Efficient Resource Allocation under Acceptant Substitutable Priorities∗ Taro Kumano

†

Department of Economics, Washington University in St.Louis,

December, 2009

Abstract The paper considers an assignment problem with acceptant substitutable priorities, and analyzes efficiency properties of the deferred acceptance rule. As a generalization of Ergin (2002), we come up with the acyclicity condition, and show that the priority structure is acyclical if and only if the deferred acceptance rule is Pareto efficient. The class of acceptant substitutable priorities is a larger domain than that of responsive priorities, and Abdulkadiro˘glu and S¨onmez (2003) implicitly point out that priorities we observe in practice violate responsiveness, but are still acceptant substitutable. JEL classification: C78; D61 Keywords: assignment, deferred acceptance algorithm, acceptant substitutable priorities, Pareto efficient rule, acyclicity

1. Introduction An assignment problem with priorities consists of five components: the set of agents, the set of objects, the preference profile of agents, the priority structure of objects, and the fixed number of quotas of each object. School choice and the specific labor market such as National Residency Matching Program in the U.S. and Japan are major real world examples of the assignment problem. For instance, in school choice, students are agents, schools are objects, the priority structure is the ranking profile of each school, and the number of quotas is the number of seats available at each school. In such markets, the deferred acceptance algorithm, which was introduced by Gale and Shapley (1962), (DA, ∗

I am especially grateful to Haluk Ergin and Fuhito Kojima for helpful discussions and encouragements. I would also like to thank Rohan Dutta, Atsushi Kajii, Shintaro Miura, John Nachbar, Ryosuke Okazawa, Masahiro Watabe and seminar participants at Kyoto University for comments. † Email address: [email protected]

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henceforth) is used to find an assignment. Since the outcome of the DA rule attains stable allocation (i.e. individually rational and admits no pairwise deviation) and since the rule is strategy-proof, the rule is thought of as the most desirable. There is a large literature dealing with the assignment problem. Gale and Shapley (1962) show that the outcome of the DA rule is Pareto superior to any other stable assignment. The DA outcome is, therefore, called the agent optimal stable assignment (AOSA, henceforth). It is easy to see that an AOSA is not Pareto efficient in general. Ergin (2002) establishes a necessary and sufficient condition for the DA rule to be Pareto efficient. He introduces the acyclical condition, which is now called “Ergin-acyclicity”, and shows that the priority structure is acyclical if and only if the DA rule is Pareto efficient. However, in most cases, the literature is concerned with responsive priorities. All results mentioned above are under responsive priorities. The class of responsive priorities is largely restricted for two reasons. In the sense of application, Abdulkadiro˘glu and S¨onmez (2003) implicitly point out that priorities we observe in practice violate responsiveness as they are often concerned about issues such as race and test score. Example 1 Possibility of Acceptant Substitutable priorities in school choice: A school has to choose students by their test scores AND races. Suppose there are three students, {1, 2, 3}, and a school which has two seats. Students’ characteristics are as follows: Student 1 :

score 90 & nationality U S

Student 2 :

score 80 & nationality U S

Student 3 :

score 70 & nationality JP N

Suppose that the school wants to accept them by test scores. In that case, the school has the following priorities. {1, 2} > {1, 3} > {2, 3} > {1} > {2} > {3}, which is responsive. But once an educational authority decides a racially fair admission. Then the school has the following priorities: {1, 3} > {2, 3} > {1, 2} > {1} > {2} > {3}, which violates responsiveness. Nevertheless, both priority structures satisfy acceptant substitutability. The acceptant property says that every student is at least as preferable as an unfilled position. Although it is rarely used in the assignment literature, it is implicitly assumed. The notion of substitutable priorities was first introduced by Kelso and Crawford (1982). Briefly, it says that if students chosen from a set of applicants then they must be chosen from any subset of the set. The class of responsive priorities is a proper subset of that of acceptant substitutable priorities. Echenique (2008) shows that the number of substitutable priorities are exponentially larger than that of responsive priorities. For these reasons, it is 2

important to ask what is equivalent for the DA rule to be Pareto efficient under acceptant substitutable priorities. In this paper, we characterize the Pareto efficient DA rule in light of a priority structure. As a generalization of Ergin (2002), we find a necessary and sufficient condition on the priority structure for the DA rule to be Pareto efficient. We introduce a general acyclical condition on priorities, which is equivalent to Ergin-acyclicity under responsive priorities. Note that our acyclicity condition looks quite different from Ergin-acyclicity since a source of inefficiency is not only involved with responsiveness but also with acceptant substitutability, as we will see an example. The strong appeal of acyclicity is its usefulness in application. Our characterization is helpful in the use of the DA algorithm, for instance, in school choice, the educational authority can check if the priority structure is suitable for the applying the DA algorithm. Related Literature: Under a responsive priority structure, as we already mentioned, the existence of stable assignments including the AOSA is shown by Gale and Shapley (1962). Roth and Sotomayor (1990) show that the DA outcome is not strongly Pareto dominated by any individually rational assignment. Balinski and S¨onmez (1999) introduce “adaption”, which reflects fairness of a rule and requires that in the outcome of the rule, if one agent envies another agent’s assignment, then she must be ranked lower than him. Note that the DA rule adapts to the priority. Then they show that the DA outcome is Pareto superior to not only stable assignments but also any other outcome which adapts to the priority structure. Under a substitutable priority structure, Kelso and Crawford (1982) show the existence of stable assignments. Roth and Sotomayor (1990) further show that the same result as Gale and Shapley (1962), namely, the DA outcome is Pareto superior to any other stable assignment. Moreover, Kojima (2008) shows that the DA outcome is not strongly Pareto dominated by any other individually rational assignment. In addition to the main result, we show that under acceptant substitutable priorities a result similar to Balinski and S¨onmez (1999) holds, namely, the DA rule is Pareto superior to any fair rule. Recently, Kojima and Manea (2009) characterizes the DA rule axiomatically, and this is the most related paper to ours in the sense that they are also working on an acceptant substitutable priorities and analyzes efficiency properties. For efficiency properties, they show that the DA rule is Pareto efficient if and only if the DA rule satisfies Maskin monotonicity if and only if the DA rule is group strategy proof. For a practical use of the DA rule, it is not easy to see whether one of these criteria is satisfied or not. Alternatively, we scope a priority structure, and characterize the Pareto efficient DA rule by acyclicity. The rest of the paper is organized as follows: Section 2 introduces the model and the motivating example. Section 3 presents results. In section 4, we discuss the results, and conclude the paper. The proof of Section 3 is in the Appendix.

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2. Model There are finite agents N and finite indivisible objects A. Let qa ∈ Z be a finite quota of object a, where qa ≥ 1, and let q = (qa )a∈A be whole available indivisible objects in an economy. A preference profile is a vector of linear orders, R = (Ri )i∈N , where Ri denotes agent i’s preference and is defined over A ∪ {i}. The strict part of Ri is denoted by Pi . Let R be the set of all preference profiles. A priority structure is a vector of linear orders ≥ = (≥a )a∈A , where ≥a denotes the priority of agents and defined over the subsets of the set of agents, P(N ). For any a ∈ A, let >a be a strict part of ≥a . Combining with feasibility, we use a chosen function: ∀S ⊂ N , Ca (S) ⊂ S and |Ca (S)| ≤ qa . The relation is that Ca (S) = S 0 if and only if |S 0 | ≤ qa and S 0 ≥a S 00 for all S 00 ⊂ S with |S 00 | ≤ qa . Throughout the paper, we assume preferences and priorities are complete, transitive and antisymmetric. Definition A priority structure is substitutable if ∀a ∈ A, ∀S, S 0 ⊂ N with S 0 ⊂ S, Ca (S) ∩ S 0 ⊂ Ca (S 0 ). Definition A priority structure is acceptant if ∀a ∈ A, ∀S ⊂ N , | Ca (S) |= min{|S|, qa }. An assignment is a function µ : N → A ∪ N with the following property; (1) ∀i ∈ N , µ(i) ∈ A ∪ {i}, and (2) ∀a ∈ A, |µ−1 (a)| ≤ qa . An assignment µ is individually rational if ∀i ∈ N, µ(i)Ri i, and given a priority structure, µ is blocked by a pair if there exists (i, a) ∈ N × A such that aPi µ(i) & i ∈ Ca (µ−1 (a) ∪ {i}). Given a priority structure, an assignment µ is stable if it is individually rational and there is no blocking pair, formally,1 (IR) (N B)

∀i ∈ N, µ(i)Ri i, and 6 ∃(i, a) [aPi µ(i) & i ∈ Ca (µ−1 (a) ∪ {i})].

We say that an assignment µ is Pareto efficient if there is no assignment µ0 such that some agent is better off without any agent being worse off. A rule f is a function from all preference profiles to an assignment and for each R ∈ R, each agent i is assigned with fi (R). Given a priority structure, a rule f is stable if for all R ∈ R, f (R) is stable. A rule f is Pareto efficient if for all preference profile R ∈ R, f (R) is Pareto efficient. A rule f is said to be strategy proof if for any preference profile, no agent can be better off by misrepresenting his true preference, formally, ∀R ∈ R, ∀Ri0 ∈ Ri , ∀i ∈ N , fi (R)Ri fi (RN \i , Ri0 ). A rule f is group strategy proof if there is no group of agents such that some of them can be better off by misrepresenting their true preferences, namely, 1

In a responsive class, in addition to the (IR) and (NB) conditions, non-wastefulness is imposed for stability. However, in our setting, non-wastefulness is redundant from the (NB) condition.

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0 0 ∀R ∈ R, there is no ∅ = 6 N 0 ⊂ N such that fi (RN \N 0 , RN and 0 )Ri fi (R), ∀i ∈ N 0 0 fj (RN \N 0 , RN 0 )Pj fj (R), ∃j ∈ N . In this paper, we mainly focus on the DA rule, denoted by f DA . Given a priority structure and a preference profile, the DA outcome is obtained via the DA algorithm as follows:

The Deferred Acceptance Algorithm: • step 1: Each agent applies to her most preferred object if it makes her better off than being assined nothing. Let Na1 be the set of agents applying to object a. Then object a tentatively accepts Ca (Na1 ) agents and rejects Na1 \Ca (Na1 ). • step t: Each agent who was rejected at step t − 1 applies to her next preferred object if it makes her better of than being assigned nothing. Let Nat be the set of agents applying to object a. Then object a tentatively accepts Ca (Ca (Nat−1 ) ∪ Nat ) and rejects Nat \Ca (Ca (Nat−1 ) ∪ Nat ). • The algorithm terminates when no one applies to a object. Each agent is assigned the object if she is tentatively accepted at the last step, otherwise assigned nothing. The DA algorithm produces AOSA (recall it is a stable assignment and weakly preferred by all agents).

2.2 The Motivating Example This section introduces the motivating example which demonstrates that an AOSA is not Pareto efficient under acceptant substitutable priorities. In general, AOSA is not Pareto efficient under responsive priorities, and so it is under acceptant substitutable priorities. Under responsive priorities, Ergin (2002) introduces Ergin-acyclicity and shows that it is equivalent to Pareto efficient DA rule. However, the following example shows that it is not straightforward to generalize Ergin-acyclicity under acceptant substitutable priorities, since we have an inefficient AOSA under a priority structure which is acceptant substitutable but not responsive. To point out the difference, the definition of responsive priorities is provided: Definition The priority relation ≥ra defined over P(N ) is responsive if, whenever i, j ∈ N , S ⊂ N \{i, j}, {i} ∪ S ≥ra {j} ∪ S ⇔ {i} ≥ra {j}, and whenever i ∈ N and S ⊂ N \{i}, {i} ∪ S ≥ra S.

Example 2 Consider that there are 4 agents, N = {1, 2, 3, 4}, and 3 objects, A = {a, b, c} with qa = 2, qb = qc = 1. The left and right matrices represent the priority

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structure and preference profile, respectively. We can easily verify that the priority of a is not responsive, but can be acceptant substitutable. a {1, 4} {1, 2} .. . 2 4 .. .

b c 2 2 3 3

1 2 3 4 4 a a a , 1 1 b b ∅ ∅ c

4 b a c

∅ The deferred acceptance algorithm outcome is now µ=

1 2 3 4 a b c a

!

6= P E.

This example tells us that an inefficient AOSA is obtained not only under responsive priorities but also acceptant substitutable priorities. Therefore, a sufficient and necessary condition for Pareto efficient DA rule is not easily obtained by a modification of Erginacyclicity.

3. Results We begin with the definition of acyclicity. It looks quite different from Ergin-acyclicity, however, a reader can easily figure out that it is equivalent under responsive priorities. Definition Given an acceptant substitutable priority structure and a vector of quotas, (≥, q), a cycle is constituted of distinct i, j, k ∈ N and a, b ∈ A such that ∃Sa , Sb ⊂ N \{i, j, k} with Sa ∩ Sb = ∅ such that (C)

j 6∈ Ca (Sa ∪ {i, j}) k 6∈ Ca (Sa ∪ {j, k}) i 6∈ Cb (Sb ∪ {k, i})

(S)

|Sl | = ql − 1, ∀l ∈ {a, b}

If there is no cycle, then we say a priority structure is acyclical. The (C) condition means that, for some peculiar preference profiles, there is a possibility of a rejection chain involving three agents and two objects, and the (S) condition further makes such a chain of rejection feasible in process of the DA algorithm. These two conditions are a source of inefficiency of the DA rule. Therefore, an acyclical priority 6

structure has no possibility of an inefficient DA outcome for any preference profiles. We are ready to state our main result. Proposition 1 Given an acceptant substitutable priority structure and a vector of quotas, (≥, q), f DA is Pareto efficient if and only if ≥ is acyclical. P roof : In Appendix. Though our characterization of the Pareto efficient DA rule on priorities is the first attempt under the class of acceptant substitutable priorities, Kojima and Manea (2009) characterize the Pareto efficient DA rule differently. To state their result, we borrow some terminologies. From Kojima and Manea (2009): Ri0 is a monotonic transformation of Ri at a ∈ A∪{i} Ri0 m.t. Ri at a if any object that is ranked above a under Ri0 is also ranked above a under Ri , i.e., bPi0 a ⇒ bPi a, ∀b ∈ A∪{i}. R0 is a monotonic transformation of R at allocation µ if Ri0 m.t. Ri at µ(i) for all i. Definition (Maskin monotonicity in Kojima and Manea (2009)2 ) An assignment rule f satisfies Maskin monotonicity if R0 m.t. R at f (R) ⇒ f (R0 ) = f (R). Remark 1 (Proposition 1 in Kojima and Manea (2009)) Given an acceptant substitutable priority structure and a vector of quotas, (≥, q), the following are equivalent: 1. f DA is Pareto efficient. 2. f DA satisfies Maskin monotonicity. 3. f DA is group strategy proof. The following is a combining result from Kojima and Manea (2009) and ours. Theorem 1 Given an acceptant substitutable priority structure and a vector of quotas, (≥, q), the following are equivalent: 1. f DA is Pareto efficient. 2. f DA satisfies Maskin monotonicity. 3. f DA is group strategy proof. 4. ≥ is acyclical. 2

Maskin monotonicity is originally introduced by Maskin (1999). For the sake of simplicity, we follow the version of Kojima and Manea (2009).

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3.2 Fair Rule As before, we characterize the Pareto efficient DA rule by priorities. However, the following question naturally aries: “ Is choosing a DA rule appropriate? Is there any other rule?” To address this, we would like to compare the DA rule with a fair rule, since fairness is a weak and the smallest demanding requirement. When an agent envies others’ object and she is ranked higher than him in a priority structure, we say that it is not fair. In our setting, this requirement is exactly same as the (NB) condition, and, therefore, given a priority structure, an assignment µ is fair if µ satisfies the (NB) condition, and a rule f is fair if f (R) is fair for all R ∈ R. Note that the DA rule is a fair rule. Then we have the following result. Proposition 2 Given an acceptant substitutable priorities and a vector of quotas, (≥, q), f DA is Pareto superior to any fair rule. P roof : Immediate from Lemma 1 in Kojima and Manea (2009) by replacing R0 with R. Hence, if we would like a rule to be fair, it is the best choice to use the DA rule. One remark is that the above result is similar to Theorem 2 in Balinski and S¨onmez (1999), however, an absence of non-wastefulness makes these two results different in a rigorous sense. It is simply because, in our setting, the (NB) condition and non-wastefulness are indistinguishable. Since underlined concepts of fairness in both models are the same, Proposition 2 may be seen as a generalization of Theorem 2 in Balinski and S¨onmez (1999).

4. Discussion and Conclusion In this paper, we consider an assignment problem with acceptant substitutable priorities. As of now, the class of acceptant substitutable priorities is the largest domain to guarantee for the existence of stable assignment. Our main contribution is that we introduce a general acyclical condition, and proved that the priority structure is acyclical if and only if the DA rule is Pareto efficient. Additionally, we also show a result similar to Balinski and S¨onmez (1999). Recently, Kojima and Manea (2009) characterize the DA rule axiomatically. In their proposition, they also characterize the equivalent relation among Pareto efficiency, Maskin monotonicity and group strategy proofness. In their setting, they allow any acceptant substitutable priority structure, but the set of the DA rules is restricted to those rules that satisfy Maskin monotonicity. Our characterization focuses on the DA rule from a different perspective, that is, we restrict the domain of priorities. The strong appeal of acyclicity is its usefulness in application. In Ergin (2002), there is the theorem which characterizes acyclicity, and that makes it easier to check a priority structure. However, the characterization highly depends on the responsiveness. So it is 8

not clear whether or not we can also have an analogue of such characterization, and this question is left for the future research. In a theoretical perspective, Ergin-acyclicity connects many desirable properties under responsive priorities. Especially, Haeringer and Klijn (2009) characterize the DA rule and Nash implementability by Ergin-acyclicity. In their proof, non-bossiness is essential to have their result, and it comes from consistency of the DA rule which is equivalent to Ergin-acyclicity. Though in our model, there is no relationship between acyclicity and consistency, it may be meaningful to try to generalize results of Haeringer and Klijn (2009) under acceptant substitutable priorities.

Appendix P roof of P roposition 1 (⇒): By contraposition, fix ≥ arbitrary, and suppose ∃R such that f DA (R) = µ is Pareto dominated by some µ0 , then we first want to show that there is a generalized cycle. Definition Given an acceptant substitutable priority structure and a vector of quotas, (≥, q), a generalized cycle is constituted of distinct j, i1 , i2 , . . . , in & a1 , . . . , an with n ≥ 2 such that ∃Sa1 , . . . , San ⊂ N \{j, i1 , . . . , in , } with Sa ∩ Sb = ∅, a 6= b such that (C)

j 6∈ Ca1 (Sa1 ∪ {i1 , j}), in 6∈ Ca1 (Sa1 ∪ {j, in }), in−1 6∈ Can (San ∪ {in , in−1 }), ··· i1 6∈ Ca2 (Sa2 ∪ {i2 , i1 }).

(S)

|Sal | = qal − 1, ∀l ∈ {1, . . . , n}

Since µ is Pareto dominated at R, there must be an agent i such that µ0 (i)Pi µ(i), whereas no one is worse off. Let N 0 be such that {i ∈ N |µ0 (i)Pi µ(i)}. Note that ∀j ∈ N \N 0 , µ0 (j) = µ(j). Since by the (IR) condition, µ0 (i)Pi µ(i)Ri i, we have ∀i ∈ N 0 , µ0 (i) ∈ A. Since ∀i ∈ N 0 , µ0 (i)Pi µ(i), and by the (NB) condition under µ and the acceptant property, |µ−1 (µ0 (i))| = qµ(i) , which implies that ∃j ∈ N 0 such that µ(j) = µ0 (i). If someone who is assigned µ0 (i) under µ is not in N 0 , then he is in N \N 0 , so there are already qµ0 (i) people under µ0 and we have i too. Hence, |µ0 −1 (µ0 (i))| = qµ0 (i) + 1, a contradiction. By this, we always find {i1 , i2 , . . . , in } ⊂ N 0 such that µ(i1 )Pin µ(in )Pin−1 µ(in−1 ) · · · Pi2 µ(i2 )Pi1 µ(i1 ). 9

Now suppose the DA algorithm which attains µ at R, and without loss of generality, before step r, i2 , . . . , in are assigned their final object (µ(i2 ), . . . , µ(in )). Suppose at step r, i1 is assigned a1 = µ(i1 ), then in must be rejected before step r. Then there is a j distinct from i1 , . . . , in such that he is rejected by a1 at step r when i1 applies to a1 at step r and is accepted at step r − 1, that is (At step r − 1): ∅ = 6 Ar−1 ⊂ N \{i1 , · · · , in } applies to a1 , and in 6∈ Ca1 (Ar−1 ∪ Sr−2 ∪ {in }), where Sr−2 stands for agents who are tentatively assigned a1 at step r − 2 and does not include in . Let Sr−1 be the set of agents chosen at step r−1. Note that, by substitutability, Ca1 (Sr−1 ∪ {in }) = Sr−1 (At step r): Ar ⊂ N \{i1 , · · · , in } and i1 applies to a1 , and let Sr be the set of agents who are tentatively assigned a1 at step r, and does not include i1 , that is, Ca1 (Ar ∪ {i1 } ∪ Sr−1 ) = Sr ∪ {i1 }. Suppose that Ca1 (Sr ∪ Sr−1 ∪ {in }). By path independence of chosen functions3 , Ca1 (Sr ∪ Sr−1 ∪ {in }) = Ca1 (Sr ∪ Ca1 (Sr−1 ∪ {in })) = Ca1 (Sr ∪ Sr−1 ). Clearly, in 6∈ Ca1 (Sr ∪ Sr−1 ). Moreover, |Sr | = qa1 − 1, by construction, and by the acceptant property, there must be j in Sr−1 \Sr , since |Sr−1 | > |Sr |, that is, ∃j ∈ Sr−1 \Sr [Ca1 (Sr ∪ Sr−1 ) = Sr ∪ {j}]. By substitutability, in 6∈ Ca1 (Sr ∪ {j, in }). Furthermore, j 6∈ Ca1 (Sr ∪ {i1 , j}), by the result of step r and substitutability. On the other hand, at step r, i2 , · · · , in are assigned their final object, so each of corresponding object is fully filled. (Otherwise, by the acceptant property, µ violates the (NB) condition at R.) Therefore, there are distinct Sa2 , · · · , Ssn agents who are in each waiting list, and by the acceptant property |Sal | = qal − 1. Hence, the condition (C) and (S) of a generalized cycle is satisfied. Next, we want to show that if there is a generalized cycle, then there exists a cycle. Suppose there is a generalized cycle with n ≥ 3, and it is the shortest. 3

Path independence is defined as ∀S, T ⊂ N, C(S ∪ T ) = C(C(S) ∪ T ). It is known that substitutability and IIA is equivalent to path independence. Furthermore, chosen function induced by complete, transitive, and antisymmetric priorities always satisfies IIA.

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(case 1): If i1 6∈ Ca3 (S3 ∪ {i3 , i1 }), then j, i1 , i3 , · · · , in and a1 , a3 , · · · , an constitutes a generalized cycle with n − 1, a contradiction. (case 2): If i1 ∈ Ca3 (S3 ∪ {i3 , i1 }), then we first suppose that i3 is rejected by a3 , that is, i3 6∈ Ca3 (S3 ∪ {i3 , i1 }). In this case, we have i3 6∈ Ca3 (S3 ∪ {i1 , i3 }) i2 6∈ Ca3 (S3 ∪ {i3 , i2 }) i1 6∈ Ca2 (S2 ∪ {i2 , i1 }) with |S3 | = q3 − 1 and |S2 | = q2 − 1, a cycle, which is a contradiction. Therefore, the only possibility left is that there is a k ∈ S3 who is rejected by a3 , that is, k 6∈ Ca3 (S3 ∪{i1 , i3 }). Let S¯3 be S3 \{k} ∪ {i3 }, note that |S¯3 | = q3 − 1. Then we still have k 6∈ Ca3 (S¯3 ∪ {i1 , k}) i2 6∈ Ca3 (S¯3 ∪ {k, i2 }) i1 6∈ Ca2 (S2 ∪ {i2 , i1 }) with |S¯3 | = q3 − 1 and |S2 | = q2 − 1, a cycle, which is a contradition.

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(⇐): By contradiction. Assume there is a cycle. Consider the following preference profile R ∈ R such that Ri : b, a, i Rj : a, j Rk : a, b, k ∀m ∈ Sa ,

Rm : a, m,

∀n ∈ Sb , ¯, ∀o ∈ N

Rn : b, n, Ro : o, ,

¯ = N \ [{i, j, k} ∪ Sa ∪ Sb ]. where N Then the DA algorithm produces µ = f DA (R) as follows: µ=

¯ Sa Sb i j k N ¯ a b a j b N

!

¯ Sa Sb i j k N ¯ a b b j a N

!

.

But there exists µ0 such that µ0 =

.

By the preference profile R, ∀l ∈ N,

µ0 (l)Rl µ(l)

and for i and k, b = µ0 (i)Pi µ(i) = a and a = µ0 (k)Pk µ(k) = b. This contradicts to the fact that f DA is Pareto efficient. 11

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Reference Abdurkadiro˘glu, A., S¨onmez, T., 2003. School Choice: A Mechanism Design Approach, American Economic Review, 93, 729-747 Balinski, M., S¨onmez, T., 1999. A Tale of Two Mechanisms: Student Placement, Journal of Economic Theory, 84, 73-94 Echeique, F., 2007. Counting Combinatorial Choice Rules, Games and Economic Behavior, 58, 231-245 Ergin, H., 2002. Efficient Resource Allocation on the Basis of Priorities, Econometrica, 70, 2489-2498 Gale, D., Shapley, L., 1962. College Admissions and the Stability of Marriage, American Mathematical Monthly, 69, 9-15 Haeringer, G., Klijn, F., 2009. Constrained School Choice, Journal of Economic Theory, 144, 1921-1947 Maskin, E., 1999. Nash Equilibrium and Welfare Optimality, Review of Economic Studies, 66, 23-38 Kelso, A., Crawford, V., 1982. Job Matching, Coalition Formation, and Gross Substitutes, Econometrica, 50, 1483-1504 Kojima, F., 2007. The Law of Aggregate Demand and Welfare in the Two-Sided Matching Market, Economics Letters, 99, 581-584 Kojima, F., Manea, M., 2009. Axioms for Deferred Acceptance, forthcoming, Econometrica Roth, A., Sotomayor, M., 1990. Two Sided Matching: A Study in Game-Theoretic Modeling and Analysis,” Econometric Society Monograph ¨ S¨onmez, T., Unver, M. U., 2008. Matching, Allocation, and Exchange of Discrete Resource, forthcoming, Handbook of Social Economics

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