j\in {\overline{I}}\diagdown \{k\}},\)\({\overline{q}}, {\overline{\succ }}|{\overline{I}}\diagdown \{k\})=\varphi _{i}({\overline{I}}, {\overline{S}},{\overline{P}},{\overline{q}},{\overline{\succ }})\) where \(\overline{P }_{k}:\emptyset {\overline{P}}_{k}x\) for all \(x\in {\overline{S}}\) and \(i\in {\overline{I}}\diagdown \{k\}\).

In Table 1, we summarize the performance of the well-known mechanisms based on the axioms defined.Footnote 29

Table 1 Performance of mechanisms

Non-wastefulness, individual rationality, fairness, strategy-proofness, and independence of irrelevant agents properties of these five mechanisms are commonly known in the literature. Mutual fairness, mutual best and favoring higher ranks properties of these mechanisms follow from the definitions of these mechanisms. It is well known that TTC, BM, and SD are non-bossy and DA is not weakly non-bossy. Afacan and Dur (2017) show that oDA is non-bossy. Kojima and Ünver (2014) show that BM is population monotonic. Kesten (2006) shows that DA is population monotonic, but TTC is not. Since TTC is non-bossy and individually rational, it satisfies weak population monotonicity. It is easy to verify that oDA satisfies population monotonicity.

1.2 Proof of Proposition 1

1.2.1 Proof of Part (a)

We argue by contradiction. Suppose \(\Psi =(\varphi ^{1},\varphi ^{2})\) is straightforward and non-wasteful and \(\varphi ^{1}\) is non-wasteful. We consider the following sequential problem. There are three schools \( S=\{s_{1},s_{2},s_{3}\}\) with one available seat and two agents \( I=\{i_{1},i_{2}\}\). Let \(S^{1}=\{s_{2},s_{3}\}\) and \(i_{1}\succ _{s}i_{2}\succ _{s}\emptyset \) for all \(s\in S\) and the true preferences be as follows: \(s_{2}P_{i_{1}}s_{3}P_{i_{1}}s_{1}P_{i_{1}}\emptyset \) and \( s_{1}P_{i_{2}}s_{2}P_{i_{2}}s_{3}P_{i_{2}}\emptyset \).Footnote 30

In the first round, we have the assignment problem \( (I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\) where \( s_{2}P_{i_{1}}^{1}s_{3}P_{i_{1}}^{1}\emptyset \) and \( s_{2}P_{i_{2}}^{1}s_{3}P_{i_{2}}^{1}\emptyset \). Any non-wasteful mechanism will assign both agents to a school in the first round when agents act straightforwardly. Hence, in the final outcome the seat at \(s_{1}\) is wasted.

1.2.2 Proof of Part (b)

We prove by showing that a mechanism selects two different outcomes for the same assignment problem.

We consider the following sequential problem. There are three schools \(S=\{s_{1},s_{2},s_{3}\}\) with one available seat and three agents \( I=\{i_{1},i_{2},i_{3}\}\). Let \(S^{1}=\{s_{2},s_{3}\}\). Let \(i_{1}\succ _{s}i_{2}\succ _{s}i_{3}\succ _{s}\emptyset \) for each \(s\in S\).Footnote 31 Let the preferences be as follows: \(s_{2}P_{i_{1}}s_{3}P_{i_{1}}s_{1}P_{i_{1}} \emptyset \), \(s_{1}P_{i_{2}}s_{2}P_{i_{2}}s_{3}P_{i_{2}}\emptyset \), and \( s_{1}P_{i_{3}}s_{2}P_{i_{3}}s_{3}P_{i_{3}}\emptyset \). When all agents act straightforwardly, any fair system composed of non-wasteful mechanisms will select \(\mu _{1}=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{3} \end{array} \right) \).

In the first round, we have the following assignment problem \( (I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\) where \( s_{2}P_{i_{1}}^{1}s_{3}P_{i_{1}}^{1}\emptyset \), \( s_{2}P_{i_{2}}^{1}s_{3}P_{i_{2}}^{1}\emptyset \), and \( s_{2}P_{i_{3}}^{1}s_{3}P_{i_{3}}^{1}\emptyset \). For \(\Psi \) to select \(\mu _{1}\) in this problem, the matching selected in Round 1 should be \(\varphi _{i_{1}}^{1}(I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=s_{2}\), \(\varphi _{i_{2}}^{1}(I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=\emptyset \), and \(\varphi _{i_{3}}^{1}(I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=s_{3}\).

Now, consider the following preference profile: \(s_{2}{\overline{P}} _{i_{1}}s_{3}{\overline{P}}_{i_{1}}s_{1}{\overline{P}}_{i_{1}}\emptyset \), \( s_{2}{\overline{P}}_{i_{2}}s_{3}{\overline{P}}_{i_{2}}s_{1}{\overline{P}} _{i_{2}}\emptyset \), and \(s_{1}{\overline{P}}_{i_{3}}s_{2}{\overline{P}} _{i_{3}}s_{3}{\overline{P}}_{i_{3}}\emptyset \). When all agents act straightforwardly, any fair system composed of non-wasteful mechanisms will select \(\mu _{2}=\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{2} \end{array} \right) \).

In the first round, we have the following assignment problem \((I,S^{1}, {\overline{P}}^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\) where \( s_{2}{\overline{P}}_{i_{1}}^{1}s_{3}{\overline{P}}_{i_{1}}^{1}\emptyset \), \(s_{2} {\overline{P}}_{i_{2}}^{1}s_{3}{\overline{P}}_{i_{2}}^{1}\emptyset \), and \(s_{2} {\overline{P}}_{i_{3}}^{1}s_{3}{\overline{P}}_{i_{3}}^{1}\emptyset \). For \(\Psi \) to select \(\mu _{2}\) in this problem, the matching selected in Round 1 should be \(\varphi _{i_{1}}^{1}(I,S^{1},{\overline{P}}^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=s_{2}\), \(\varphi _{i_{2}}^{1}(I,S^{1}, {\overline{P}}^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=s_{3}\), and \( \varphi _{i_{3}}^{1}(I,S^{1},{\overline{P}}^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=\emptyset \). Note that \((I,S^{1},{\overline{P}} ^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})=(I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\) but \( \varphi ^{1}(I,S^{1},{\overline{P}}^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\ne \varphi ^{1}(I,S^{1},P^{1},(q_{s})_{s\in S^{1}},(\succ _{s})_{s\in S^{1}})\). This contradicts the fact that a mechanism selects a unique matching under each assignment problem.

1.2.3 Proof of Part (c)

We argue by contradiction. Suppose \(\Psi =(\varphi ^{1},\varphi ^{2})\) is straightforward, \(\varphi ^{1}\) is non-wasteful and weakly fair, \(\varphi ^{2}\) is non-wasteful, and \(\Psi \) respects improvements. We consider the first sequential problem used in the Proof of Proposition 1(b). When all agents act straightforwardly, any system with mechanisms as described in the statement will select \(\mu _{1}=\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{2} \end{array} \right) \).

Now, consider the following priority order: \(i_{1}\succ _{s}^{\prime }i_{3}\succ _{s}^{\prime }i_{2}\succ _{s}^{\prime }\emptyset \) for all \( s\in S\). Note that \(\succ ^{\prime }\) is an improvement in the priorities of \(i_{3}\) compared to \(\succ \). When all agents act straightforwardly, any system with mechanisms as described in the statement will select \(\mu _{2}=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{3} \end{array} \right) \). Since \(\mu _{1}(i_{3})P_{i_{3}}\mu _{2}(i_{3}),\)\(\Psi \) does not respect the improvement of \(i_{3}\) in priorities.

1.2.4 Proof of Part (d)

We consider the sequential problem used in the Proof of Proposition 1(a). In the first round, if both agents act straightforwardly, then there are two non-wasteful and individually rational matchings: \(\mu _{1}^{1}=\left( \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{2} \end{array} \right) \) and \(\mu _{1}^{2}=\left( \begin{array}{c} s_{2} \\ i_{2} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \right) \). No matter which of these two matchings is selected in the first round, none of the agents can participate the second round, and \(s_{1}\) is available in the second round. Therefore, the unique matching selected in the second round is \(\mu _{2}^{1}=\mu _{2}^{2}=\left( \begin{array}{c} s_{1} \\ \emptyset \end{array} \right) \). Any system satisfying conditions mentioned in the statement assigns \(i_{2}\) to either \(s_{2}\) or \(s_{3}.\) Let \({\widetilde{\Psi }}\) and \( {\overline{\Psi }}\) be systems selecting \((\mu _{1}^{1},\mu _{2}^{1})\) and \( (\mu _{1}^{2},\mu _{2}^{2})\), respectively. The outcomes of \(\widetilde{\Psi }\) and \({\overline{\Psi }}\) are \(\mu ^{1}=\left( \begin{array}{c} s_{1} \\ \emptyset \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{2} \end{array} \right) \) and \(\mu ^{2}=\left( \begin{array}{c} s_{1} \\ \emptyset \end{array} \begin{array}{c} s_{2} \\ i_{2} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \right) \), respectively.

Suppose \(i_{2}\) deviates and submits the following preference list for the first round: \(\emptyset P_{i_{2}}^{\prime }s_{2}P_{i_{2}}^{\prime }s_{3}\). There is a unique individually rational and non-wasteful matching in Round 1: \(\mu _{1}^{\prime }=\left( \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ \emptyset \end{array} \right) \), and both \({\widetilde{\Psi }}\) and \({\overline{\Psi }}\) select \(\mu _{1}^{\prime }.\) Based on the matching selected in the first round, \(i_{2}\) can participate in the second round and \(s_{1}\) is available in the second round. When \(i_{2}\) submits \(P_{i_{2}}^{\prime \prime }:s_{1}P_{i_{2}}^{\prime \prime }\emptyset \) for Round 2, then there is a unique non-wasteful matching: \(\mu _{2}^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \right) \). That is, the pair \((P_{i_{2}}^{\prime },P_{i_{2}}^{\prime \prime })\) is a profitable deviation for \(i_{2}\) under \({\widetilde{\Psi }}\) and \( {\overline{\Psi }}\). Note that \(P_{2}^{\prime \prime }\) is \(i_{2}\)’s true relative order over the available schools in the second round.

1.3 Proof of Proposition 2

The proof follows from the problems used in the Proof of Proposition 1. In particular, wastefulness and Pareto-inefficiency follow from the problem used in the Proof of Part (a), unfairness follows from the problem used in the Proof of Part (b), and not respecting improvement in the priority order follows from the problem used in the Proof of Part (c).

1.4 Proof of Proposition 3

Suppose \(\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))P_{i}\Psi _{i}(P^{1},P^{2})\). Since both \(\varphi ^{1}\) and \(\varphi ^{2}\) are individually rational, \(\Psi _{i}(P^{1},P^{2})R_{i}\emptyset \). Hence, \(\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))P_{i}\emptyset \). Let \( \Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))=s\). We consider two possible cases.

Case 1i does not participate in Round 2 under \( ((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))\). Then, \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})=s\in S^{1}\). Since \(\varphi ^{1}\) is strategy-proof and \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})P_{i}\emptyset \), \( \varphi _{i}^{1}(P^{1})R_{i}\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})P_{i}\emptyset \). Hence, \(\Psi _{i}(P^{1},P^{2})=\varphi _{i}^{1}(P^{1})R_{i}\)\(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})\)\(\ =\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))\). This is a contradiction.

Case 2i participates in Round 2 under \( ((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))\). Hence, \(\varphi _{i}^{2}(Q_{i}^{2},P_{-i}^{2})=s\) and \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})=\)\(\emptyset \). By individual rationality, in Round 1 i is unassigned when he submits \({\widetilde{Q}}_{i}^{1}=\emptyset {\widetilde{Q}}_{i}^{1}x\) for all \(x\in S^{1}\). If \(\varphi _{j}^{1}(Q_{i}^{1},P_{-i}^{1})\in S^{1}\) then \(\varphi _{j}^{1}({\widetilde{Q}} _{i}^{1},P_{-i}^{1})\in S^{1}\) due to the individual rationality and the weak population monotonicity (\(\varphi _{j}^{1}({\widetilde{Q}} _{i}^{1},P_{-i}^{1})R_{j}\varphi _{j}^{1}(Q_{i}^{1},P_{-i}^{1})P_{j}\emptyset \)). Moreover, if \(\varphi _{j}^{1}(Q_{i}^{1},P_{-i}^{1})\ne \varphi _{j}^{1}({\widetilde{Q}} _{i}^{1},P_{-i}^{1})\), then \(\varphi _{j}^{1}({\widetilde{Q}} _{i}^{1},P_{-i}^{1})\) should have filled all its available seats in matching \(\varphi ^{1}(Q_{i}^{1},P_{-i}^{1}).\) Otherwise non-wastefulness of \(\varphi ^{1}\) would be violated. That is, only the agents who are assigned to schools in \(\varphi ^{1}(Q_{i}^{1},P_{-i}^{1})\) become better off in \( \varphi ^{1}({\widetilde{Q}}_{i}^{1},P_{-i}^{1})\). Hence, the same set of agents is assigned to schools in \(S^{1}\) in \(\varphi ^{1}(Q_{i}^{1},P_{-i}^{1})\) and \(\varphi ^{1}({\widetilde{Q}} _{i}^{1},P_{-i}^{1})\). In other words, the same set of agents will participate in the second round when i submits \(Q_{i}^{1}\) and \(\widetilde{ Q}_{i}^{1}\). By strategy-proofness, \(\varphi _{i}^{2}({\widetilde{Q}} _{i}^{2}=P_{i}^{2},P_{-i}^{2})R_{i}\varphi _{i}^{2}(Q_{i}^{2},P_{-i}^{2})\). Therefore, \(\Psi _{i}(({\widetilde{Q}}_{i}^{1},P_{-i}^{1}),({\widetilde{Q}} _{i}^{2},P_{-i}^{2}))R_{i}\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))\).

1.5 Proof of Proposition 4

By contradiction, we show that there does not exist a profitable deviation in which all the acceptable schools under the true preferences are ranked above \(\emptyset \) in the reported preference list Round 1. That is, we assume if s\(P_{i}^{1}\)\(\emptyset \) for school \(s\in S^{1}\), then s\( Q_{i}^{1}\emptyset \). We consider two cases: \(\varphi _{i}^{1}(P^{1})=\emptyset \) and \(\varphi _{i}^{1}(P^{1})\ne \emptyset \).

Case 1 By strategy-proofness and individual rationality, \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})=\emptyset \). Consider the preference profile \( {\widehat{Q}}_{i}^{1}:\emptyset \)\({\widehat{Q}}_{i}^{1}\)s for all \(s\in S^{1} \). By non-wastefulness and weak population monotonicity, the same set of agents is assigned to schools in \(S^{1}\) under \(\varphi ^{1}({\widehat{Q}} _{i}^{1},P_{-i}^{1})\), \(\varphi ^{1}(P^{1})\) and \(\varphi ^{1}(Q_{i}^{1},P_{-i}^{1})\). Then, the same set of agents participate in the second round when i submits \(P_{i}^{1},\)\(Q_{i}^{1}\), and \({\widehat{Q}} _{i}^{1}\). Since \(\varphi ^{2}\) is strategy-proof, i cannot be assigned to a better school than \(\Psi _{i}(P^{1},P^{2})=\varphi _{i}^{2}(P^{2})\). This is a contradiction.

Case 2 By strategy-proofness, if \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})=\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2})),\) then \(\varphi _{i}^{1}(P^{1})=\Psi _{i}(P^{1},P^{2})\)\(R_{i}\)\(\Psi _{i}((Q_{i}^{1},P_{-i}^{1}),(Q_{i}^{2},P_{-i}^{2}))\). That is, there does not exist a profitable deviation in which i is assigned to a better school in the first round. On the other hand, \(\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})=\emptyset \) cannot be true by the strategy-proofness of \(\varphi ^{1}\). Note that \(\varphi _{i}^{1}(P^{1})\)\( P_{i}^{1}\)\(\emptyset \) since \(\varphi ^{1}\) is individually rational. By our construction \(\varphi _{i}^{1}(P^{1})\)\(Q_{i}^{1}\)\(\emptyset =\varphi _{i}^{1}(Q_{i}^{1},P_{-i}^{1})\). Hence, an agent with preference profile \( Q_{i}^{1}\) can be better off by submitting \(P^{1}\) under \(\varphi _{{}}^{1}\).

1.6 Proof of Theorem 1

Instead of proving the theorem for the mechanisms listed in the statement, we prove it for a general class of mechanisms by using axioms.Footnote 32

Claim 1

Let \(\Psi =(\varphi ^{1},\varphi ^{2})\) be a system such that

  1. (a)

    \(\varphi ^{1}\) is individually rational, non-wasteful, and either fair or [population monotonic and weak non-bossy] and

  2. (b)

    \(\varphi ^{2}\) is individually rational, non-wasteful, and either fair or [population monotonic, independent of irrelevant agents and weak non-bossy].

Every NE outcome of the preference revelation game associated with \(\Psi \) leads to a non-wasteful and individually rational matching.

Proof

Consider an arbitrary sequential problem \((I,S,S^{1},P,q,\succ )\). Let \( Q=(Q_{i}^{1},Q_{i}^{2})_{i\in I}\) be an NE and \(\mu \) be the associated NE outcome. Here, \(Q_{i}^{t}\) is a strict ranking over the available schools, including \(\emptyset \), at round \(t\in \{1,2\}\). First note that for any \( i\in I\) we cannot have \(\emptyset \succ _{\mu (i)}i\) because both \(\varphi ^{1}\) and \(\varphi ^{2}\) are individually rational. If \(\mu \) is not individually rational, then there exists \(i\in I\) such that \(\emptyset \)\( P_{i}\)\(\mu (i)\). If \(\emptyset \)\(P_{i}\)\(\mu (i)\), then individual rationality of both \(\varphi ^{1}\) and \(\varphi ^{2}\) implies that listing \( \emptyset \) at the top of the submitted preference lists for both rounds is a profitable deviation for i. Therefore, Q cannot be NE profile, which is a contradiction.

Suppose \(\mu \) is wasteful. Then, there exists \(i\in I\) such that \( sP_{i}\mu (i),\)\(i\succ _{s}\emptyset \) and \(|\mu ^{-1}(s)|<q_{s}.\) We consider two cases and we show that if \(\mu \) is wasteful, then i can profitably deviate, i.e., Q is not an NE.

Case 1 Suppose \(s\in S^{1}\). Since \(\varphi ^{1}\) is non-wasteful, \(\varphi _{i}^{1}(Q^{1})\)\(Q_{i}^{1}\)s. Consider the preference profile \(P_{i}^{\prime }:s\)\(P_{i}^{\prime }\)\(\emptyset \)\( P_{i}^{\prime }\)x for all \(x\in S^{1}\backslash \{s\}\). Denote \(\varphi ^{1}(P_{i}^{\prime },Q_{-i}^{1})\) by \(\mu _{1}.\) By individual rationality, either \(\mu _{1}(i)=s\) or \(\mu _{1}(i)=\emptyset \). We consider the following subcases:

\(\varphi ^{1}\)is individually rational, non-wasteful, and fair. By the rural hospital theorem (Roth 1986), the same set of students is assigned to schools and each school fills the same number of seats at all fair, non-wasteful, and individually rational matchings. Then, we consider the outcome of the sequential implementation of DA mechanism (McVitie and Wilson 1971) where i applies after all agents are tentatively assigned. Since DA is population monotonic and non-wasteful, the number of agents tentatively assigned to s before i’s turn is less than \(q_{s}\). When it is i’s turn, he will be assigned to s under DA. By rural hospital theorem, \(\mu _{1}(i)\in S^{1}\) and this school is s,  which is the only acceptable one in \(P_{i}^{\prime }\).

\(\varphi ^{1}\)is individually rational, non-wasteful, population monotonic, and weak non-bossy: Let \(\varphi ^{1}(Q_{i}^{1},Q_{-i}^{1})={\widetilde{\mu }}\). By the definition of the system, \(\mu ^{-1}(s)={\widetilde{\mu }}^{-1}(s)\). If \(\mu _{1}(i)=s\), then \( (P_{i}^{^{\prime }},Q_{i}^{2})\) is a profitable deviation for i. If \(\mu _{1}(i)=\emptyset ,\) then \(|\mu _{1}^{-1}(s)|=q_{s}\). Otherwise, \(\mu _{1}\) is wasteful. Let \({\widetilde{I}}=\{j\in I|{\widetilde{\mu }}(j)\ne \mu _{1}(j)=s\}.\) Since \(|\mu _{1}^{-1}(s)|=q_{s}\) and \(|{\widetilde{\mu }} ^{-1}(s)|<q_{s}\), \({\widetilde{I}}\ne \emptyset \). For all \(j\in {\widetilde{I}} \) we have \({\widetilde{\mu }}(j)Q_{j}^{1}sQ_{j}^{1}\emptyset \). Otherwise, \( \varphi ^{1}\) cannot be non-wasteful or individually rational. Now consider problem \(({\overline{P}}_{i},Q_{-i}^{1})\) where \(\emptyset {\overline{P}}_{i}x\) for all \(x\in S^{1}\). By weak non-bossiness and individual rationality, \( \varphi ^{1}({\overline{P}}_{i},Q_{-i}^{1})=\mu _{1}\). When we consider problems \(({\overline{P}}_{i},Q_{-i}^{1})\) and \((Q_{i}^{1},Q_{-i}^{1}),\) all the other students should weakly prefer \(\mu _{1}\) to \({\widetilde{\mu }}\) by population monotonicity. However, agents in \({\widetilde{I}}\) prefer \( {\widetilde{\mu }}\) to \(\mu _{1}\). This is a contradiction.

Case 2 Suppose \(s\notin S^{1}\). If \(\mu (i)\notin S^{1}\), then by using the same steps in Case 1, one can show that \((Q_{i}^{1},P_{i}^{\prime \prime })\) is a profitable deviation for i where \(P_{i}^{\prime \prime }:s\)\(P_{i}^{\prime \prime }\)\(\emptyset \)\(P_{i}^{\prime \prime }x\) for all \( x\in S^{2}\setminus \{s\}\). If \(\mu (i)\in S^{1}\), then we show that \(( {\overline{P}}_{i},P_{i}^{\prime \prime })\) is a profitable deviation for i. Since \(\varphi ^{1}\) is individually rational, non-wasteful, and fair and/or population monotonic, if \(\varphi _{j}^{1}(Q^{1})\in S^{1}\), then \(\varphi _{j}^{1}({\overline{P}}_{i},Q_{-i}^{1})\in S^{1}\). Therefore, in the second round the set of agents is a subset of \(I^{2}\cup \{i\}\). Let \(I_{2}^{\prime }\ \) and \(\mu _{2}\), respectively, be the set of agents and the selected matching in Round 2 when i submits \(({\overline{P}}_{i},P_{i}^{\prime \prime })\). By individual rationality, either \(\mu _{2}(i)=s\) or \(\mu _{2}(i)=\emptyset .\) We show that \(\mu _{2}(i)\) cannot be \(\emptyset \). Suppose \(\mu _{2}(i)=\emptyset \). Let \({\widetilde{I}}_{2}=\{j\in I_{2}^{\prime }|\mu (j)\ne \mu _{2}(j)=s\}.\) Since \(\varphi ^{2}\) is non-wasteful, \(|\mu _{2}^{-1}(s)|=q_{s}\), \({\widetilde{I}}_{2}\ne \emptyset ,\) and \(\mu (j)\)\(Q_{j}^{2}\)\(\mu _{2}(j)\) for all \(j\in {\widetilde{I}}_{2}\). We consider two subcases:

\(\varphi ^{2}\)is individually rational, non-wasteful, and fair: The agents who participate in Round 2 except i cannot fill all the available seats of s. Hence, the result follows from the same argument in Case 1.

\(\varphi ^{2}\)is individually rational, non-wasteful, weak non-bossy, population monotonic, and independent of irrelevant agents: By weak non-bossiness and individual rationality, \(\mu _{2}\) will be selected when i ranks \(\emptyset \) at the top of his submitted list for round 2. Let \( \mu _{2}^{\prime }\) be the outcome of \(\varphi ^{2}\) when we consider only agents in \(I_{2}^{\prime }\setminus \{i\}\), keeping everything else the same. Note that \(I_{2}^{\prime }\setminus \{i\}\subseteq I_{2}\). By population monotonicity and independence of irrelevant agents, \(\mu _{2}^{\prime }(j)=\mu _{2}(j)Q_{j}^{2}\mu (j)\) for all \(j\in I_{2}^{\prime }\setminus \{i\}\) where \(\mu (j)\ne \mu _{2}(j)\). This contradicts the fact that for all \(k\in {\widetilde{I}}_{2}\subseteq I_{2}^{\prime }\setminus \{i\}\) , \(\mu (k)\)\(Q_{k}^{2}\)\(\mu _{2}(k)\). \(\square \)

1.7 Proof of Theorem 2

Instead of proving the theorem for the listed mechanisms, we prove it for a general class of mechanisms by using axioms.

Claim 2

Let \(\Psi =(\varphi ^{1},\varphi ^{2})\) be a system such that \( \varphi ^{1}\) is individually rational, non-wasteful, and

  1. (a)

    Mutually fair, and favors higher ranks, or

  2. (b)

    Population monotonic, weakly non-bossy, and fair.

Every NE outcome of the preference revelation game associated with \(\Psi \) leads to a matching \(\mu \) in which there does not exist an agent pair (ij) such that \(\mu (j)\in S^{1},\)\(\mu (j)\)\(P_{i}\)\(\mu (i)\) and \( i\succ _{\mu (j)}j.\)

Proof

Consider an arbitrary sequential problem \((I,S,S^{1},P,q,\succ )\). Let \( Q=(Q_{i}^{1},Q_{i}^{2})_{i\in I}\) be an NE profile and \(\mu \) be the associated equilibrium outcome. Here, \(Q_{i}^{t}\) is a strict ranking over the available schools, including \(\emptyset \), at round \(t\in \{1,2\}\). Suppose there exist two agents \(i,j\in I\) such that \(\mu (j)\in S^{1},\)\(\mu (j)P_{i}\mu (i)\) and \(i\succ _{\mu (j)}j\). By individual rationality, \( j\succ _{\mu (j)}\emptyset \). Let \(\mu (j)=s\), \(I^{\prime }=\{i^{\prime }\in I|sP_{i^{\prime }}\mu (i^{\prime })\) and \(i^{\prime }\succ _{s}j\}\) and \({\widehat{i}}\in I^{\prime }\) be the agent who has the highest priority for s among the ones in \(I^{\prime }\). We claim that submitting \( (Q^{\prime },Q_{i}^{2})\) where \(Q^{\prime }:s\)\(Q^{\prime }\)\(\emptyset \)\( Q^{\prime }\)\(x\ \)for all \(x\in S^{1}\diagdown \{s\}\) is a profitable deviation for \({\widehat{i}}\).

Let \(\varphi ^{1}(Q^{1})=\mu _{1}\), \(\varphi ^{1}(Q^{\prime },Q_{-{\widehat{i}} }^{1})={\widetilde{\mu }}_{1},\) and \({\widetilde{I}}_{1}=\{k\in I|\mu _{1}(k)\ne {\widetilde{\mu }}_{1}(k)=s\}\). Since \(\varphi ^{1}\) is individually rational, either \({\widetilde{\mu }}_{1}({\widehat{i}})=s\) or \( {\widetilde{\mu }}_{1}({\widehat{i}})=\emptyset \). If \({\widetilde{\mu }}_{1}( {\widehat{i}})=s\), then \((Q^{\prime },Q_{i}^{2})\) is a profitable deviation for \({\widehat{i}}\). If \({\widetilde{\mu }}_{1}({\widehat{i}})=\emptyset \), then \(|{\widetilde{\mu }}_{1}^{-1}(s)|=q_{s}\), \(k\succ _{s}{\widehat{i}}\) for all \( k\in \)\({\widetilde{\mu }}_{1}^{-1}(s)\), \({\widetilde{\mu }}_{1}(j)\ne s\) and \( {\widetilde{I}}_{1}\ne \emptyset \). Otherwise, mutual fairness and/or non-wastefulness of \(\varphi ^{1}\) would be violated. Suppose \(\widetilde{ \mu }_{1}({\widehat{i}})=\emptyset \).

\(\varphi ^{1}\)favors higher ranks and is mutually fair: Since \(\varphi ^{1}\) favors higher ranks, s\(Q_{k}^{1}\)x for all \(x\in S^{1}\cup \{\emptyset \}\) and \(k\in {\widetilde{I}}_{1}\). Then, \(\mu _{1}\) cannot be mutually fair or non-wasteful.

\(\varphi ^{1}\)is fair, weakly non-bossy, and population monotonic: Consider the profile \((Q^{\prime \prime },Q_{-{\widehat{i}}}^{1})\) where \(Q^{\prime \prime }:\emptyset \)\(Q^{\prime \prime }\)x for all \(x\in S^{1}\). By weak non-bossiness and individual rationality, \(\varphi ^{1}(Q^{\prime \prime },Q_{-{\widehat{i}}}^{1})={\widetilde{\mu }}_{1}\). By population monotonicity and individual rationality, \({\widetilde{\mu }} _{1}(l)=sQ_{l}^{1}\mu _{1}(l)\) for all \(l\in {\widetilde{I}}_{1}\). Since \( l\succ _{s}{\widehat{i}}\succ _{s}j\) and \(sQ_{l}^{1}\mu _{1}(l)\) for all \(l\in {\widetilde{I}}_{1}\), fairness or non-wastefulness is violated in matching \( \mu _{1}\). \(\square \)

1.8 Proof of Theorem 3

Instead of proving the theorem for the listed mechanisms, we prove it for a general class of mechanisms by using axioms.

Claim 3

Let \(\Psi =(\varphi ^{1},\varphi ^{2})\) be a system such that \(\varphi ^{1}\) is mutually best, individually rational, non-wasteful, population monotonic, and

  1. (a)

    \(\varphi ^{2}\) is individually rational, non-wasteful, population monotonic, independent of irrelevant agents, weakly non-bossy, and fair, or

  2. (b)

    \(\varphi ^{2}\) is individually rational, non-wasteful, mutually fair, and favors higher ranks.

Every NE outcome of the preference revelation game associated with \(\Psi \) leads to a matching \(\mu \) in which there does not exist an agent pair (ij) such that \(\mu (j)\in S\setminus S^{1}=S^{2},\)\(\mu (j)\)\(P_{i}\)\( \mu (i)\) and \(i\succ _{\mu (j)}j.\)

Proof

Consider an arbitrary sequential problem \((I,S,S^{1},P,q,\succ )\). Let \( Q=(Q_{i}^{1},Q_{i}^{2})_{i\in I}\) be an NE profile and \(\mu \) be the associated equilibrium outcome. Here, \(Q_{i}^{t}\) is a strict ranking over the available objects, including \(\emptyset \), at round \(t\in \{1,2\}\). Suppose there exist two agents \(i,j\in I\) such that \(\mu (j)\)\(P_{i}\)\(\mu (i)\), \(\mu (j)\in S^{2}\) and \(i\succ _{\mu (j)}j.\) By individual rationality, \(j\succ _{\mu (j)}\emptyset \). Let \(\mu (j)=s\). Under strategy profile Q, there are two cases: (1) i does not participate in Round 2 because i is assigned in the first round;  or (2) i participates in Round 2.

Suppose that i participates in Round 2. Without loss of generality, let i be the agent with the highest priority for s among the ones who prefer s to his assignment and participating in Round 2. Note that when no agent deviates from his strategy for the first-round schools, the set of agents in the second round do not change. Therefore, we can prove that i can benefit from deviating to \((Q_{i}^{1},Q^{\prime })\) where s\(Q^{\prime }\)\( \emptyset \)\(Q^{\prime }\)x for all \(x\in S^{2}\setminus \{s\}\) by following the same steps in the Proof of Theorem 2.

Now, suppose that under Q there does not exist an agent \(i^{\prime }\) such that \(i^{\prime }\in I^{2}\), s\(P_{i^{\prime }}\)\(\mu (i^{\prime })\) and \( i^{\prime }\succ _{s}j\). Then, \(\mu (i)\in S^{1}\). We claim that submitting \(( {\widetilde{Q}},Q^{\prime })\) is a profitable deviation for i where \( {\widetilde{Q}}:\emptyset \)\({\widetilde{Q}}\)x for all \(x\in S^{1}\) and \( Q^{\prime }:s\)\(Q^{\prime }\)\(\emptyset \)\(Q^{\prime }\)x for all \(x\in S^{2}\setminus \{s\}\). By individual rationality, \(\varphi _{i}^{1}( {\widetilde{Q}},Q_{-i}^{1})=\emptyset \). By population monotonicity and individual rationality, if \(\varphi _{{\overline{j}}}^{1}(Q^{1})\in S^{1}\), then \(\varphi _{{\overline{j}}}^{1}({\widetilde{Q}},Q_{-i}^{1})\in S^{1}\) for each \({\overline{j}}\in I\). Therefore, in the second round the set of agents is a subset of \(I^{2}\cup \{i\}\). Let \(I_{2}^{\prime }\) and \({\overline{\mu }} _{2}\), respectively, be the set of agents and the selected matching in Round 2, when i submits \(({\widetilde{Q}},Q^{\prime })\) and all other agents play their strategies under Q. By individual rationality, \({\overline{\mu }} _{2}(i)\) is either s or \(\emptyset .\) If \({\overline{\mu }}_{2}(i)=s\), then we are done. Suppose \({\overline{\mu }}_{2}(i)=\emptyset \). Let \({\widetilde{I}} _{2}=\{j\in I_{2}^{\prime }|\mu (j)\ne {\overline{\mu }}_{2}(j)=s\}.\) Since \( \varphi ^{2}\) is non-wasteful and mutually fair, \(|{\overline{\mu }} _{2}^{-1}(s)|=q_{s}\), \({\widetilde{I}}_{2}\ne \emptyset \), \(k\succ _{s}i\) for all \(k\in {\overline{\mu }}_{2}^{-1}(s)\).

\(\varphi ^{2}\)favors higher ranks and is mutually fair: Since \(\varphi ^{2}\) favors higher ranks, s\(Q_{k}^{2}\)x for all \(x\in {\widetilde{S}}^{2}\cup \emptyset \) and \(k\in {\widetilde{I}}_{2}\). Then, \(\mu _{2}\) cannot be mutually fair.

\(\varphi ^{2}\)is fair, weakly non-bossy, population monotonic, and independent of irrelevant agents: By weak non-bossiness and individual rationality, \({\overline{\mu }}_{2}\) will be selected by \(\varphi ^{2}\) when i submits \({\widetilde{P}} _{i}^{2}=\emptyset \)\({\widetilde{P}}_{i}^{2}\)x for all \(x\in S^{2}\). Let \( \mu _{2}^{\prime }\) be the outcome of \(\varphi ^{2}\) when we consider only agents in \(I_{2}^{\prime }\setminus \{i\},\) keeping everything else the same. By the independence of irrelevant agents, \(\mu _{2}^{\prime }(l)= {\overline{\mu }}_{2}(l)\) for all \(l\in I_{2}^{\prime }\setminus \{i\}\). Due to population monotonicity and independence of irrelevant agents, \(\mu (l)Q_{l}^{2}\mu _{2}^{\prime }(l)={\overline{\mu }}_{2}(l)\) cannot be true for any \(l\in (I_{2}^{\prime }\setminus \{i\})\subseteq I_{2}\). This contradicts the fairness of \(\varphi ^{2}\). \(\square \)

1.9 Proof of Theorem 4

Instead of proving the theorem for the listed mechanisms, we prove it for a general class of mechanisms by using axioms.

Claim 4

Let \(\Psi =(\varphi ^{1},\varphi ^{2})\) be a system such that both \( \varphi ^{1}\) and \(\varphi ^{2}\) are individually rational, non-wasteful, and mutually fair. Every stable matching under agents’ true preferences can be supported as an NE outcome of the game associated with \(\Psi \).

Proof

Consider an arbitrary sequential problem \((I,S,S^{1},P,q,\succ )\). Let \(\mu \) be a stable matching under agents’ true preferences in grand assignment problem. Then, consider the strategy profile \(Q=(Q_{i}^{1},Q_{i}^{2})_{i\in I} \) where \(Q_{i}^{k}\) is the submitted preferences over schools available in Round \(k\in \{1,2\}\), including \(\emptyset \), such that

  • If \(\mu (i)\in S^{1},\) then \(\mu (i)Q_{i}^{1}\emptyset Q_{i}^{1}x\) for all \(x\in S^{1}\setminus \{\mu (i)\}\) and \(\emptyset Q_{i}^{2}x\) for all \( x\in S^{2}\); and

  • If \(\mu (i)\in S\setminus S^{1},\) then \(\emptyset Q_{i}^{1}x\) for all \( x\in S^{1}\) and \(\mu (i)Q_{i}^{2}\emptyset Q_{i}^{2}x\) for all \(x\in S\setminus \{\mu (i)\}\).

By non-wastefulness and individual rationality, the outcome of \(\Psi \) under this strategy profile is \(\mu \). We claim that Q is an NE strategy profile. On the contrary, suppose an agent i can gain when he submits strategy \({\widetilde{Q}}_{i}=({\widetilde{Q}}_{i}^{1},{\widetilde{Q}}_{i}^{2})\). First note that, by individual rationality, if \(\mu (i)\in S\), then i cannot be better off by submitting a preference profile that leaves him unassigned. Moreover, i cannot be assigned to a school s such that \( \emptyset \succ _{s}i\). By our construction, for each \(s\in S^{1}\) with \( \emptyset \succ _{s}i\) that i prefers to \(\mu (i)\) the number of agents ranking s as top choice under \(Q^{1}\) and having higher priority than i is \(q_{s}\). Due to mutual fairness of \(\varphi ^{1}\), \(\varphi _{j}^{1}( {\widetilde{Q}}_{i}^{1},Q_{-i}^{1})=s\) for all \(j\in \mu ^{-1}(s)\). Hence, \( \mu (i)R_{i}\varphi _{i}^{1}({\widetilde{Q}}_{i}^{1},Q_{-i}^{1})\). If \(\varphi _{i}^{1}({\widetilde{Q}}_{i}^{1},Q_{-i}^{1})\ne \emptyset \), then \(\widetilde{ Q}_{i}\) cannot be profitable deviation for i. Hence, we take \(\varphi _{i}^{1}({\widetilde{Q}}_{i}^{1},Q_{-i}^{1})=\emptyset \).

Since each agent ranks at most one school acceptable under \(Q^{1}\) and \( \varphi ^{1}\) is individually rational, each agent in \(I\setminus \{i\}\) participating to Round 2 under problem \(Q^{1}\) participates to Round 2 under problem \(({\widetilde{Q}}_{i}^{1},Q_{-i}^{1})\). Then, by our construction, for each \(s\in S^{2}\) with \(\emptyset \succ _{s}i\) that i prefers to \(\mu (i) \) the number of agents ranking s as top choice under \(Q^{2}\) and having higher priority than i is \(q_{s}\). Due to mutual fairness of \( \varphi ^{1}\), \(\varphi _{j}^{1}({\widetilde{Q}}_{i}^{1},Q_{-i}^{1})=s\) for all \(j\in \mu ^{-1}(s)\). Hence, \(\mu (i)R_{i}\varphi _{i}^{2}({\widetilde{Q}} _{i}^{2},Q_{-i}^{2})\) and Q is an NE strategy profile. \(\square \)

1.10 Assignment systems in Boston and NYC

In Boston, there are three exam schoolsFootnote 33 that enroll around 25% of the seventh-grade students.Footnote 34 In a given year, sixth-grade students take the centralized exam before December and apply to one of these schools in the following year. A ranking of students is then obtained based on a combination of the exam scores and GPAs from the previous year. The assignment of students to exam schools is determined via the serial dictatorship mechanism induced by this ranking. Admitted students receive their acceptance letters from exam schools by mid-March,Footnote 35 and the assignment for regular schools is determined via DA.

In New York City, there are nine exam schools.Footnote 36 The assignments to the exam and regular schools are also implemented sequentially although students submit their preferences over both types of schools at the same time. Every year between 25,000 and 30,000 students take the Specialized High School Admission Test (SHSAT), which is then used to determine the assignments to the exam schools, which enroll only about 5000 annually. Students who take this test submit two different rank-order-lists to the central authority. In the first one, they rank-list only the exam schools, whereas in the second one they rank-list only the regular schools (which do not require any test score). The admission decisions for the exam schools are determined based on the scores on SHSAT, while the admissions for regular schools follow the outcome of DA. Both decisions are made concurrently. The central authority aims to make placements to the exam schools first. Therefore, initially only those students who have been admitted to both an exam and a regular school are informed, and they are asked to make a choice between the two schools they are admitted to. Subsequently, students who are not assigned in this round are considered and they are assigned to regular schools once again via DA.

1.11 Turkish assignment system

The number of teachers assigned to tenured and contractual positions in 2009 and 2010 is presented in Table 2. For instance in December 2009, 8850 tenured positions were filled by applicants in the first round. A total of 6323 of these applicants were existing teachers working in contractual positions. These contractual positions which became available as a consequence of assignments of existing teachers to the tenured positions were filled in the same month.

Table 2 Number of teachers assigned to tenured and contractual positions (2009–2010)

1.12 Examples

In Examples 5 and 6, we illustrate how the SD-DA and TRSD mechanisms fail to satisfy the desired properties. In Example 7, we show that a system featuring any of the mechanisms we know leads to an unfair and wasteful subgame perfect Nash equilibrium outcome under true preferences.

Example 5

Let \(S=\{s_{1},s_{2},s_{3},s_{4}\}\), \(S^{e}=\{s_{3},s_{4} \},\)\(S^{r}=\{s_{1},s_{2}\},\)\(I=\{i_{1},i_{2},i_{3},i_{4}\},\) and \( h(i)=\emptyset \) for all \(i\in I\). For each \(s\in S\), \(q_{s}=1\). Let true preferences and exam scores be as follows:

$$\begin{aligned} s_{2}P_{i_{1}} s _{3}P_{i_{1}}s_{1}P_{i_{1}}s_{4}P_{i_{1}}\emptyset \quad c_{i_{1}}= & {} 90 \\ s_{1}P_{i_{2}}s_{4}P_{i_{2}}s_{2}P_{i_{2}}s_{3}P_{i_{2}}\emptyset \quad c_{i_{2}}= & {} 88 \\ s_{3}P_{i_{3}}s_{1}P_{i_{3}}s_{2}P_{i_{3}}s_{4}P_{i_{3}}\emptyset \quad c_{i_{3}}= & {} 85 \\ s_{4}P_{i_{4}}s_{2}P_{i_{4}}s_{1}P_{i_{4}}s_{3}P_{i_{4}}\emptyset \quad c_{i_{4}}= & {} 70 \end{aligned}$$

Let \(i_{1}\succ _{s}i_{2}\succ _{s}i_{3}\succ _{s}i_{4}\) for all \(s\in S^{r}\) . The set of available schools in Round 1 is \(S^{e}=\{s_{3},s_{4}\}.\) The outcome selected in Round 1 when all the agents act straightforwardly is \( \mu _{1}(i_{1})=s_{3},\)\(\mu _{1}(i_{2})=s_{4},\)\(\mu _{1}(i_{3})=\emptyset , \) and \(\mu _{1}(i_{4})=\emptyset .\) In Round 2, the set of the available schools and the set of the applicants allowed to participate are \( S^{r}=\{s_{1},s_{2}\}\) and \(I^{2}=\{i_{3},i_{4}\}.\) The outcome selected in Round 2 when all the agents act straightforwardly is \(\mu _{2}(i_{3})=s_{1}\) and \(\mu _{2}(i_{4})=s_{2}.\) The final outcome of the SD-DA mechanism is \( \mu =\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{2} \end{array} \right) \).

SD-DA is not Pareto-efficient There exists another matching \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \begin{array}{c} s_{2} \\ i_{1} \end{array} \begin{array}{c} s_{3} \\ i_{3} \end{array} \begin{array}{c} s_{4} \\ i_{4} \end{array} \right) \) that Pareto-dominates the outcome of the SD-DA mechanism, \(\mu .\) It is worth noting that \(\mu ^{\prime }\) is a fair matching.

SD-DA is not straightforward If \(i_{2}\) ranks \(s_{4}\) below \( \emptyset \) in his list in Round 1, then the final outcome will be \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{3} \end{array} \right) \) and \(i_{2}\) will be strictly better off.

SD-DA does not respect improvements If \(c_{i_{2}}^{\prime }=75\), then the outcome will be \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{2} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{3} \end{array} \right) \) and \(\mu ^{\prime }(i_{2})P_{i_{2}}\)\(\mu (i_{2}).\) That is, when \( i_{2}\) gets a higher score he is assigned to a less preferred school.

SD-DA is not fair\(\mu (i_{4})P_{i_{1}}\mu (i_{1})\) and \(i_{1}\) has higher priority for \(\mu (i_{4})=s_{2}.\)

SD-DA is wasteful Consider the same example with only one agent, \( I=\{i_{1}\}\). SD-DA assigns \(i_{1}\) to \(s_{3}.\) But \(i_{1}\) prefers \(s_{2}\) to his match \(s_{3}\) and \(s_{2}\) has an empty seat under the outcome of SD-DA . \(\square \)

Example 6

Consider Example 5 with the following modifications: \(h(i_{1})=s_{1},\)\(h(i_{2})=s_{2},\)\(h(i_{3})=h(i_{4})= \emptyset .\) Take the same exam scores for \(i_{1},\)\(i_{3},\) and \(i_{4}.\) Only change the exam score of \(i_{2}\) to \(c_{i_{2}}=80.\) When all agents act straightforwardly, TRSD selects matching: \(\mu =\left( \begin{array}{c} s_{1} \\ i_{4} \end{array} \begin{array}{c} s_{2} \\ i_{2} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{3} \end{array} \right) .\)

TRSD is not Pareto-efficient There exists another matching \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{2} \end{array} \right) \) that Pareto-dominates the outcome of the TRSD mechanism \(\mu .\)

TRSD is not straightforward If \(i_{3}\) ranks \(s_{4}\) below \( \emptyset \) in the submitted preferences in Round 1, then the final outcome will be \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{2} \end{array} \right) \) and \(i_{3}\) will be strictly better off.

TRSD does not respect improvements If \(c_{i_{3}}^{\prime }=75\), then the outcome will be \(\mu ^{\prime }=\left( \begin{array}{c} s_{1} \\ i_{3} \end{array} \begin{array}{c} s_{2} \\ i_{4} \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{2} \end{array} \right) \) and \(\mu ^{\prime }(i_{3})P_{i_{3}}\)\(\mu (i_{3}).\) That is, when \( i_{3}\) gets higher score he is assigned to a less preferred school.

TRSD is not fair\(\mu (i_{4})P_{i_{3}}\mu (i_{3})\) and \(i_{3}\) has higher priority than \(i_{4}\) for \(\mu (i_{4})=s_{1}.\)

TRSD is wasteful Consider the same example with only two agents, \( I=\{i_{1},i_{3}\}\) and \(S=\{s_{1},s_{3},s_{4}\}\) where \(h(i_{1})=s_{1}\) and \( h(i_{3})=\emptyset .\) Let true preferences and exam scores be as follows:

$$\begin{aligned} s _{3}P_{i_{1}}s_{1}P_{i_{1}}s_{4}\quad c_{i_{1}}= & {} 90 \\ s_{3}P_{i_{3}}s_{1}P_{i_{3}}s_{4}\quad c_{i_{3}}= & {} 85 \end{aligned}$$

The matching selected by the TRSD is \(\mu ^{\prime \prime }=\left( \begin{array}{c} s_{1} \\ \emptyset \end{array} \begin{array}{c} s_{3} \\ i_{1} \end{array} \begin{array}{c} s_{4} \\ i_{3} \end{array} \right) \). But \(i_{3}\) prefers \(s_{1}\) to his match \(s_{4}\) and \(s_{1}\) has an empty seat under \(\mu ^{\prime \prime }.\)\(\square \)

Example 7

Let \(S=\{s_{1},s_{2},s_{3},s_{4},s_{5}\}\), \( S^{1}=\{s_{1}\}\), \(q=(1,1,1,1,1)\), and \(I=\{i_{1},i_{2},i_{3},i_{4},i_{5}\}\) . Priorities and preferences for the grand problem are given as

\(\succ _{s_{1}}\)

\(\succ _{s_{2}}\)

\(\succ _{s_{3}}\)

\(\succ _{s_{4}}\)

\( \succ _{s_{5}}\)

\(i_{1}\)

\(i_{3}\)

\(i_{2}\)

\(i_{1}\)

\(i_{4}\)

\(i_{5}\)

\(i_{1}\)

\(i_{1}\)

\(i_{4}\)

\(i_{1}\)

 : 

 : 

 : 

 : 

 

\(P_{i_{1}}\)

\(P_{i_{2}}\)

\(P_{i_{3}}\)

\(P_{i_{4}}\)

\(P_{i_{5}}\)

\(s_{5}\)

\(s_{2}\)

\(s_{3}\)

\(s_{4}\)

\(s_{1}\)

\(s_{2}\)

\(s_{3}\)

\(s_{2}\)

\(s_{5}\)

\(\emptyset \)

\(s_{1}\)

\(\emptyset \)

\(\emptyset \)

\(\emptyset \)

 

Let \(\Psi =(\varphi ^{1},\varphi ^{2})\) be a system such that both \(\varphi ^{1}\) and \(\varphi ^{2}\) are mutually best, non-wasteful, and individually rational. Consider the following strategy profile: In the first round, students submit their true preferences over \(s_{1}\) and \(\emptyset \) and in any subgames in the second round in which \(i_{1}\), \(i_{2}\), \(i_{3}\), and \( i_{4}\) participate each student reports only the school that he has the highest priority acceptable and in any subgames in the second round in which \(i_{1}\) does not participate each student reports only his top choice acceptable. One can easily verify that this strategy profile is SPNE and the outcome of this profile is \(\mu =\left( \begin{array}{c} s_{1} \\ i_{1} \end{array} \begin{array}{c} s_{2} \\ i_{2} \end{array} \begin{array}{c} s_{3} \\ i_{3} \end{array} \begin{array}{c} s_{4} \\ i_{4} \end{array} \begin{array}{c} s_{5} \\ \emptyset \end{array} \begin{array}{c} \emptyset \\ i_{5} \end{array} \right) \). Under \(\mu \), seat at \(s_{5}\) is wasted and \(i_{1}\) has justified envy for school \(s_{2}\). \(\square \)

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Dur, U., Kesten, O. Sequential versus simultaneous assignment systems and two applications. Econ Theory 68, 251–283 (2019). https://doi.org/10.1007/s00199-018-1133-9

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