STABLE SOLUTIONS ON MATCHING MODELS WITH QUOTA RESTRICTION

Stable Solutions on Matching Modelswith Quota Restriction�

byDel�na Femeniay, Mabel Maríy, Alejandro Nemez

andJorge OviedozNovember 2008

abstract: In this paper, we present a matching market in which an institutionhas to hire a set of pairs of complementary workers, and has a quota that is themaximum number of candidates pair positions to be �lled. We de�ne a naturalstable solution and �rst show that in the unrestricted institution preferences do-main, the set of stable solution may be empty and second we obtain a completecharacterization of the stable sets under responsive restriction of the institution�spreference.

. Keywords: Matching. Quota restriction. q-stable.

�This work is partially supported by the Consejo Nacional de Investigaciones Cientí�casy Técnicas CONICET, through Grant PICT-02114, by the Agencia Nacional de PromociónCientí�ca y Técnica, through Grant 03-10814, by the Universidad Nacional de San Luisthrough Grant 319502, and by the Universidad Nacional de San Juan through Grant21/F776.

yUniversidad Nacional de San Juan. San Juan, Argentina. (E-mails: del�[email protected], [email protected].)

zInstituto de Matemática Aplicada. Universidad Nacional de San Luis and CONICET.Ejército de los Andes 950. 5700, San Luis, Argentina. (E-mails: [email protected],[email protected].)

1 Introduction

One-to-one matching models have been useful for studying assignment prob-lems with the distinctive feature that agents can be divided from the verybeginning into two disjoint subsets of complementary workers: the set ofworkers of type I and the set of workers of type II. The nature of the assign-ment problem consists of matching each agent (workers of type I) with anagent from the other side of the market (workers of type II).On one hand, the fundamental question of this assignment problem con-

sists of matching each worker, with a worker from on the other side. Roth(1984, 1986, 1990, and 1991), Mongell and Roth (1991), Roth and Xing(1994), and Romero-Medina (1998) are examples of papers studying partic-ular matching problems like entry-level professional labor markets, studentadmissions at colleges, etc.The agents have preferences on the potential partners. Stability has been

considered the main property to be satis�ed with any sensible matching.A matching is called stable if all the agents are matched to an acceptablepartner and there is no matched pair of workers that would prefer the otherpartner to their current one.Sometime an institution will hire the sets of pairs of complementary work-

ers (matching). This institution has preference on this potential set of pairs.Most often, the institution has a quota that is the maximum number of in-dividuals that must be matched because there are more pair of candidatesthan positions to be �lled by the institution (quota q). This limitation mayarise from, for example, technological, legal, or budgetary reasons. Since,only matching such as their cardinality is smaller or equal to q is acceptable,the assignment problem consists of matching each worker, on one side, witha worker, on the other side, such as the pair of workers work for the insti-tution, and the number of worker pair of the matching acceptable for theinstitution it its most q: If the number of pair of workers of the matching issmaller than the quota q, then this matching is acceptable for the institutionand in this case the theory of two sided matching is applied. If the numberof pair of workers of the matching is greater than the quota, this matchingis not acceptable for the institution and (at most), q pair of workers must bechosen according to their preference.Because the institution have a preference on the potential set of pairs

assigned, this problem is di¤erent to the three side matching problems intro-duced by Alkan (1988).

1

We will re-de�ne the stability property by considering the quota restric-tion: a matching is called q�stable if the following conditions are satis�ed;i) all the pair of workers that are chosen for the institution have acceptablepartners, ii) there is no a matched pair of workers that is not matched toeach other and both would prefer to be matched to each other rather thanstaying with their current partners, and iii) there is no unmatched pair ofworkers, at least one is not chosen by the institution, who both would preferto be matched to each other rather than staying with their current partnersand the institution prefers the new matching to the current. First we showthat in the unrestricted institution preferences domain the set of stable solu-tion may be empty and second we obtain a complete characterization of thestable sets under responsive restriction of the institution�s preferences.The paper is organized as follows. In Section 2 we present the notation,

the most important de�nition of the model, and we show that the set ofq�stable matching of the assignment market with quota restriction may beempty. In Section 3, we introduce the preference responsive over matching.Finally, in section 4 we will consider a restriction over the institution�s pref-erences under which the existence of q�stable matching is guaranteed, thenwe characterize the q-stable set in contrast with as a stable set of standardmatching submarket.

2 The models

Our models consist of two disjoint sets of agents, the set of n workers of typeI, and the set of m workers of type II and an institution which we denote byD = fd1; :::; dng, E = fe1; :::; emg and U , respectively.Each worker of type I has preference over the set of workers of type II

and each worker of type II has preference over the set of workers of typeI. These preferences are such that each worker, say d 2 D; prefer to remainunassigned to work with a worker, e 2 E; who is not of his interest. Formally,each worker d 2 D has a strict, transitive, and complete preference relationPd over E[f;g ; and each worker e 2 E has a strict, transitive, and completepreference relation Pe over D [ f;g.Notice that we are considering only strict preferences. Similarly results

may be obtained if indi¤erence is allowed.Preference pro�les are (n+m)-tuples of preference relations and they are

represented by P = (Pd1 ; :::; Pdn ;Pe1 ; :::; Pem) = (PD; PE).

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Given a preference pro�le P, we denote the standard matching marketby M = (D;E;P):Given a preference relation Pd the subsets of workers preferred to the

empty set by d are called acceptable. Similarly, given a preference relation Pethe subsets of workers preferred to the empty set by e are called acceptable.To express preference relations concisely, and since only acceptable part-

ners will matter, we will represent preference relations as a lists of acceptablepartners only. For instance,

Pdi = e1; e3; e2 Pej = d1; d3:

indicate that e1Pdie3Pdie2Pdi; and d1Pejd3Pej;.The assignment problem consists of matching workers of type I to workers

of type II keeping the bilateral nature of their relationship and having thepossibility that both types of workers may remain unmatched. Formally:

De�nition 1 A matching � is a mapping from the set D [ E into the setD [ E [ f;g such that for all d 2 D and e 2 E:

1. Either � (d) 2 E or else � (d) = ;:

2. Either � (e) 2 D or else � (e) = ;.

3. � (d) = e if and only if � (e) = d:

LetM be the set of all possible matching �:Given a matching market M = (D;E;P); a matching � is blocked by a

single agent f 2 D [ E if ;Pf �(f): We say that a matching is individuallyrational if it is not blocked by any single agent. A matching � is blocked bya pair of workers (d; e) if dPe �(e) and e Pd �(d):

De�nition 2 A matching � is stable if it is not blocked by any individualagent or by any pair of workers.

Given a matching market M = (D;E;P); S(M) denote the set of stablematching.Given a matching �, we denote the cardinality of � as

#� = #fd : �(d) 2 Eg = #fe : �(e) 2 Dg:

We present, as a Remark below, three properties of stable matchings.

Remark 1 Let M = (D;E;P); M 0 = (D0; E;P0) been matching markets.Then:

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1. S (M) 6= ;:

2. For all �; �0 2 S (M) ; fd : �(d) 2 Eg = fd : �0(d) 2 Eg; and fe :�(e) 2 Dg = fe : �0(e) 2 Dg:

3. IfD0 � D, andP agrees withP0 onD0 and E; and �D (�E) and �0D (�

0E)

are the D (E)�optimal stable matching for M and M 0 respectively.Then for all d 2 D0 and e 2 E; we have that �0D(d)Rd�D(d) and�D(e)Re�

0D(e):

Properties 1 and 2 are due Gale and Shapley, (1962) and Mc Vitie andWilson (1970) respectively. Property 3 are due Kelso and Crawford (1982),for more general market model and Gale and Sotomayor (1985)1.Now, we are assuming that the pair of workers will work for the institution

U and it has a maximum number of positions, quota q; to be �lled, thenonly matching such that their cardinality is smaller or equal to q may beacceptable. We denoteMq = f� 2M : #� � qg:Institution U; has a preference over the set of pairs who are working for.

It formally, institution U has a re�exive, transitive, and complete binaryrelation RU over the set of all possible matching M, including the emptymatching. As usual, let PU and IU denote the strict and indi¤erent preferencerelations induced by RU , respectively. The institution may choose somematching ofM according to their preference PU and their quota restrictionq: This new matching market with quota restriction is denoted by M q

U =(M ;RU ; q):Notice that if q � minfm;ng; the set of all the matching may be accept-

able, i.e.,M =Mq:The we assume that, of now in more, q � minfm;ng:Amatching � is acceptable for institution U according to their preferences

RU if � 2 Mq and �RU�;, where �; is the matching such that �;(f) = ;,for every f 2 D [ E.In this model, the criteria for excluding potential matching has to take

into account the institution preference. Then we have to exclude a matching� if:i) The matching � is blocked by a single agent.ii) The matching � is blocked by a pair of workers and the new matching

1See Theorem 2.25 pp 44, and pp 50 of Roth and Sotomayor (1990) .

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formed with the blocking pair is preferred by the institution.iii) The matching � such that #� > q is not accepted by the institution.Given MU and a quota q � minfn;mg. The institution only may accept

matching � 2 M which they prefer to �; the empty matching accordingto their preference PU , and its cardinality is not bigger than the number ofpositions allowed, #� � q: A matching is acceptable if the partner assignedin the matching is preferred to beginning single. Formally,

De�nition 3 Given a matching market MU and a quota q � minfn;mg; amatching � is q�individually rational if #� � q; �PU�

; and � (f)Rf;for every worker f 2 D [ E:

Given a matching � 2 Mq and a pair of workers (d; e); we can de�ne�(d;e) as follows:

�(d;e)(f) =

8>><>>:�(f) if f =2 fd; e; �(e); �(d)gd if f = ee if f = d; otherwise.

Notice that, if �(d) = e, then �(d;e) = �:Remark 2 The matching �(d;e) may be not q�individually rational. Con-sider a matching � such that #� = q and let (d; e) be such that �(d) = ; =�(e); then #�(d;e) > q and �(d;e) is not q�individually rational.Usually, in the standard models, (d; e) is blocking pair if they are un-

matched and both would prefer to be matched to each other rather thanstaying with their current partners. Notice that in our models, we may havea blocking pair (d; e) such that the new matching formed by satisfying thisblocking pairs is not acceptable for institution U: Then, we will consider twotype of blocking pairs of matching �. One type is when both workers arematched by � and in this case the workers would prefer to be matched toeach other rather than staying with their current partners. The other type iswhen at least one worker is unmatched by � and both workers would prefer tobe matched to each other rather than staying with their current partners andthe institution prefers the new matching obtained by satisfying the blockingpair to the current one. Formally:

De�nition 4 A matching � is q� blocked by pair of workers (d; e) if

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1. ePd� (d) ; dPe� (e) and

2. either

(a) �(d) 2 E and �(e) 2 D or

(b) �(d;e) is q�individually rational and �(d;e)RU�:

A matching � is q� stable if is not blocked by a single agent, workers orinstitution, and pair of workers.

De�nition 5 A matching � is q� stable if it is q�individually rational andis not q�blocked by any pair of workers.

Given a matching market M qU ; with q � minfn;mg, we denote S(M q

U)the set of all q�stable matching.

Proposition 1 Let M qU = (M ;RU ; q) be a matching market with quota re-

striction and q = minfm;ng: Then Sq(M) 6= ;

The proposition 1 follows immediately from 1 of Remark 1 and the fol-lowing lemma:

Lemma 1 LetM qU = (M ;RU ; q) be a matching market with quota restriction

and q = minfm;ng: Then S(M) � S(M qU):

Proof Let � 2 S(M), then � is individually rational and becauseq = minfm;ng we have that � is q�individually rational.Assume that � =2 S(M q

U), then exists a pair (d; e) such that � is q�blockedby pair (d; e), then either ePd� (d) ; dPe� (e) and �(d) 2 E and �(e) 2 D;or ePd� (d) ; dPe� (e), �(d;e) is q�individually rational and �(d;e)RU�: In bothcases ePd� (d) ; dPe� (e) which implies that the matching � is blocked by(d; e): This contradict that � 2 S(M): �

The following example shows that S(M)may be a proper subset of S(M qU)

for q = minfm;ng:

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Example 1 Let M2U = (M ;RU ; 2) be the matching market with quota re-

striction such that D = fd1; d2g and E = fe1; e2; e3g are the two set ofworkers with the preference pro�le (Pd1 ; :::; Pe3), where:

Pd1 = e1; e2 Pe1 = d1

Pd2 = e3; e2 Pe2 = d2; d1

Pe3 = d2:

and RU satis�es:�d1 d2 ;e1 e2 e3

�PU

�d1 d2 ;e1 e3 e2

�PU

�d1 d2 ;e2 e3 e1

�:

Consider the following individual rational matching of cardinality two :

�1 =

�d1 d2 ;e1 e3 e2

�; �2 =

�d1 d2 ;e1 e2 e3

�; and �3 =

�d1 d2 ;e2 e3 e1

��3, is blocked by (e1; d1), because e1Pd1�3(d1) = e2 and d1Pe2�3(e1) = ;:

�2 is blocked by (e3; d2), because e3Pd2�2(d2) = e2 and d2Pe3�2(e3) = ;:

We have that S(M) = f�1g:

Since q = 2; then �2 is not 2�blocked by (e3; d2), because �2RU�1: ThenS(M2

U) � f�1; �2g, which implies that S(M) S(M2U):

A natural question for this model arises: The set of q-stable matchings isnot empty for any q < minfn;mg: Unfortunately the answer for the modelswith unrestricted institution preferences, is a NO. The following exampleshows that for q < minfn;mg; the set S(M q

U) may be empty.

Example 2 LetM2U = (M ;RU ; 2) be the matching market with quota restric-

tion such that the preference relation of RU satis�es the following condition:For every �; �0 2Mq:

�PU�0 , #� > #�0: (1)

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Let D = fd1; d2; d3g and E = fe1; e2; e3; e4g be the two sets of workers withthe preference pro�le (Pd1 ; :::; Pe4), where:

Pd1 : e1; e3 Pe1 : d2; d1

Pd2 : e3; e2 Pe2 : d2; d3

Pd3 : e2; e4; e3 Pe3 : d1; d3; d2

Pe4 : d1; d3:

By condition (1) every matching � such that #� 6= 2 is not 2-stable.Consider the followings 2-individual rational matching:

�1 =

�d1 d2 d3 ; ;e1 e3 ; e2 e4

�is 2-blocked by (d3; e3)

�2 =

�d1 d2 d3 ; ;e1 e2 ; e3 e4


�3 =

�d1 d2 d3 ; ;e1 ; e2 e3 e4


�4 =

�d1 d2 d3 ; ;e1 ; e4 e2 e3


�5 =

�d1 d2 d3 ; ;; e3 e2 e1 e4


�6 =

�d1 d2 d3 ; ;e1 ; e3 e2 e4


�7 =

�d1 d2 d3 ; ;e3 ; e2 e1 e4


�8 =

�d1 d2 d3 ; ;e3 ; e4 e1 e2


�9 =

�d1 d2 d3 ; ;e3 e2 ; e1 e4


8

�10 =

�d1 d2 d3 ; ;; e3 e4 e1 e2

�is 2-blocked by (d1; e3):

�11 =

�d1 d2 d3 ; ;; e2 e4 e1 e3


�12 =

�d1 d2 d3 ; ;; e2 e3 e1 e4

�is 2-blocked by (d1; e3):

Which implies that S(M2U) = ;:

From now on, we will denote F 2 fD;Eg and F c 2 fD;Eg such thatfF; F cg = fD;Eg; and denote f 2 F a generic worker.Given F 0 � F; we denote PjF 0 the restriction of PF to F 0: Given M =

(F; FC ;P), we denote MF 0 = (F0; FC ; PjF 0 ; PFC ) the restriction of M to F 0.

To make it simple, we are going to denote MF 0 = (F0; FC ;P); where we

have to understood that P =(PjF 0 ; PFC )

Lemma 2 Given M = (D;E;P) and F 0 � F; let � and �0 be the stablematching for M and MF 0 respectively. Then #�0 � #� � #�0 +#(FnF 0):

Proof Without losing of generality we are assuming that F = D: Let �Dand �0D be the D�optimal stable matching for M and MD0 respectively. By(Theorem 2.25 pp44), Roth and Sotomayor (1990) we have that

�0D(d)Rd�D(d) for every d 2 D0 (2)

and�D(e)Re�

0D(e) for every e 2 E (3)

Since �D and �0D are individually rational, then (2) implies that �

0D(d) 6=

;; for every d 2 D0 such that �D(d) 6= ; and (3) implies that �D(e) 6= ;; forevery e 2 E such that �0D(e) 6= ;: Hence we have that

fd 2 D0 : �D(d) 6= ;g � fd 2 D0 : �0D(d) 6= ;g (4)

andfe 2 E : �0D(e) 6= ;g � fe 2 E : �D(e) 6= ;g: (5)

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Since

#�D = #fd 2 D : �D(d) 6= ;g = #fe 2 E : �D(e) 6= ;g;

#�0D = #fe 2 E : �0D(e) 6= ;g = #fd 2 D : �0D(d) 6= ;g;and (4) implies that

#�D = #fd 2 D0 : �D(d) 6= ;g+#fd 2 DnD0 : �D(d) 6= ;g

� #fd 2 D0 : �D(d) 6= ;g+ fd 2 DnD0 : �0D(d) 6= ;g� #�0D +#(DnD0):

Similarly, using (5), we have that

#�0D = #fe 2 E : �0D(e) 6= ;g � #fe 2 E : �D(e) 6= ;g = #�D:

Thus,#�0D � #�D � #�0D +#(DnD0): (6)

By 2 of Remark 1, we have that #� = #� 0 for every stable matching � and� 0; so (6) implies that

#�0 � #� � #�0 +#(DnD0);

for every � 2 S(M) and �0 2 S(MD0): �

3 Preferences responsive over matching

In this section we are going to consider the matching market models underthe restriction that the institution preferences are responsive.At this point in our description of the matching market, we are going

to assume that the institution has preferences over each set of workers andtheir preferences over matchings are directly connected to its preferences overworkers.A preference of the institution will be called responsive to its individual

preferences if for any matching that di¤er in only one worker, the institutionprefers the matching that has the most preferable worker according to theindividual preferences.

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We can state the description formally, as follows.Given a matching market M = (D;E; P ); for every matching � consider

the following subset of D � E :

B� = f(d; e) 2 D � E : �(d) = eg

Now, consider the following special matching, for every f 2 D [ E:

�(d;e)(f) =

8<:; si f =2 fd; egd si f = ee si f = d:

Notice that �(d;e) = �;(d;e):

De�nition 6 A preference relation RU is responsive extension of prefer-ences �D and �E over D[f;g and E[f;g respectively, such that it satis�esthe following conditions:i) �(d;e)PU�; if and only if d �D ; and e �E ;:ii) �PU�; if and only if �(d;e)PU�; for every (d; e) 2 B�:iii) �(d;e)PU�(d;e

0) if and only if e �E e0:iv) �(d;e)PU�(d

0;e) if and only if d �D d0:v) For every �; �0 2M such that #� = #�0 and B� = B�0nf(d0; e0)g[f(d; e)gwe have that:

�PU�0 if and only if �(d;e)PU�(d

0;e0):

vi) For every �; �0 2M such that B�0 � B� and �PU�;; then �PU�0.vii) For every �; �0 2 M such that �(E) = �0(E) and �(D) = �0(D); then�IU�

0:

We refer a preference RU as responsive if there are two individual prefer-ences �D and �E over D [ f;g and E [ f;g respectively, such that RU is aresponsive extension.

Remark 3 Given two preferences �D and �E; over D[f;g and E[f;grespectively, we can construct a responsive preference relation RU over thesetM. Moreover, this extension is not unique.

The following example shows one way to extend a responsive preferencesover the setM , from two individual preferences �D and �E :

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Example 3 Let E = fe1; e2g and D = fd1; d2; d3g be two sets of workers.Let �D and �E be the following preferences over D and E :

e1 �E e2 �E ; and d3 �D d2 �D d1 �D ;:

Consider the following matchings:

�1 =

�d1 d2 d3e1 e2 ;

�; �2 =

�d1 d2 d3e1 ; e2

�; �3 =

�d1 d2 d3e2 e1 ;

�;

�4 =

�d1 d2 d3e2 ; e1

�; �5 =

�d1 d2 d3; e1 e2

�; �6 =

�d1 d2 d3; e2 e1

�;

�7 =

�d1 d2 d3 ;e1 ; ; e2

�, �8 =

�d1 d2 d3 ;e2 ; ; e1

�, �9 =

�d1 d2 d3 ;; e1 ; e2

�,

�10 =

�d1 d2 d3 ;; e2 ; e1

�, �11 =

�d1 d2 d3 ;; ; e1 e2

�, �12 =

�d1 d2 d3 ;; ; e2 e1

�.

Then by condition i) we have that:

�7PU�; �8PU�

; �9PU�; �10PU�

; �11PU�; �12PU�

;:

By condition ii) we have that:

�1PU�; �2PU�

; �3PU�; �4PU�

; �5PU�; �6PU�

;:

By condition iii) we have that:

�11PU�9PU�7 �12PU�10PU�8:

By condition iv) we have that

�7PU�8 �9PU�10 �11PU�12:

By condition iv) we have that:

�;PU�1 �;PU�2 �;PU�3 �;PU�4:

By condition vi) we have that:

�6PU�10 �6PU�11 �5PU�9 �5PU�12:

By condition vii) we have that:

�6IU�5 �1IU�3 �2IU�4:

Notice that there are many unde�ned relations that we are free to choose.For example between �9 and �12, �11 and �1; etc:

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4 Existence of stable solution

Now, we are going to consider the model M qU ; where RU is a responsive pref-

erence. Without losing of generality and in order to avoid adding notationalcomplexity to the model M q

U ; we are assuming that all the agents of sets Dand E are acceptable for the institution, i.e. for every d 2 D and e 2 E; wehave that d �D ; and e �E ;:For every t 2 N; we can de�ne the following subset F t � F such that

#F t = t; and for every f 2 F t and f 0 =2 F t we have that f �F f 0: Note thatF 1 � F 2 � ::: � F l = F; where #F = l:We denote by d = f1; 2; :::;#Dg and e = f1; 2; :::;#Eg, for every (t1; t2) 2

d� e, we denote M (t1;t2), the restriction of M to Dt1 and Et2 , i.e., M (t1;t2) =(Dt1 ; Et2 ;P).Since 2 of Remark 1 we have that every stable matching has the same

cardinality. Then, for a given MU = (M;RU); (t1; t2) 2 d� e; and q; we cande�ne the following sets of matchings:

Tq(M(t1;t2)) =

8<: S(M (t1;t2)) if #� = q for every � 2 S(M (t1;t2))

; otherwise

andTq(M) = f� : 9 (t1; t2) such that � 2 Tq(M (t1;t2))g:

Notice thatTq(M) =

[(t1;t2)2d�e

Tq(M(t1;t2)):

We �rst de�ne the following lemmas which will be used to prove the next

proposition.

Lemma 3 Let � 2 Tq(M (t1;t2)) and (t01; t02) be such that fd 2 D : �(d) 6= ;g �

Dt01 ; fe 2 E : �(e) 6= ;g � Et02, t01 � t1 and t02 � t2: Then � 2 Tq(M (t01;t02)):

Proof Because fd 2 D : �(d) 6= ;g � Dt01 and fe 2 E : �(e) 6= ;g � Et02, �is a matching on M (t01;t

02): Since Et

02 � Et2 and Dt01 � Dt1 any individual or

blocking pair of � on M (t01;t02) is individual or blocking pair of � on M (t1;t2).

Which implies that � 2 Tq(M ((t01;t02)): �

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Lemma 4 Let (t1; t2) and (t01; t02) be such that Tq(M

(t1;t2)) \ Tq(M (t01;t02)) = ;

with t01 � t1 and t02 � t2: Then Tq(M (t1;t2)) � Tq(M (t01;t02)):

Proof Because Tq(M (t1;t2))\Tq(M (t01;t02)) 6= ;; let � 2 Tq(M (t1;t2))\Tq(M (t01;t

02)):

Since � 2 Tq(M(t01;t

02)) we have that fd 2 D : �(d) 6= ;g � Dt01 and

fe 2 E : �(e) 6= ;g � Et02 :Let �� 2 Tq(M (t1;t2)) be, since � 2 S(M (t1;t2)) and �� 2 S(M (t1;t2)); by 2 of

Remark 1, we have that

fd 2 D : �(d) 6= ;g = fd 2 D : ��(d) 6= ;g � Dt01

andfe 2 E : �(e) 6= ;g = fe 2 E : ��(e) 6= ;g � Et02 :

Which implies that, by lemma 3, �� 2 S(M (t01;t02)) and consequently �� 2

Tq(M(t01;t

02)): �

Lemma 5 Let (t1; t2) and (t01; t02) be such that Tq(M

(t1;t2))\Tq(M (t01;t02)) 6= ;:

Then exists (�t1; �t2) such that Tq(M (t1;t2)) � Tq(M(�t1;�t2)) and Tq(M (t01;t

02)) �

Tq(M(�t1;�t2)):

Proof Let � 2 Tq(M (t1;t2)) \ Tq(M (t01;t02)): De�ne �t1 = min(t1; t01) and �t2 =

min(t2; t02): Since � 2 Tq(M (t1;t2)) we have that fd 2 D : �(d) 6= ;g � Dt1

and fe 2 E : �(e) 6= ;g � Et2 : Because � 2 Tq(M(t01;t

02)) we have that

fd 2 D : �(d) 6= ;g � Dt01 and fe 2 E : �(e) 6= ;g � Et02 : Hence we have

thatfd 2 D : �(d) 6= ;g � D�t1 and fe 2 E : �(e) 6= ;g � E�t2

By Lemma 3, we have that

� 2 Tq(M (�t1;�t2)):

Which implies that

Tq(M(t1;t2)) \ Tq(M (�t1;�t2)) 6= ; and Tq(M (t01;t

02)) \ Tq(M (�t1;�t2)) 6= ;:

Then Lemma 4 implies that

Tq(M(t1;t2)) � Tq(M (�t1;�t2)) and Tq(M (t01;t

02)) � Tq(M (�t1;�t2)):

�

The following proposition gives us some information about the structureof the set Tq(M):

14

Proposition 2 Let M qU = (M ;RU ; q) be a matching market with quota re-

striction. Then

Tq(M) =

�[(t1;t2)2K

Tq(M(t1;t2));

where

K = f(t1; t2) 2 d� e : 8 (t01; t02) 6= (t1; t2) t01 � t1; t02 � t2then Tq(M (t1;t2)) \ Tq(M (t01;t

02)) = ;

:

Proof First we show that Tq(M) =S

(t1;t2)2KTq(M

(t1;t2)); and second that

Tq(M(t1;t2))\Tq(M (t01;t

02)) = ; for every (t01; t02); (t1; t2) 2 K such that (t01; t

02) 6=

(t1; t2):By de�nition of Tq(M) we have that:[

(t1;t2)2K

Tq(M(t1;t2)) � Tq(M): (7)

Now we prove that Tq(M) �S

(t1;t2)2KTq(M

(t1;t2)):

Let � 2 Tq(M) =S

(t1;t2)2d�eTq(M

(t1;t2)) be, then exists (t1; t2) 2 d�e such

that � 2 Tq(M (t1;t2)): Assume that (t1; t2) =2 K then, there (t01; t02) 6= (t1; t2)

exists, t01 � t1 and t02 � t2; such that Tq(M (t1;t2)) \ Tq(M (t01;t02)) 6= ;.

Let (t�1; t�2) 2 d� e be the minimal pair such that

� 2 Tq(M (t�1;t�2)); (8)

That is, for every (t1; t2) 6= (t�1; t�2); with t1 � t�1 and t2 � t�2, then � =2

Tq(M(t1;t2)); which implies that Tq(M (t�1;t

�2)) \ Tq(M (t1;t2)) = ;:

We are going to show that (t�1; t�2) 2 K: Assume otherwise, that (t�1; t�2) 62

K is, then exists (t01; t02) 6= (t�1; t�2) , with t01 � t�1, t

02 � t�2 and Tq(M

(t01;t02)) \

Tq(M(t�1;t

�2)) 6= ;: Lemma 4, implies that

Tq(M(t�1;t

�2)) � Tq(M (t01;t

02)): (9)

By (8) and (9), � 2 Tq(M(t01;t

02)), but this contradict the minimality

of (t�1; t�2). Consequently � 2 Tq(M

(t�1;t�2)), with (t�1; t

�2) 2 K, thus � 2

15

S(t1;t2)2K

Tq(M(t1;t2)): Hence we have that

Tq(M) =[

(t1;t2)2K

Tq(M(t1;t2)):

In order to conclude the proof, we would demonstrate that Tq(M (t1;t2))\Tq(M

(t01;t02)) = ;; for every (t01; t02); (t1; t2) 2 K such that (t01; t

02) 6= (t1; t2):

Assume otherwise, let (t01; t02) 2 K and (t001; t

002) 2 K be such that Tq(M (t01;t

02))\

Tq(M(t001 ;t

002 )) 6= ;: By Lemma 5, exists (t1; t2), with ti � minft0i; t00i g, i = 1; 2,

such that

Tq(M(t01;t

02)) � Tq(M (t1;t2)) and Tq(M

(t001 ;t002 )) � Tq(M (t1;t2));

contradicting that (t01; t02) 2 K and (t001; t

002) 2 K; which concludes the proof.�

The following proposition show that the set that we are characterized inproposition 2, are a subset of the set of q-stable matching.

Proposition 3 IfM qU = (M ;RU ; q) is a matching market with quota restric-

tion. Then Tq(M) � S(M qU):

Proof Let � 2 Tq(M) be, then exists (t1; t2) be such that � 2 Tq(M (t1;t2)):Since � 2 S(M (t1;t2)) and #� = q; we have that � is q�individually rational.Assume that � 62 S(M q

U): Let (d; e) be a q�blocking pairs to �; that is:ePd� (d)Rd;; dPe� (e)Re; and eitheri) �(d) 2 E and �(e) 2 D orii) �(d;e) is q�individually rational and �(d;e)RU�:We are going to consider the following cases:

Case 1. d 2 Dt1 and e 2 Et2:Because �(Et2) � Dt1 and �(Dt1) � Et2 , then (d; e) 2 Dt1 � Et2 : BecauseePd� (d)Rd; and dPe� (e)Re;; we have that (d; e) is a blocking pairs of � inM (t1;t2): Contradicting that � 2 S(M (t1;t2)):

Case 2. d =2 Dt1 or e =2 Et2 :Since (d; e) is a q�blocking pair of �; and d =2 Dt, we have that � (d) =2 E;by condition ii) we have that �(d;e) is q�individual rational, and �(d;e)RU�:Notice that �(e) 2 Dt1 : Otherwise #�(d;e) > q: Then

B�(d;e) = B�nf(�(e); e)g [ f(d; e)g:

16

Because RU is responsive and � (e) 2 Dt1 : we have

�(�(e);e)PU�(d;e);

thus, �PU�(d;e); which implies that (d; e) is not a q-blocking pair of �:

Case 3. d =2 Dt1 and e =2 E:Because d =2 Dt1 and e =2 Et2, then �(d) = �(e) = ;. Thus #�(d;e) > q,contradicting that (d; e) is a q-blocking pair of �: �

Given M qU = (M ;RU ; q); and (t1; t2) 2 d� e; we de�ne the following sets

of stable matchings:

T<q(M(t1;t2)) =

�� 2 S(M (t1;t2)) : #� < q; either ;Ped or ;Pdefor every d 2 Dn�(Et1) and e 2 En�(Dt2)g

Notice that, by Gale and Sotomayor [5] and Roth [12] , we have thatT<q(M

(t1;t2)) = S(M (t1;t2)) or T<q(M (t1;t2)) = ;:De�ne

T<q(M) =[

(t1;t2)2d�e

T<q(M(t1;t2)):

Now de�ne the following lemmas which will be used to prove the nextproposition.

Lemma 6 Let � 2 T<q(M (t1;t2)) and (t01; t02) been such that fd 2 D : �(d) 6=

;g � Dt01 ; fe 2 E : �(e) 6= ;g � Et02, t01 � t1 and t02 � t2: Then � 2

T<q(M(t01;t

02)):

Proof Let � 2 T<q(M (t1;t2)) be, because fd 2 D : �(d) 6= ;g � Dt01 ; andfe 2 E : �(e) 6= ;g � Et

02 ; � is a matching on M (t01;t

02): Because Et

02 � Et2

and Dt01 � Dt1 any individual or pair blocking of � onM (t1;t2) is individual orpair blocking of � on M (t01;t

02). Moreover, if d 2 Dn�(Et1) and e 2 En�(Dt2);

we have that either ;Ped or ;Pde. Because �(Et1) = �(Et01) and �(Dt2) =

�(Dt02); then d 2 Dn�(Et1) and e 2 En�(Dt2) we obtain that either ;Ped or;Pde: Which implies that � 2 T<q(M (t01;t

02)): �

17

Lemma 7 Let (t1; t2) and (t01; t02) be such that T<q(M

(t1;t2))\T<q(M (t01;t02)) =

; with t01 � t1 and t02 � t2: Then T<q(M (t1;t2)) � T<q(M (t01;t02)).

Proof Because T<q(M (t1;t2)) \ T<q(M (t01;t02)) 6= ;; let � 2 T<q(M

(t1;t2)) \T<q(M

(t01;t02)): Since � 2 T<q(M (t01;t

02)) we have that fd 2 D : �(d) 6= ;g � Dt01

and fe 2 E : �(e) 6= ;g � Et02 :Let �� 2 T<q(M (t1;t2)) be, since � 2 S(M (t1;t2)) and �� 2 S(M (t1;t2)); by 2

of Remark 1, we have that

fd 2 D : �(d) 6= ;g = fd 2 D : ��(d) 6= ;g � Dt01

andfe 2 E : �(e) 6= ;g = fe 2 E : ��(e) 6= ;g � Et02

Which implies that, by lemma 6 , �� 2 S(M (t01;t02)). Moreover, if d 2 Dn�(Et1)

and e 2 En�(Dt2) we have either ;Ped or ;Pde. Since �(Et1) = �(Et01) and

�(Dt2) = �(Dt02); then d 2 Dn�(Et01) and e 2 En�(Dt02) we obtain either;Ped or ;Pde: Which implies that �� 2 T<q(M ((t01;t

02)): �

Lemma 8 Let (t1; t2) and (t01; t02) be such that T<q(M

(t1;t2))\T<q(M (t01;t02)) 6=

;: Then exists (�t1; �t2) such that T<q(M (t1;t2)) � T<q(M (�t1;�t2)) andT<q(M

(t01;t02)) � T<q(M (�t1;�t2)):

Proof Let � 2 T<q(M(t1;t2)) \ T<q(M (t01;t

02)): De�ne �t1 = min(t1; t

01) and

�t2 = min(t2; t02): Because � 2 T<q(M (t1;t2)) we have that fd 2 D : �(d) 6=

;g � Dt1 and fe 2 E : �(e) 6= ;g � Et2 : Because � 2 Tq(M (t01;t02)) we have

that fd 2 D : �(d) 6= ;g � Dt01 and fe 2 E : �(e) 6= ;g � Et02 :Which impliesthat

fd 2 D : �(d) 6= ;g � D�t1 and fe 2 E : �(e) 6= ;g � E�t2

By Lemma 6 we have that

� 2 T<q(M (�t1;�t2)):

Which implies that

T<q(M(t1;t2)) \ T<q(M (�t1;�t2)) 6= ; and T<q(M (t01;t

02)) \ T<q(M (�t1;�t2)) 6= ;:

Then the Lemma 7 implies that

T<q(M(t1;t2)) � T<q(M (�t1;�t2)) and T<q(M (t01;t

02)) � T<q(M (�t1;�t2)):

�

The following proposition gives us some information about T<q(M):

18

Proposition 4 Let M qU = (M ;RUq) be a matching market with quota re-

striction. Then

T<q(M) =

�[(t1;t2)2K̂

T<q(M(t1;t2));

where bK = f(t1; t2) 2 d� e : 8 (t01; t02) 6= (t1; t2); t01 � t1 t02 � t2 thenT<q(M

(t1;t2)) \ T<q(M (t01;t02)) = ;

:

Proof We �rst show that T<q(M) =S

(t1;t2)2 bK T<q(M(t1;t2)); and second that

T<q(M(t1;t2)) \ T<q(M (t01;t

02)) = ; for every (t01; t02); (t1; t2) 2 bK such that

(t01; t02) 6= (t1; t2):By de�nition of the set T<q(M) it is clear that:[

(t1;t2)2 bKT<q(M

(t1;t2)) � T<q(M): (10)

Now we prove that T<q(M) �S

(t1;t2)2 bK T<q(M(t1;t2)):

Let � 2 T<q(M) =S

(t1;t2)2d�eT<q(M

(t1;t2)); then exists (t1; t2) 2 d�e such

that � 2 T<q(M (t1;t2)): Assume that (t1; t2) =2 bK, then exists (t01; t02) 6= (t1; t2)such that T<q(M (t1;t2)) \ T<q(M (t01;t

02)) 6= ;.

Let (t�1; t�2) 2 d� e be the minimal pair such that

� 2 T<q(M (t�1;t�2)); (11)

That is, for every (t1; t2) 6= (t�1; t�2); with t1 � t�1 and t2 � t�2, then � =2

T<q(M(t1;t2)); which implies that T<q(M (t�1;t

�2)) \ T<q(M (t1;t2)) = ;:

We are going to show that (t�1; t�2) 2 bK: Assume otherwise, that is exists

(t�1; t�2) 62 bK, then (t01; t02) 6= (t�1; t�2) , with t01 � t�1, t02 � t�2 and T<q(M (t01;t

02)) \

T<q(M(t�1;t

�2)) 6= ;: Lemma 7 implies that

T<q(M(t�1;t

�2)) � T<q(M (t01;t

02)): (12)

By (11) and (12), � 2 T<q(M(t01;t

02)), but this contradict the minimal-

ity of (t�1; t�2). Consequently � 2 T<q(M (t�1;t

�2)) with (t�1; t

�2) 2 bK, thus � 2

19

S(t1;t2)2 bK T<q(M

(t1;t2)): This implies that

T<q(M) �[

(t1;t2)2 bKT<q(M

(t1;t2)):

In order to conclude the proof, we would demonstrate that T<q(M (t1;t2))\T<q(M

(t01;t02)) = ; for every (t01; t02); (t1; t2) 2 bK such that (t01; t

02) 6= (t1; t2): As-

sume otherwise, let (t01; t02) 2 bK and (t001; t

002) 2 bK be such that T<q(M (t01;t

02))\

T<q(M(t001 ;t

002 )) 6= ;: By Lemma 8, exists (t1; t2) , with ti � minft0i; t00i g such

that

T<q(M(t01;t

02)) � T<q(M (t1;t2)) and T<q(M

(t001 ;t002 )) � T<q(M (t1;t2));

contradicting that (t01; t02) 2 bK and (t001; t

002) 2 bK:Which conclude the proof.�

The following proposition show that the set that we are characterized inproposition 4, are a subset of the set of q-stable matching.

Proposition 5 If M qU = (M ;RU ; q) is a matching market which quota re-

striction. Then T<q(M) � S(M qU):

Proof Assuming that there exists � and (t1; t2) such that � 2 T<q(M (t1;t2))and � =2 S(M q

U): By de�nition � 2 S(M (t1;t2)) and #� < q: Moreover, forevery d =2 Dt1, e =2 Et2 ; the pair of workers (d; e) are not mutually acceptable.This implies that either ;Pde or ;Ped:Since � 2 S(M (t1;t2)); and #� < q; we have that � is q�individual ratio-

nal.Since � =2 S(M q

U); there exists a q�blocking pairs (d; e) to �, that is:ePd� (d) and dPe� (e)and eitheri) �(d) 2 E and �(e) 2 D orii) �(d;e) is q�individually rational and �(d;e)RU�:We are going to consider the following cases:

Case 1. d 2 Dt1 and e 2 Et2:Then ePd� (d)Rd; and dPe� (e)Re;; which implies that (d; e) is a blockingpair to � in M (t1;t2): This contradicts that � 2 S(M (t1;t2)):

Case 2. d =2 Dt1 or e =2 Et2 :We are assuming that d =2 Dt1 : Assume �rst that � (e) = ; this means that

20

e =2 � (Dt1) and (d; e) are not mutually acceptable which is a contradiction ofconditions ii). If � (e) 6= ;; and � is a matching in M (t1;t2) then e 2 � (Dt1) :Hence � (e) �D d which implies that �(d;e)RU� contradicts that (d; e) is aq�blocking pair of �: Then � 2 S(M q

U): The case e =2 Et2 is similar.Case 3. d =2 Dt1 and e =2 Et2:Since d =2 Dt1 and e =2 Et2, then d and e are not mutually acceptable,

contradicting that (d; e) is a q-blocking pair of �: �

The next Theorem show that the set of q-stable matching is non-empty.

Theorem 1 If M qU = (M ;RU ; q) is a matching market which quota restric-

tion where RU is responsive. Then

S(M qU) 6= ;:

Proof Let � be a stable matching on M = (D;E;P): We are going toconsider the following cases:Case 1: If #� = q; clearly the matching � 2 S(M q

U) and theorem follows.Case 2: #� < q:Let (t1; t2) be the minimum such that �(E) � Dt1 and �(D) � Et2 : Because� 2 S(M) we have that � 2 S(M (t1;t2)): By 3 of Remark 1, each stablematching and every pairs (d; e) such that �(d) = �(e) = ;; we have that(d; e) are not mutually acceptable. Then, every pairs (d; e) such that �(d) =�(e) = ; and either d =2 Dt1 or e =2 Et2, its are not mutually acceptable, so� 2 T<q(M (t1;t2)).Case 3: #� > q:Let (t1; t2) be such that � 2 S(M (t1;t2)). Notice that for every �0 2 S(M (t1;t2))we have that #�0 = #�:Consider the following sequence of matching �1; :::; �t; �t+1; :::; �k such

that if �t 2 S(M (s1;s2)); then either �t+1 2 S(M (s1+1;s2)) or �t+1 2 S(M (s1;s2+1))and �1 2 S(M (1;1)). By Lemma 2, we have that:

#�1 � ::: � #�t � #�t+1 � ::: � #�

and#�t�1 � #�t � #�t�1 + 1:

This implies that either #�t�1 = #�t or �t�1 = #�t�1; for every t: Because#� > q and #�1 � 1; we have that there exists bt such that #�bt = q and

21

�bt is stable on the market M (t1;t2); i.e., �bt 2 Sq �M (t1;t2)�and by de�nition

�bt 2 Tq (M) : By Proposition 3, the matching �bt is stable onMU , this impliesthat �bt 2 S(M q

U): �

The following theorem is a complete characterization of the q�stable setsS(M q

U):

Theorem 2 If M qU = (M ;RU ; q) is a matching market which quota restric-

tion where RU is responsive. Then

S(M qU) = Tq(M) [ T<q(M)

Proof From Propositions 3 and 5 this is what follows immediately:

Tq(M) [ T<q(M) � S(M qU):

To complete the proof we will show that S(M qU) � Tq(M) [ T<q(M):

Let � 2 S(M qU) be a q�stable matching on the market MU : Let t1; t2 be

the minimum such that �(E) � Dt1 and �(D) � Et2 : We are going to showthat either � 2 Tq(M) or � 2 T<q(M):First, we will assume that #� < q and � =2 T<q(M (t1;t2)):Consider the following cases:

Case 1: � =2 S(M (t1;t2)):Then there exists a blocking pair (d; e) 2 Dt1 � Et2 ; which implies thatthe pair (d; e) is a blocking pairs of � on MU and this is a contradiction of� 2 S(M q

U):Case 2: � 2 S(M (t1;t2)) and there exist d =2 Dt1 and e =2 Et2 mutuallyacceptable. Because d =2 Dt1 and e =2 Et2 we have that �(d) = ; = �(e):Since (d; e) are mutually acceptable �(d;e) is q�individual rational and byresponsiveness of RU ; �(d;e)RU�; which contradict that � 2 S(M

qU):

Finally, we will consider that #� = q. Then, by de�nition of S(M qU); we

have that � 2 S(M (t1;t2)), which implies that � 2 Tq(M): �

Theorem 2 is a complete characterization of the q-stable sets of a matchingmarket models under the restriction that the institutions preferences areresponsive. The mentioned characterizations are as a disjoint union of astable matching of a classical matching market without restriction or a voidset. Observe that theorem 1 shows that ones of this sets are non empty.

22

Theorems 1 and 2 have multiple implications. These results not onlyshow the existence and characterization of the set of q-stable matching, butalso give a methodology for calculating all q-stable solutions. That is, bysimilar reasoning to that used in the proof of Theorem 1 we can calculateall the matching market models M (t1;t2) for which the set of stable matchingsatis�es the following conditions: either all the stable matching have cardi-nality q, or all the stable matching has cardinality strictly less than q anypair of unassigned agents are not mutually acceptable. Roth (1980) providesan algorithm to compute the set of all stable matching in these traditionalmodels. Finally, Theorem 2 characterizes the set of q-stable matching as theunion of these sets of stable allocations.

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Documents

STABLE SOLUTIONS ON MATCHING MODELS WITH QUOTA RESTRICTION