Matching with the simplest semiorder preference relations: stability and Pareto-efficiency National Research University Higher School of Economics SING, Paris, 20.07.2011
Gale-Shapley matching model Players A - set of students, B - set of universities Each student a A can be admitted to one university Each university b B can admit no more than q b students. Each player has linear order preferences over players on the other side Definition Matching is a mapping from A B to subsets of A B such that: 1 a A µ(a) = {b} (b B) or µ(a) = a 2 b B µ(b) A or µ(b) = b 3 b B µ(b) q b
Stability of matching Matching is stable if the following is satisfied (Gale, Shapley): individual rationality of students: no student is matched to an unacceptable university, individual rationality of universities: no universitiy admits an unacceptable student, empty seats stability: no university-student pair exists such that an applicant prefers this university to her current match and the university finds applicant acceptable and has empty seats pairwise stability: no unversity-student pair exists such that an applicant prefers this university to her current match and the university prefers this student to at least one of currently admitted students.
Deferred acceptance procedure Gale and Shapley provided a constructive proof of existance of a stable matching. Deferred acceptance procedure (students proposing) Step 1. Each student applies to her most preferred university. Each university temporary admits no more than q b best students and rejects the others.... Step k. Each rejected student applies to her second most preferred university. Each universiy consideres all currently applying students (both remaining from the previous steps and applied at the k-th), temporary admits no more than q b among them and rejects the others. When each student is temporary assgined to university or is rejected by all acceptable universities, procedure stops.
Properties of DA procedure DA procedure always generates a stable matching µ For students µ is a unique Pareto-optimal stable matching For universities µ is weakly Pareto-dominated by any other stable matching Unfortunately, when preferences of agents are not strict: DA procedure is not well-defined Even If DA procedure is redefined, two last properties do not hold (Abdulkadiroglu, Sonmez, 2003)
The simplest semiorder preferences: motivating example Three students apply to a university b a 1 has 92 points exam score a 2 has 91 points exam score a 3 has 90 points exam score University b has the following preferences: a 1 is indifferent to a 2, a 2 is indifferent to a 3, but a 1 is preffered to a 3. This is an example of very natural situation, where negative transitivity property for preference relation does not hold.
The simplest semi-order preference relation Formally, simplest semi-order preference relation P can be characterized as irreflexive xpx transitive xpy,ypz xpz not negatively transitive: for some x,y,z xpy,ypx,ypz and zpy, but xpz x, z there exists no more than one y for which negative transitivity condition does not hold. This less rigid formulation of preference in some cases corresponds better to real preferences of participants
The Model Players Each student a A can be admitted to one university Each university b B can admit no more than q b students. Preferences R - the profile of students preferences over universities. a A R a is a linear order on B a. - the profile of universities preferences over students. b B b is the simplest semi-order on A b. b b satisfies no indifference with empty set property, i.e. b B, a A either a b or a b. preference relation of each university over the sets of applicants satisfies the responsiveness to the preference relation over individuals: b B, a 1, a 2 A, A A if a 1 b a 2 then A a 1 b A a 2.
Existance of stable matching Theorem Stable matching always exists in case, when universities have simplest semi-order preferences over students Theorem For each stable matching µ there exists such universities preference profile, which consists of linear order preferences and doesn t contradict original profile, such that matching µ is also stable under this strict preference profile.
Pareto-optimal stable matching Erdil and Ergin, 2006, propose a procedure which allows to find student Pareto-optimal stable matching in case of universities weak order preferences. We modify it for the case of the simplest semi-order preferences and prove, that it works in our model. 1 Preferences of universities are aritrary transformed to linear orders and Deferred Acceptance procedure with students proposing is applied. Result is a matching µ, stable but not necessarily Pareto-efficient. 2 We try to find so-called Stable Improvement Cycle. If SIC exists, then we can improve µ. 3 Procedure ends, when we arrive to the matching µ which does not have Stable Improvement Cycle.
Stable Improvement Cycle C(b, µ) = {a : br a µ(a)}, D(b, µ) = {a C : a Ca b a }. Definition Stable Improvement Cycle consists of distinct applicants a 1,..., a n a 0 (n 2) such that µ(a i ) B (each student in a cycle is assigned to a university), a i µ(a i+1 )R ai µ(a i ) a i a i D(µ(a i+1 ), µ)(a i is one of the best students among those who prefer µ(a i+1 ) to her current match)
Pareto-efficience: necessary and sufficient condition Theorem Fix and R, and let µ be a stable matching. If µ is student-side Pareto dominated by another stable matching, the it admits a Stable Improvement Cycle.
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Proof of the theorem Consider two stable matchings µ and ν and a ν a µ. A = {a : ν(a) a µ(a)} and B = µ(a ). b - transformed preference relation of university b, such that if a 1 b a 2, a 2 b a 3, a 1 b a 3, then a 1 b a 2, a 2 b a 3, a 1 b a 3. G(V, E) - an oriented graph, where V = B and an edge e(b 1, b 2 ) E if a µ(b 1 ) such that a A and a A : a b 2 a. Graph G always contains a cycle, as each university in B is preferred to her university under µ by at least one student. Furthermore, this cycle will always be a Stable Improvement Cycle: first and second properties of the SIC hold by construction. Let s show that third also holds. Evidently, a A, b SIC(b) b a. Let us prove the remaining by contradiction. Suppose that c A/A such that b c µ(c) and c b SIC(b) (blocking pair). But under stable matching ν ν(b) b c, as ν is the stable matching. As SIC(b) is the best student among those, who desire b under µ according to b, c b SIC(b) contradicts stability of ν.