One-Sided Matching Problems with Capacity. Constraints.

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1 One-Sided Matching Problems with Capacity Constraints Duygu Nizamogullari Ipek Özkal-Sanver y June,1, 2016 Abstract In this paper, we study roommate problems. The motivation of the paper comes from our daily routine; how to allocate xed number of identical o ces to university professors where, due to capacity constraint, at least one professor has to share his/her room with another professor. The outside option here is just resigning and nding a job in better physical conditions. Although it is literally a roommate problem; it doesn t t to the generic roommate problems. We draw special attention to the capacity constraint and modify the model introducing the outside option, beside the option of being single. Key Words: Roommate problems, Individual Rationality, Capacity Constraints. JEL Classi cation Number: C78 It is very preliminary version, please do not quote. Department of International Logistics and Transportation, Piri Reis University, dnizamogullari@pirireis.edu.tr y Department of Economics, Istanbul Bilgi University; Murat Sertel Center for Advanced Economic Studies, Istanbul Bilgi University: isanver@bilgi.edu.tr 1

2 1 Introduction The motivation of the paper comes from our daily routine; how to allocate xed number of identical o ces to university professors where, due to capacity constraint, at least one professor has to share his/her room with another professor. The outside option here is just resigning and nding a job in better physical conditions. Although it is literally a roommate problem; it doesn t t to the generic roommate problems since the outside option is not clearly de ned there, but the option of being single is considered only. A roommate problem di ers from the allocation problem, because the agents have no preferences over the rooms. Since the agents have preferences over their potential roommates, it is a de netely a matching problem. However, it is more than just a nickname for one-sided matching problems. Because of the availibility of rooms- xed number of the rooms-, it describes one sided matching problems with capacity. Recalling the well-known scenerio of a roommate problem, there is a nite set of dormitory rooms and a nite set of students who are placed in these rooms in pairs. Here the number of rooms compared to the number of students is a crucial parameter. Now consider a di erent scenario: A class of 5-Grade primary school is forming teams for their science-project. Their teacher alllows at most two students forming a team; wheras a student might do the project by his/her own. This problem is also a one sided matching problem with capacity; althougt there is no capacity constraint The number of teams might equal to the number of students. The option of staying single has a very di erent avour for the matching problems with capacity constraint. By its nature, the individual rationality axiom is a commonly agreed, very mild axiom in matching problems. It is some kind of liberty axiom which allows you to be not a part of the matching process: You are not forced to being matched with someone if staying single is a better option for you. However, the option of staying single has a very di erent avour for the matching problems with capacity constraint. A clear de nition of outside option is necessary; being single might not be the outside option for several cases. The aim of the paper is, after rede ning the basic axioms in the context of one-sided matching problems with capacity constraints; to study several basic results in this framework. We wish to show the similarities and the di erences of the results in the models with and without capacity constraints, clari ying the logic behind, as well. 2

3 2 Model We write A for the universal set of agents. A society A is a nonempty and nite subset of A. 1 For each agent i 2 A the set of potential mates of i equals A: Each agent i 2 A has a strict preference relation over his potential mates that is denoted by P i : We will use tables throughout the paper to represent the preferences of the agents. Let P denote the set of all possible preference pro les P (P i ) i2a. A (roommate) problem is a pair p (A; P ), where A is an arbitrary set of agents and P is the pro le of their preferences over potential mates in A. Let P denote the set of all problems. Now we introduce two new concepts, the set of rooms(capacity constraints) and outside option (not assigning a room). Let C denote the set of the rooms. We assume that rooms are identical; furthermore we assume that jaj=2 jcj < jaj:let c 0 denote the outside option, not assigned a room. A (roommate) problem with capacity constraint is a triple ep (A; P e ; jcj), where A is an arbitrary set of agents, P e is the pro le of their preferences over their potential mates in A and c 0 and jcj is the number of the rooms.let P jcj denote the set of all problems with capacity constraints. Note that if jcj = jaj and for all i 2 A if jp i c 0 for any j then any roommate problem with capacity constraint can be seen as roommate problem. A roommate problem p (A; P ) is equivalent to a roommate problem with capacity constraint ep (A; P e ; jcj) if (A; P )=(A; P e ja ) where P e ja is the restirected preference pro le to A: A matching for a set of agents A is a function : A! A[c 0 such that for all j; k 2 A; (j) = k implies (k) = j. Note that (j) = j means that agent j stays in a room as a single; (j) = c 0 means that agent j is assigned to no room. For any agent i 2 A; if (i) = j for some j 2 A; then (i; j) is called a matched pair under : 2 A matching 2 M (A) is individually rational for a roommate problem p = (A; P ) if and only if for all i 2 A; (i) P i i or (i) = i: Let I(p) denote the set of all such matchings. A pair of agents (i; j) with i 6= j; blocks a matching 2 M (A) if and only if j P i (i) and i P j (j). A matching 2 M (A) is stable for p = (A; P ) if and only if it is individually rational for p and there is no pair (i; j) blocking. Let S(p) denote the set of all such 1 For any society A, jaj denotes the cardinality of A: 2 Note that j can be i; in which case i is matched with him/herself under : 3

4 matchings. A matching 2 M (A) is in the core of p if there is no A 0 A and no matching 0 2 M (A) such that [ f(i)g = A0 and 0 P i2a 0 i for all i 2 A 0 : The core of p is the set CO(p) of all matchings which are in the core of p. The core of any roommate problem p = (A; P ) equals the set of stable matchings at p, that is CO(p) = S(p) for any p 2 P. As Gale and Shapley (1962) show the S(p) may be empty for some problems. A matching 2 M (A) is constraint individually rational for a roommate problem with capacity constraint ep (A; P e ; jcj) 2 P jcj : if and only if for all i 2 A; (i) P i c 0 :Let CI(ep) denote the set of all such matchings. For roommate problems with capacity constraints we will de ne blocking in two ways, blocking by an agent and blocking by a pair. An agent i blocks a matching with constraints 2 M (A) if and only if i P i (i) and #fmatched pairs under g < jcj: A pair of agents (i; j) with i 6= j; blocks a matching with constraint 2 M (A) if and only if j P i (i) and i P j (j) and either (i) = i or (j) = j or both. (i) 6= i and (j) 6= j and #{matched pairs under g < jcj (i) 6= i and (j) 6= j and #{matched pairs under g = jcj and (j) P (i) i and (i) P (j) j: 3 A matching 2 M (A) is constraint stable for ep (A; P e ; jcj) if and only if it is constraint individually rational for ep and there is no agent i and no pair (i; j) constraint blocking. Let CS(ep) denote the set of all such matchings. A matching 2 M (A) is in the constraint core of ep (A; P e ; jcj) if there is no A 0 A; no C 0 C and no matching 0 2 M (A) such that 0 P i for all i 2 A 0 : [ i2a 0 f0 (i)g = A 0 [ fc 0 g jc 0 j = #fi 2 A 0 : (i) = ig + #fi2a0 :(i)6=i and (i)2a 0 g 2 + jcj #{matched pairs under g The constraint core of ep is the set CCO(ep) of all matchings which are in the core of ep. 3 This condition is equivalent to the one used for de ning exchange stability in Alcalde (1995): 4

5 3 Some Results under General Model In marriage and roommate models, the core of a problem equals to the set of stable matchings at that problem. In roommate problems with capacity constraint, in the following proposition we show that CCO(ep) CS(ep) for all ep 2 P jcj : Proposition 3.1 For any ep 2 P jcj ; CCO(ep) CS(ep) Proof. Let ep (A; P e ; jcj) 2 P jcj be a roommate problem with capacity constraint. Let 2 CCO(ep). Assume that =2 CS(ep). If is not constraint individually rational then there is an agent i 2 A such that c 0 P i (i): Then fig forms blocking coalition with the matching 0 and jc 0 j = 0 where 0 (i) = c 0 and 0 ((i)) = (i) and 0 (j) = (j) for all j 6= i; (i): If there is an agent i that blocks then there is another agent i P i (i) and #fmatched pairs under g < jcj: Then fig forms a blocking coalition with the matching 0 and jc 0 j = 1 where 0 (i) = i and 0 ((i)) = (i) and 0 (j) = (j) for all j 6= i; (i): If there is a pair of agents (i; j) with i 6= j that blocks then j P i (i) and i P j (j) with one of the three cases: Case1: either (i) = i or (j) = j or both. Then A 0 = fi; jg forms a blocking coaliton with the matching 0 and jc 0 j 1 where 0 (i) = j and 0 (k) = (k) for all k 6= i; j: Case2: (i) 6= i and (j) 6= j and #{matched pairs under g < jcj: Then A 0 = fi; jg forms a blocking coaliton with the matching 0 and jc 0 j 1 where 0 (i) = j and 0 ((i)) = (i) and 0 ((j)) = (j) and 0 (k) = (k) for all k 6= i; (i); j; (j): But for the reverse inclusion we have a negative result. That is, in roommate problems with capacity constraint there are some problems ep = (A; P e ; jcj) 2 P jcj and 2 M (A) such that 2 CS(ep) but =2 CCO(ep):To see that consider the following example. Example 3.1 Consider the problem ep = (A; e P ; jcj); where A = fa; b; c; d; e; fg; jcj = 3 and e P as follows: ep a e Pb e Pc e Pd e Pe e Pf c e a f b d b a d c f e f d e b c a

6 Consider the matching = f(a; b); (c; d); (e; f)g: Since #{matched pairs under g = jcj; a and c or b and e or d and f do not form blocking pairs. Hence, 2 CS(ep) But the matching =2 CCO(ep) since for all i 2 A, 0 Pi e where 0 = f(a; c); (b; e); (d; f)g Remark 3.1 In contrast to marriage and roommate models the core of a problem does not equal to the set of stable matchings for roommate problems with capacity constraints. Remark 3.2 There is a roommate problem with capacity constraint ep (A; e P ; jcj) and an equivalent roommate problem p = (A; P ) of ep such that S(p) * CS(ep): To see that consider the following example. Example 3.2 Consider the problem ep = (A; e P ; jcj); where A = fa; b; c; dg; jcj = 2 and e P as follows: ep a e Pb e Pc e Pd b c d a c c 0 a c d a b b c 0 c c c 0 d c 0.. Consider the matching = f(a; b); (c; d)g: Then 2 S(p) where p is the equivalent problem of ep: But =2 CS(ep) since is not constraint individually rational for ep: Remark 3.3 There is a roommate problem with capacity constraint ep (A; e P ; jcj) and an equivalent roommate problem p = (A; P ) of ep such that CS(ep) * S(p): To see that consider the following example. Example 3.3 Consider the problem ep = (A; e P ; jcj); where A = fa; b; c; d; e; fg; jcj = 3 and e P as follows: ep a e Pb e Pc e Pd e Pe e Pf c e a f b d b a d c f e f d e b c a

7 Consider the matching = f(a; b); (c; d); (e; f)g: Since #{matched pairs under g = jcj; a and c or b and e or d and f do not form blocking pairs. Hence, 2 CS(ep) But the matching =2 S(p) where p is the equivalent problem of ep since (a; c) forms a blocking pair. Remark 3.4 There is a roommate problem with capacity constraint ep (A; e P ; jcj) and an equivalent roommate problem p = (A; P ) of ep such that CCO(ep) * CO(p) to see that consider the following example: Example 3.4 Consider the problem ep = (A; e P ; jcj); where A = fa; b; cg; jcj = 2 and e P as follows: ep a e Pb e Pc a b c b a b c c a c 0 c 0 c 0 Consider the matching = f(a; b); (c)g: Then 2 CCO(ep): For the coalition fa; bg we have jc 0 j = 1 then they can not form a blocking coalition with the matching 0 (a) = a and 0 (b) = b: But =2 CO(p) since fa; bg form a blocking coalition with the matching 0 where 0 (a) = a and 0 (b) = b: Remark 3.5 There is a roommate problem with capacity constraint ep (A; e P ; jcj) and an equivalent roommate problem p = (A; P ) of ep such that CO(p) * CCO(ep) to see that consider the following example: Example 3.5 Consider the problem ep = (A; e P ; jcj); where A = fa; b; c; dg; jcj = 2 and e P as follows: ep a e Pb e Pc e Pd b c d a c c 0 a c d a b b c 0 c c c 0 d c 0.. Consider the matching = f(a; b); (c; d)g: Then 2 CO(p) where p is the equivalent problem of ep: But =2 CCO(ep) since fbg forms a blocking coalition with matching 0 where 0 = f(a; d); (b; c 0 ); (c)g: 7

8 4 Some Results under Restricted Model Throughout this section we assume that jaj = jcj: We de ne two new axioms which transforms a roommate problem to constraint roommate problem and vice versa. Let ' be any solution and ep (A; e P ; jcj) be a constraint problem and p = (A; P ) be equivalent transformation of ep: Axiom A: A solution ' satis es AxiomA if for any 2 '(ep) there is 0 2 '(p) such that (i) = 0 (i) for all i with (i) 6= c 0 and if (i) = c 0 then 0 (i) = i: Axiom B: A solution ' satis es AxiomB if for any 2 '(p) there is 00 2 '(ep) such that (i) = 00 (i) for all i with (i) 6= i and if (i) = i then 00 (i) = i or 00 (i) = c 0 : Remark 4.1 CCO satis es AxiomA and AxiomB. We also need another axiom that says if there are available rooms and being single is better than outside option then any matching that is recommended by a solution will not match this agent by the outside option. More formally: No Constraint Axiom: A solution ' satis es no constraint axiom if ip i c 0 then for any matching 2 '(ep) then (i) 6= c 0 : 4 Proposition 4.1 If CO is characterized any set of axioms and if CCO satis es all these axioms then CCO can also be characterized by using these axioms, AxiomA, AxiomB, constraint individual rationality and no constraint. Proof. Let be a solution that satis es all these axioms. Assume for a contradiction that CCO 6= : Then there is a constraint problem ep (A; e P ; jcj) and 2 CCO(ep) but =2 (ep): Since 2 CCO(ep) and CCO satis es AxiomA then there is 0 2 CCO(p) (where p is equivalent transformation of ep) such that (i) = 0 (i) for all i with (i) 6= c 0 and if (i) = c 0 then 0 (i) = i:note that CCO(p)=CO(p). Since CO is characterized by these axioms we have CO(p) = (p): Since satis es AxiomB there is 00 2 (ep) such that (i) = 00 (i) for all i with (i) 6= i and if (i) = i then 00 (i) = i or 00 (i) = c 0 : After showing = 00 4 It is easy to see that CCO satis es no constraint axiom. 8

9 we get a contradiction. If (i) 6= i or c 0 then (i) = 00 (i): So there are two cases to consider; (i) = i or (i) = c 0 : Case1: If (i) = i: Then 0 (i) = i: Since CCO satis es constraint individual rationality then we have i = (i) P e i c 0 : Then 00 (i) = i or 00 (i) = c 0 : Also satis es "no constraint" then 00 (i) can not be c 0 : which means 00 (i) = i Case2: If (i) = c 0 : Then 0 (i) = i: Since CCO satis es "no constraint" then we have c 0 P i (i) = _i. Also satis es constraint individual rationality then 00 (i) can not be i: which means 00 (i) = c 0 : Hence = 00 so we have a contradiction. For the other side assume that there is a constraint problem ep (A; P e ; jcj) and 2 (ep) but =2 CCO(ep): Since satis es AxiomA then there is a matching 0 2 (p) (where p is equivalent transformation of ep) such that (i) = 0 (i) for all i with (i) 6= c 0 and if (i) = c 0 then 0 (i) = i: Since CO is characterized by these axioms we have CO(p) = (p): So 0 2 CO(p) = CCO(p): Since CCO satis es AxiomB there is 00 2 CCO(ep) such that (i) = 00 (i) for all i with (i) 6= i and if (i) = i then 00 (i) = i or 00 (i) = c 0 : Again we will show = 00 and get a contradiction. If (i) 6= i or c 0 then (i) = 00 (i): So there are two cases to consider; (i) = i or 00 (i) = c 0 : Case1: If (i) = i: Then 0 (i) = i:since satis es constraint individual rationality then we have i = (i) P e i c 0 : Then 00 (i) = i or 00 (i) = c 0 : Also CCO satis es "no constraint" then 00 (i) can not be c 0 : which means 00 (i) = i Case2: If (i) = c 0 : Then 0 (i) = i: Since satis es "no constraint" then we have c 0 P i (i) = _i. Also CCO satis es constraint individual rationality then 00 (i) can not be i: which means 00 (i) = c 0 : Hence = 00 so we have a contradiction. 5 Final Remarks The results of the standard roommate problems 5 can be directly transformed to the restricted model of roommate problems with capacity constraints. However, in the general model the constraint core (respectively the set constraint stable matchings) di er from the core (respectively the set of stable matchings) of the standard roommate model, as we showed in section three. Our aim is to inverstigate the general model where the constraint (the availability of vacant rooms) is binding. Currently, we are working on adapting 5 For characterization results, see Klaus (2011); Can and Klaus (2012): 9

10 the well-known axioms to this general model in a meaningful way and characterizing the constraint core in this set up. 6 References Alcalde, J. (1995); "Exchange-proofness or divorce-proofness? Stability in onesided Matching Markets ", Economic Design, 1, : Can, B. and Klaus, B., (2012), "Consistency and Population Sensitivity Properties in Marriage and Roommate Markets," Social Choice and Welfare, 41; : Klaus, B., (2011), "Competition and Resource Sensitivity in Marriage and Roommate Markets," Games and Economic Behaviour, 72; : 10