Unilateral substitutability implies substitutable completability in many-to-one matching with contracts

Transcription

1 Unilateral substitutability implies substitutable completability in many-to-one matcing wit contracts Sangram Vilasrao Kadam April 6, 2015 Abstract We prove tat te unilateral substitutability property introduced in Hatfield and Kojima [2010] implies te substitutable completability property from Hatfield and Kominers [2014]. Tis paper provides a novel linkage between tese two sufficient conditions for te existence of a stable matcing in many-to-one matcing markets wit contracts. A substitutable completion of a preference is a substitutable preference created by adding some sets of contracts to te original preference order. We provide an algoritm wic wen operated on te unilaterally substitutable preferences produces suc a substitutable completion. Tus it provides a constructive proof of te connection between te two properties. I am deeply indebted to Scott Kominers and Eric Maskin for extremely elpful discussions and detailed comments on tis paper. I would also like to tank Mira Frick, Drew Fudenberg, Ben Golub, Yuta Isii, Divya Kirti, Fuito Kojima, Maciej Kotowski, Assaf Romm, Al Rot, Ran Sorrer, Tayfun Sönmez, Neil Takral for elpful discussions and comments and te two anonymous referees for valuable feedback wic improved te exposition and te relevance of te results. All te remaining errors are mine. Department of Economics, Harvard University svkadam@fas.arvard.edu. 1

2 1 Introduction Te literature on many-to-one matcing markets wit contracts started wit te seminal contributions by Kelso and Crawford [1982] 1 and Hatfield and Milgrom [2005]. 2 3 Te practical applications of tese markets wit contracts ave recently been investigated in some interesting contexts like cadet brancing (Sönmez [2013], Sönmez and Switzer [2013]), matcing wit regional caps (Kamada and Kojima [2012, 2015a]), and diversity design in scool coice (Kominers and Sönmez [2013]). Rot [1990] described te importance of stability for practical matcing markets. He observed tat te markets wic generated a stable outcome continued to operate over longer periods of time tan te ones wic did not guarantee tis property. For many-to-one matcing wit contracts, te literature as provided many conditions on te agents preferences wic are sufficient for stability, e.g. substitutability (Kelso and Crawford [1982], Rot [1984]), unilateral substitutability (Hatfield and Kojima [2010]), bilateral substitutability (Hatfield and Kojima [2010]), and substitutable completability (Hatfield and Kominers [2014]). However, te literature as not fully explored te connections between tese sufficient conditions, wic migt be useful for practical applications. Tis paper sows tat unilateral substitutability implies substitutable completability. Te preference of an agent on te many-side of te market satisfies te substitutability condition wen te agent does not ave any complementarities between contracts. 4 In oter words, te agent views eac contract independently and never finds a contract tat is rejected from some set of contracts to be acceptable only in te presence of anoter contract. A many-to-one preference of an agent satisfies te substitutable completability condition if tere is a substitutable completion, i.e. a certain related substitutable preference in te many-to-many setting for tat agent. Te preference of an agent as te unilateral substitutability property wen te preference exibits complementarities, if any, of only a certain kind; put differently, tere may be certain permissible violations of te substitutability condition. Allowing for a broader class of complementarities gives te bilateral substitutability condition, and furter expanding te allowed set of violations of te substi- 1 Kelso and Crawford [1982] builds on te analysis of Crawford and Knoer [1981]. 2 Tis generalized te many-to-one matcing market from Gale and Sapley [1962]. Tat as in turn been furter extended and generalized to interesting domains, viz. supply cain networks (Ostrovsky [2008], Hatfield and Kominers [2012]) and many-to-many matcing markets wit contracts (Hatfield and Kominers [2013]). 3 Ecenique [2012] as sown te surprising isomorpism between Kelso and Crawford [1982] and Hatfield and Milgrom [2005]. 4 All te relevant conditions are defined in section

3 tutability condition yields weak substitutability condition (abbreviated as W.Sub.). Te weak substitutability condition is te strongest known necessary condition 5 for stability. Te following description and te two Venn diagrams summarize te relationsips between tese conditions tat are known from te existing literature. substitutability [Sub.] unilateral substitutability [U.Sub.] bilateral substitutability [B.Sub.] substitutability substitutable completability [Sub.Comp.] bilateral substitutability substitutable completability substitutable completability bilateral substitutability W.Sub. W.Sub. Sub. U.Sub. Sub. B.Sub. B.Sub. Sub.Comp. By proving tat unilateral substitutability implies substitutable completability, we are able to provide te following unified Venn Diagram. 5 Te necessary condition implies tat if tere is an agent wit a preference not satisfying te weak substitutability condition ten tere exists a setup of weak substitutable preferences for oter agents suc tat no stable matcing exists as proved in Hatfield and Kojima [2008]. 3

4 W.Sub. Sub. U.Sub. B.Sub. Sub.Comp. To fix ideas, consider te many-to-one matcing setting of ospitals and doctors were a ospital can sign at most one contract wit eac of (possibly) many doctors. Eac doctor can sign at most one contract. A set of contracts wic involves multiple contracts wit at least one doctor is termed an infeasible set. A substitutable completion of a preference, more precisely, is defined as a substitutable preference, i.e. a preference witout any violations of te substitutability condition, created by te addition of infeasible sets to te original preference order. A preference tat as a substitutable completion is defined as substitutably completable. Hatfield and Kominers [2014] give tis definition involving te addition of infeasible sets to preference orders and provide tecniques tat work for some class of preferences, namely, slot-specific preferences and task-specific preferences. In our proof, we provide an algoritm wic for any unilaterally substitutable preference arrives at a substitutable completion. Te algoritm ensures tat te additions of infeasible sets, as allowed by te definition of completion, are at te rigt places in te preference order, ensuring tat te existing violations of te substitutability property are sequentially eliminated and tat no new violations are created. Tus wit te algoritm, we provide a constructive proof tat unilateral substitutability implies substitutable completability. Te unilateral substitutability property was found to be relevant in te case of te cadet brancing market (Sönmez and Switzer [2013]). It guaranteed te existence of a stable matcing in tat setting altoug te strongest of te sufficient conditions above, i.e. substitutability, was not satisfied. Te substitutable completability property gives te intuitive understanding tat in te setting of many-to-one matcing market wit contracts, tere exists a stable matcing wic is a projection from te many-to-many matcing market wit contracts setting to te many-to-one case. Moreover, in te case of matcing wit regional caps, Kamada and Kojima [2015a] (essentially) use te substitutable completability property wile connecting teir 4

5 setting to te matcing wit contracts framework. 6 Tey view regions and doctors as te two sides of te matcing market instead of ospitals and doctors. Te ospital in a particular region being assigned to te doctor is specified in te contract and a region may coose a particular doctor for two separate ospitals. Te two most important practical applications of te matcing wit contracts framework ave used two different sufficient conditions. Tis paper provides a syntesis for te two conditions being used by proving tat one is implied by te oter. Te rest of te paper is organized as follows. Section 2 introduces te preliminaries of te many-to-one matcing market wit contracts. Section 3 gives te teorem, and a constructive proof using an algoritm. Section 4 gives an example wit te details of te algoritm. Section 5 discusses te properties of stable matces. In te last section, I summarize te contribution of tis paper. 2 Preliminaries We use te model from Hatfield and Milgrom [2005], and Hatfield and Kojima [2010] and extend te notation from Kominers and Sönmez [2013]. A many-to-one matcing market wit contracts is parametrized by a finite set of doctors, D, a finite set of ospitals, H, and a finite set of contracts, X. Eac contract x X is associated wit a doctor d(x) D and a ospital (x) H. We extend tis notation to sets of contracts by defining d(x ) x X {d(x)} and (X ) x X {(x)} for X X. Definition 2.1. For a doctor d, a ospital and a set of contracts X, te subset of contracts wit only a specific agent are defined as follows. X d {x X d(x) = d} X {x X (x) = } Eac doctor d as access to a d contract wic represents being not matced to anybody. We say tat X X is feasible for a ospital if X X and for all x, x X, we ave d(x) = d(x ) implies x = x. A set of contracts X X is called an allocation if x, x X and d(x) = d(x ) implies x = x. Tese conditions ensure tat eac doctor can sign at most one contract. 6 More precisely, Kamada and Kojima [2015a] Teorem 2 follows directly from Kamada and Kojima [2015b] Teorem 1 wic tey prove using (many-to-many) matcing wit contracts framework. 5

6 2.1 Preferences of ospitals and doctors For eac d D, P d is a strict preference relation on contracts in X d { d }. For eac H, P is a strict preference relation on all feasible subsets of contracts of X (including te null set). Te primitives of tis model are preferences over contracts or sets of contracts (and not coice functions over subsets). Hence, te Aygün and Sönmez [2013a,b] s irrelevance of rejected contracts (IRC) condition olds in our setting. 7 Te IRC condition requires tat te removal of rejected contracts sould not affect te cosen set of contracts. Definition 2.2. Given a set of contracts X, a coice function C : 2 X 2 X satisfies te irrelevance of rejected contracts (IRC) condition if for all Y X and for all z X \ Y, we ave z / C(Y {z}) implies C(Y ) = C(Y {z}). We define a coice function C d (X ) (and C (X )) based on te strict preference ordering P d (and P respectively). Te coice function C d (X ) will be a singleton tat represents te cosen contract (possibly d ) by te doctor d from X { d }. Similarly, C (X ) will represent te cosen set of contracts (possibly ) by te ospital from X based on its strict preference ordering P over its feasible subsets of contracts. Formally, for any set X X, d D, and H let C d (X ) max {x X d = d(x)} { d } P d (1) C (X ) max {Z X x Z = (x)}. P (2) Te rejection sets are defined as te contracts not cosen by te agents, i.e. R d (X ) X \ {C d (X )} and R (X ) X \ C (X ). Furter, te cosen set for all doctors is defined as te union of te cosen sets for eac of te doctors, i.e. C D (X ) = d D C d (X ). Likewise, we define te cosen sets for all ospitals, i.e. C H (X ) = H C (X ). 2.2 Stability and previous results Definition 2.3. An allocation X contracts) if 1. (Individual Rationality) C D (X ) = C H (X ) = X, and X is a stable allocation (or a stable set of 7 Wen te primitives are coice functions instead of preference orders over sets of contracts, ten tis condition is not automatically guaranteed. See Aygün and Sönmez [2013b] for an illuminating discussion. 6

7 2. (No Blocking) tere does not exist a ospital and a set of contracts X C (X ) suc tat X = C (X X ) C D (X X ). Kelso and Crawford [1982] and Hatfield and Milgrom [2005] sowed tat te substitutability condition is sufficient for stability. Definition 2.4. Contracts are substitutes for a ospital H if X X X, te rejection sets are isotone, i.e. R (X ) R (X ). In oter words, contracts are substitutes for a ospital H if for all contracts x, z X and all sets Y X, we ave z C (Y {z, x}) implies z C (Y {z}). 8 Definition 2.5. A tuple (x, z, Y ) were x, z X and Y X, tat fails te substitutes requirement, i.e. z C (Y {z, x}) but z / C (Y {z}), is called a substitutability violation. 9 Te contract x in tis case is called an alterer contract and te contract z is termed as a recalled contract. 10 Te ospital preferences satisfy te substitutability condition if tere are no substitutability violations. By permitting some specific violations, Hatfield and Kojima [2010] arrived at te following weaker conditions, wic tey proved to be sufficient for stability. Preferences are unilaterally substitutable if te only substitutability violations, (x, z, Y ), tat exist are wit sets Y suc tat d(z) d(y ), i.e. te doctor involved in te recalled contract as at least some contract in te set Y. Te bilateral substitutability condition is weaker tan te unilateral substitutability condition and te permissible substitutability violations, (x, z, Y ), are wit sets Y suc tat d(x) d(y ) or d(z) d(y ), i.e. eiter te doctor involved in te alterer contract or te recalled contract as some contract in te set Y. Definition 2.6. Contracts are unilateral substitutes for a ospital H if for all contracts x, z X and all sets Y X suc tat d(z) / d(y ), we ave z C (Y {z, x}) implies z C (Y {z}). 11 Remark 2.1. If (x, z, Y ) is a substitutability violation in a preference wic as te unilateral substitutability property ten d(z) d(y \ {z}). 8 Te requirement specified as a bite only wen x C (Y {z, x}) oterwise te requirement of [ z C (Y {z, x}) z C (Y {z}) ] trivially olds due to IRC (definition 2.2). 9 Te definition was introduced and named as suc in Hatfield, Immorlica, and Kominers [2012]. 10 Te following preference P fails te substitutability condition were d(z) = d(z ) d(x) and all contracts are wit ospital. P : {x, z} {x} {z } {z} as (x, z, {z }) is a substitutability violation. x is te alterer contract and z is te recalled contract. 11 Te requirement specified as a bite only wen x C (Y {z, x}). See footnote 8 7

8 Note tat te definition of unilateral substitutability explicitly requires tat for all te substitutability violations we sould ave d(z) d(y ). Te above remark wic is stronger tan te explicit requirement implies tat for any substitutability violation (x, z, Y ) in unilaterally substitutable preferences, te doctor involved wit te recalled contract as at least some contract oter tan z in Y. If tis were not true, i.e. suppose d(z) / d(y \ {z}) ten we will get a contradiction to te unilateral substitutability property. Define Ỹ Y \ {z} and it follows tat z C (Y {z, x}) = C (Ỹ {z, x}) and z / C (Y {z}) = C (Ỹ {z}). Tus (x, z, Ỹ ) is a substitutability violation and by assumption d(z) / d(ỹ ). Tis would imply te contradiction. Definition 2.7. Contracts are bilateral substitutes for a ospital H if for all contracts x, z X and all sets Y X suc tat d(x), d(z) / d(y ), we ave z C (Y {z, x}) implies z C (Y {z}). 12 Remark 2.2. If (x, z, Y ) is a substitutability violation in a preference wic satisfies bilateral substitutability property ten d(z) d(y \ {z}) or d(x) d(y \ {x}). Te above remark wic is stronger tan te explicit requirement of te definition implies tat for any substitutability violation, (x, z, Y ), in bilaterally substitutable preferences, te doctor involved wit te recalled contract as at least some contract oter tan z in Y or te doctor involved wit te alterer contract as at least some contract oter tan x in Y. Tis olds for exactly te same reasons as in Remark 2.1 above. Hatfield and Kominers [2014] provided a condition termed substitutable completability as a sufficient condition for te existence of a stable matcing. Tey use te many-to-many matcing markets setup and preferences to give tis sufficient condition. Definition 2.8. A many-to-many preference for a ospital is a preference relation over all subsets X X suc tat X = X. 13 Te process of preference completion involves adding infeasible sets tat include multiple contracts wit some doctors, to a many-to-one preference relation (wic is only over feasible sets of contracts). Definition 2.9. A completion of a many-to-one coice function C, corresponding to te preference P for a ospital H, is a coice function C suc tat for all Y X eiter 12 Te requirement specified as a bite only wen x C (Y {z, x}). See footnote 8 13 A preference profile in te many-to-one matcing market wit contracts was over all feasible subsets X X were as te many-to-many preference is over all subsets X X. 8

9 1. C (Y ) = C (Y ) or 2. distinct z, ẑ C (Y ) suc tat d(z) = d(ẑ) If a coice function C as a completion tat is substitutable and satisfies te IRC condition, ten we say tat te C is substitutably completable. In our setting were we take te preference P as te primitive for te ospitals, te equivalent definition of substitutable completability is as follows. Definition A preference P is substitutably completable for ospital H if tere exists a substitutable many-to-many preference P suc tat P satisfies te following conditions: For feasible X, X X, if X X under P ten X X under P. For feasible X X, if X under P ten it remains unacceptable, i.e. X under P. P is defined as a substitutable completion of te preference P. Since te resulting completion is substitutable, tere exists a stable matcing in te many-to-many setting due to te sufficiency of te substitutability condition (Hatfield and Kominers [2012, 2013]). Tis stable matcing also respects te restrictions of te many-to-one setting, as te preferences for te doctors were not canged and tey still find only singleton contracts acceptable Main Result We now present te main result of tis paper. We provide te intuition beind te constructive proof and describe te algoritm before proceeding to te proof of tis teorem. Teorem 1. If te ospital preferences are unilaterally substitutable ten tey are substitutably completable, i.e. for any P wit te unilateral substitutability property, tere exists a completion P wic satisfies te substitutability property. 14 A stable matc in te many-to-many setting sould be individually rational for eac doctor wic rules out a subset of contracts tat is not an allocation. 9

10 We now present te main idea beind ow te constructive proof acieves a substitutable completion of preferences. Recall tat a preference as te substitutability property if and only if tere are no substitutability violations. Furter recall tat a preference is unilaterally substitutable if and only if tere are no substitutability violations OR tere are only permissible substitutability violations. Specifically, all violations (x, z, Y ) tat exist are te permissible ones were te doctor of te recalled contract, i.e. d(z), as some contract in Y (oter tan z). 15 For example, consider te following preferences P for a ospital. Here we ave, d(z) = d(z ) d(x) and all te contracts listed are wit ospital. (x, z, {z }) P : {x, z} {x} {z } {z} is te only substitutability violation and it is permissible. Te algoritm uses te recalled contract and te existing contract(s) wit d(z) in set Y and adds tem to te cosen set of contracts under te original preferences from Y {z} to create te new cosen set under te new preference. Te cange in te cosen set is acieved by making te new cosen set (just) better tan te older cosen set. Te presence of at least two contracts wit d(z) ensures tat te new cosen set being added in te preference relation is an infeasible set. Tis ensures tat te resulting new preference is a completion. To ensure tat te completion process does not create any new substitutability violations wile correcting te existing ones, te algoritm may also cange a cosen set tat includes a contract involving d(z) under te original preference. Tis furter ensures tat te final completion created by te algoritm is substitutable. Te following example elucidates tis process. 15 See remark 2.1. C ({z, z }) = {z } as per te preference relation P Ĉ ({z, z }) = {z, z } by completing te preference to P P : {x, z} {x} {z, z} {z } {z} (x, z, {z }) is not a substitutability violation 10

11 Before proceeding to te proof, we introduce te following concept wic will be used in one of te steps of te algoritm, and provide an example wic clarifies te relevance of tis concept. Definition 3.1. Te maximal subset of Ỹ wit only unrejected contracts wit a doctor, say d 1, under a preference P, is defined as te set Ŷ Ỹ suc tat 1. All non-d 1 contracts of Ỹ are present in Ŷ, i.e. Ŷ \ Ŷd 1 = Ỹ \ Ỹd 1 2. Y Ŷ, d 1 d(y ) d 1 d(c (Y )) and Ŷ is te maximal suc subset under te partial order of set inclusion.16 Note te maximal set defined above is unique due to te many-to-one nature of te setting. 17 Tis maximal subset only as contracts wic are, loosely speaking, not always rejected in te original preference. Consider te following preference wic will explain te above concept. Example 3.1. P : {x, z} {x} {z } {z} {ẑ} d(z) = d(z ) = d(ẑ) d(x) Let Ỹ = {z, z, ẑ}. Te maximal subset of Ỹ wit only unrejected contracts wit doctor d 1 = d(ẑ) under P is Ŷ = {z, z} because d(ẑ) / d(c ({ẑ})). Now we are ready to present te algoritm wit te understanding of te underlying intuition presented above. 3.1 Algoritm Step 0 Define P 0 P and C 0 ( ) C ( ) for te ospital. Go to step 1. Step i Does tere exist a substitutability violation (x i, z i, Y i ) for P i 1? 18 If yes define C i (Ỹ ) as follows for all Ỹ X. If d(z i ) / d(c i 1 [Ỹ ]) ten Ci (Ỹ ) Ci 1 (Ỹ ). 16 Maximality of Ŷ implies tat any oter set Ỹ Ỹ but not a subset of Ŷ for wic te following conditions olds, Ỹ \ [Ỹ ] d1 = Ỹ \ Ỹd 1 and Y Ỹ, d 1 d(y ) d 1 d[c (Y )] 17 See Appendix Claim 1 for te proof. 18 Witout loss of generality, we can restrict our attention to x i, z i X and Y i X. 11

12 If d(z i ) d(c i 1 [Ỹ ]) ten find te maximal subset Ŷ of te set under consideration, Ỹ wit only unrejected contracts wit doctor d(z i ) under P i 1. Define te coice function as C i (Ỹ ) Ci 1 (Ỹ ) [Ŷ ] d(z i ). Te coice function C i as te underlying preference relation P i. Te existence of suc a preference relation over subsets of X is proved in Claim 2 of te Appendix wit a constructive proof. Go to step i + 1. If not ten I i 1 and P P I and terminate te algoritm.19 We start wit te following four observations about te algoritm wic will be useful at various steps of te proofs tat will follow. 1. During step i, preferences are canged to P i only if a substitutability violation (x i, z i, Y i ) is identified in P i Te cosen sets are weakly increasing in te order of set inclusion at all steps of te algoritm. Specifically, for all Ỹ C i 1 (Ỹ ) Ci (Ỹ ). We refer to tis property as te property of a weak order over te cosen sets. 3. Te cosen set C i (Ỹ ) could be different from Ci 1 (Ỹ ) only if d(z i) d(c i 1 (Ỹ )). 4. Moreover, C i (Ỹ ) could be different from Ci 1 (Ỹ ) only in contracts wit doctor d(z i ). We use C i ( ) and P i intercangeably, above and for te rest of te proof, as may be appropriate for te context. Te existence of P i is guaranteed in te algoritm as proved in Claim 2 of te Appendix. 3.2 Lemmas We define a new violation as a substitutability violation tat exists in P i but not in P i 1 for some i 1 in te above algoritm. We now prove tat no suc new violations are created by te algoritm. Lemma 3.1. No new violations are created in te algoritm above wen preferences of ospitals satisfy te irrelevance of rejected contracts condition. 19 Section 4 as an illustrative example worked out. 12

13 Proof We prove tis by contradiction. Suppose tere are new violations introduced in te algoritm above. Let us consider te first step i were a new violation, (x, z, Y ) is introduced. We ave z C i (Y {x, z}) but z / Ci (Y {z}). We ten prove te following. (I) Te contract z is added to te cosen set of Y {x, z} in step i, i.e. z / (Y {x, z}) and z C i (Y {x, z}). C i 1 (II) Te doctor d(z) as some contract in te cosen set from Y {z} under P i 1. (III) Te contract z is a part of te cosen set under P i of Y {z} at step i wic contradicts te assumption tat (x, z, Y ) is a substitutability violation. (I) To prove te first claim recall tat z C i (Y {x, z}) but z / Ci (Y {z}). Along wit observation 2 about te weak order over te cosen sets, tis implies tat z / C i 1 (Y {z}). As (x, z, Y ) is not a violation in step i 1, we necessarily ave z / C i 1 (Y {x, z}). 20 Tus te cosen set from Y {x, z} is modified during step i. We can furter claim tat z is added to te cosen set from Y {x, z}. Let Ŷ1 be te maximal subset of Y {x, z} as described in definition 3.1. From te algoritm above and observations 1, 2, and 4, we know tat step i identified a substitutability violation (x i, z i, Y i ) and we ave te following. d(z) = d(z i ) and z Ŷ1 (3) (II) We prove te second claim by contradiction. Suppose not, i.e. d(z) / d[c i 1 (Y {z})]. We ave maintained te assumption tat te primitives in tis setting are preferences over contracts and not coice correspondences, wic guarantees te IRC condition. Define Ỹ = Y \ Y d(z) {z}. By IRC, we ave te following. d(z) / d[c i 1 (Ỹ )] (4) Recall tat te maximal subset Ŷ1 of Y {x, z} satisfies te following properties. (a) Ŷ1 \ [Ŷ1] d(z) = Y {x, z} \ [Y {x, z}] d(z) = Y {x} \ Y d(z) (b) Y 1 Ŷ1, d(z) d(y 1 ) d(z) d[c (Y 1 )] 20 We know tat z C i 1 (Y {x, z}) z C i 1 (Y {z}) is true. 13

14 We can make te following set of conclusions using te definition of Ỹ above and te two conditions on Ŷ1. Ỹ \ {z} Ŷ1 \ [Ŷ1] d(z) z Ŷ1 terefore Ỹ Ŷ1 Since d(z) d(ỹ ) ; d(z) d[c (Ỹ )] However, tis is at odds wit equation 4 above and observation 2. Tis leads to te required contradiction. Hence we ave d(z) d[c i 1 (Y {z})]. (III) Let te maximal subset of Y {z} as per definition 3.1 be Ŷ2. Since d(z) d[c i 1 (Y {z})], te cosen set is (possibly) altered at step i using Ŷ2. We want to prove tat z Ŷ2. Suppose not. Ten Ŷ2 {z} is a larger set tan Ŷ2 but is not te required maximal subset. Let us observe tat all non-d(z) contracts of Y {z} are present in Ŷ2 and consequently in Ŷ2 {z} as well. Hence Ŷ2 {z} must fail te second requirement of te maximal subset. Tere must exist a set Y 2 Ŷ2 {z} suc tat te following is true. d(z) d(y 2 ) but d(z) / d[c (Y 2 )] (5) However, Y 2 Ŷ2 oterwise tat would be a violation of te second requirement for te maximal subset, Ŷ2 in definition 3.1. Tis implies tat z Y 2. Define Ỹ2 = Y 2 \ [Y 2 ] d(z) {z}. Clearly z Ỹ2 and from 5 and IRC we ave, d(z) / [C (Ỹ2)] (6) Recall tat Ŷ1 is te maximal subset of Y {x, z}. We can compare Ỹ2 wit Ŷ 1 and Ŷ2 more closely. Ỹ 2 \ {z} = Y 2 \ [Y 2 ] d(z) Ŷ2 \ [Ŷ2] d(z) = Y \ Y d(z) Y {x} \ Y d(z) = Ŷ1 \ [Ŷ1] d(z) Ŷ1 We ave Ỹ2 Ŷ1 {z} = Ŷ1 (from equation 3). Since we ave z Ỹ2 by te second condition on te maximal subset, Ŷ1 we ave d(z) [C (Ỹ2)] wic contradicts te conclusion above in equation 6. Hence we ave z Ŷ2 and tus z is a part of te cosen set from Y {z} at step i, i.e. z C i [Y {z}]. Tis provides te required contradiction to (x, z, Y ) being a substitutability violation. 14

15 An immediate corollary is tat te unilateral substitutability property is preserved at eac step in te algoritm above. Oterwise, tere would be a new violation in P i tat was not present in te original preference P. Tis would contradict te lemma. Corollary 3.1. Te preference P i created in te algoritm above at eac step i as te unilateral substitutability property if P is unilaterally substitutable. Te next lemma sows tat te algoritm eliminates te substitutability violations one by one. Lemma 3.2. Suppose preferences satisfy te unilateral substitutability condition. For eac step i, te substitutability violation (x i, z i, Y i ) under P i 1 is corrected and is not a violation under P i. Proof We prove tis in two steps. In te first part, we establis tat for te substitutability violation (x i, z i, Y i ), ẑ i C i 1 (Y i {z i }) suc tat d(ẑ i ) = d(z i ). In te second part, we prove tat z i is added to te cosen set from Y i {z i } in step i and tus te violation is fixed. 21 Part I By Lemma 3.1, we know tat te violation considered in step i also existed in te original preferences P. Furter note tat z i C (Y i {z i, x i }) and ence tere does not exist any oter d(z i ) contract in C (Y i {z i, x i }) as te original preferences were defined only over feasible subsets of contracts. Using irrelevance of rejected contracts, we will ave C (Y i {z i, x i }) = C (Y i \ [Y i ] d(zi ) {z i, x i }) were [Y i ] d(zi ) is te set of contracts in Y i involving doctor d(z i ). If ẑ 1 i C (Y i {z i }) suc tat d(ẑ 1 i ) = d(z i ) ten C (Y i {z i }) = C (Y i \ [Y i ] d(zi ) {z i }). Define Ŷ = Y i \ [Y i ] d(zi ) and we ave z i / C (Ŷ {z i}) z i C (Ŷ {z i, x i }) (x i, z i, Ŷ ) is a substitutability violation of P and d(z i ) / d(ŷ ) wic implies te preferences do not satisfy unilateral substitutability condition. Hence, our initial assumption must be incorrect and in fact, d(z i ) d(c (Y i {z i })). Moreover, by te weak order over te cosen sets tere exists a ẑ i C i 1 (Y i {z i }) suc tat d(ẑ i ) = d(z i ). 21 Te claims in tese two parts are very similar to tose of steps (II) and (III) of Lemma 3.1 but te proof is very different because for te new violation we knew tat z Ŷ1. Instead, ere we ave z i C (Y i {x i, z i }) but te unilateral substitutability condition elps us prove tese claims. 15

16 Part II Consider te maximal subset 22 Ŷ of Y i {z i } tat includes only te unrejected contracts wit d(z i ). We prove tat z i Ŷ by contradiction. Suppose not and we ave z i / Ŷ. Consider Ŷ {z i} wic contains all te non-d(z i ) contracts of Y i {z i } but is not te required maximal subset. Hence it fails te second condition of te maximal subset. Ỹ Ŷ {z} suc tat d(z i ) Ỹ but d(z i) / d[c i 1 (Ỹ )] We can conclude tat z i Ỹ. If not, Ỹ Ŷ and Ỹ will not be a (maximal) subset wit only unrejected contracts wit d(z i ). By te irrelevance of rejected contracts condition, tere also exists Ỹ1 {z i } Ỹ wit [Ỹ1] d(zi ) =, i.e. te only d(z i ) contract in Ỹ1 {z i } is z i, satisfying d(z i ) d(ỹ1 {z i }) but d(z i ) / d[c i 1 (Ỹ1 {z i })]. Tus we ave Ỹ1 {z i } Ỹ Ŷ {z i} Y i {z i } Y i {z i, x i }. We also ave te following. z i Ỹ1 {z i } and z i / C i 1 (Ỹ1 {z i }) z i Y i {z i, x i } and z i C i 1 (Y i {z i, x i }) By te weak order over te cosen sets, we also know tat z i / C (Ỹ1 {z i }) and since te violation existed in te original preferences P, z i C (Y i {z i, x i }). Moreover, we ave tat [Ỹ1] d(zi ) = and ence Ỹ1 {z i } Y i \ [Y i ] d(zi ) {z i, x i }. Tus we ave te following. z i Ỹ1 {z i } and z i / C (Ỹ1 {z i }) z i Y i \ [Y i ] d(zi ) {z i, x i } and z i C (Y i \ [Y i ] d(zi ) {z i, x i }) Define Ỹ Y i \ [Y i ] d(zi ) {z i, x i }. Label all te elements in Ỹ \ [Ỹ1 {z i }] as q 1, q 2, q 3,..., q J. It is clear tat d(q j ) d(z i ) j {1, 2,, J}. Moreover, wen we add one contract at a time to te set Ỹ1 {z i } to arrive at newer cosen sets, z i will become acceptable (eventually). Tere would be a first ĵ suc tat we ave te following z i C (Ỹ1 {q 1, q 2,..., qĵ} {z i }). We found a substitutability violation (x, z, Y ) were x = qĵ, z = z i, and Y = Ỹ1 {q 1, q 2,..., qĵ 1 } were d(z) / d(y ) and tus te original preference is not unilaterally substitutable. Tis is a contradiction and in fact we always ave z i Ŷ. If z i Ŷ ten z i C i (Y i {z i }) and te violation does not exist in P i. Lemma 3.3. P i is a completion of P i. 22 See Definition

17 Proof For i = 0 tis is trivially true as eac preference is a trivial completion of itself. For i 1, we need to prove tat in te algoritm, P i is modified from P i 1 by including infeasible sets and not canging te order of feasible sets. If we use te coice function definition of te completion instead, ten we need to prove tat te cosen contracts by te ospital are eiter te same or ave more tan one (distinct) contracts wit te same doctor. For any Ỹ suc tat d(z i) / d[c i 1 (Ỹ )], C i (Ỹ ) = Ci 1 (Ỹ ). If Ỹ is suc tat d(z i) d[c i 1 (Ỹ )], ten te corresponding maximal subset Ŷ contains eiter one contract or more tan one contracts involving d(z i ) and te ospital. If te maximal subset contains only one contract, tat contract will be a part of te cosen set in step i 1. Ten essentially we ave C i (Ỹ ) = Ci 1 (Ỹ ). Moreover, if te maximal subset contains two or more contracts involving d(z i ) and te ospital, all tose contracts would be included in te new definition of C i (Ỹ ). Tis ensures tat tere are at least two distinct contracts wit te same doctor d(z i ) wen C i (Ỹ ) Ci 1 (Ỹ ). Bot tese conditions imply tat C i is a completion of Ci 1 and by induction on i, we can say tat C i is a completion for all i. Te existence of a unique P i as guaranteed by Claim 2 in te appendix for eac C i completes te proof of te claim above. Now we are ready to give te proof of te main result. Proof of Teorem 1 Since tere is a finite number of contracts, for a given preference ordering for a ospital tere is a finite number of (possible) substitutability violations. Since at eac step no new violations are created (by Lemma 3.1) and at least one violation is reduced (by Lemma 3.2), te number of violations at eac step is strictly less tan te number of violations in te previous step. Since we ad a finite number to begin wit and it strictly decreases progressively, te algoritm is bound to end in finitely many steps. After te algoritm terminates, we ave P I defined as P. Tis preference as no violations and ence satisfies te substitutability condition. It is a completion of te original preference P by Lemma 3.3. Hence it is a substitutable completion of te preferences P wit te unilateral substitutability property. Note tat tis property was needed to guarantee tat Lemma 3.2 goes troug. Tis ensured tat te number of violations strictly decrease at eac step wic was crucial for tis algoritm to work. In te algoritm presented above at eac step i, all te substitutability violations involving te recalled contract doctor, i.e. d(z), are fixed. Tis gives te following proposition about te number of steps in te algoritm above. Proposition 3.1. If te ospital preferences are unilaterally substitutable ten te completion algoritm identifies at most D substitutability violations and tus com- 17

18 pletes in at most D steps. Tere are at most D doctors wo would ave a substitutability violation in a given ospital preference and ence te number of steps in te algoritm 3.1 is capped at te cardinality of te set of doctors. 23 We present te proof of tis in te appendix. 4 Example I explain te algoritm above using an example. Consider te following preferences wit unilateral substitutability property for a ospital over a set of contracts X = {x, x, y, y }. We also ave d(x) = d(x ) d(y) = d(y ). P : {x, y } {x, y} {x, y } {x, y} {x } {y } {x} {y} Te four violations are as follows, were te first contract denoted as z in te discussion so far is rejected from te set Y {z} but becomes acceptable wen a different contract x becomes available as well. x z Y y x {x } y x {x,y} x y {y } x y {x,y} Consider te following steps in te algoritm. Step 0 Define P 0 = P. Step 1 Consider te first violation as (y, x, {x }) and consider all te contracts Ỹ suc tat d(x) is cosen by te ospital. We ave te following list for Ỹ, C0 (Ỹ ), and C 1 (Ỹ ). 23 Tis is different from te computation time for running tis algoritm. It as exponential time complexity in te number of contracts. Eac step in te algoritm above searces troug 2 X 1 subsets of te set of contracts and alters te preferences to arrive at te preference P i at te end of te step. 18

19 Ỹ C 0 (Ỹ ) C1 (Ỹ ) {x} {x} {x} {x } {x } {x } {x, x } {x } {x, x } {x, y} {x, y} {x, y} {x, y } {x, y } {x, y } {x, y} {x, y} {x, y} {x, y } {x, y } {x, y } {x, y, y } {x, y } {x, y } {x, y, y } {x, y} {x, y} {x, x, y} {x, y} {x, x, y} {x, x, y } {x, y } {x, x, y } {x, x, y, y } {x, y } {x, x, y } Altoug te list above is long, it materially impacts at only tree places and we obtain P 1 as follows. P 1 : {x, x, y } {x, y } {x, x, y} {x, y} {x, y } {x, y} {x, x } {x } {y } {x} {y} Step 2 Te second violation was also corrected troug te first step. Now consider te tird violation (x, y, {y }) and consider all te contracts Ỹ suc tat d(y) is cosen by te ospital. We ave te following list for Ỹ, C0 (Ỹ ), and C1 (Ỹ ). Ỹ C 1 (Ỹ ) C2 (Ỹ ) {y} {y} {y} {y } {y } {y } {y, y } {y } {y, y } {x, y} {x, y} {x, y} {x, y } {x, y } {x, y } {x, y} {x, y} {x, y} {x, y } {x, y } {x, y } {x, y, y } {x, y } {x, y, y } {x, y, y } {x, y} {x, y, y } {x, x, y} {x, x, y} {x, x, y} {x, x, y } {x, x, y } {x, x, y } {x, x, y, y } {x, x, y } {x, x, y, y } 19

20 P 2 Altoug te list above is long, it materially impacts at four places and we obtain as follows. P 2 : {x, x, y, y } {x, x, y } {x, y, y } {x, y } {x, x, y} {x, y, y } {x, y} {x, y } {x, y} {x, x } {x } {y, y } {y } {x} {y} Now tere are no more violations in P 2 and it satisfies te substitutability property and is a completion of preferences P. Hence, P = P 2 is a substitutable completion of P. 5 Properties of stable matcings Te set of stable matcings satisfy certain properties, like te existence of a doctoroptimal stable matcing, a lattice structure, etc. under suitable restrictions on te preferences of te ospitals. We can summarize a few key properties in various settings. Setting Preferences Properties of all agents Many-to-many Substitutable Existence of doctor-optimal stable matcing Existence of doctor-pessimal stable matcing Existence of a Lattice + Law of Aggregate Demand Rural Hospital Teorem Many-to-one Substitutably a completion in te many-to-many completable setting wic is substitutable Existence of doctor-optimal stable matcing Many-to-one Unilaterally Existence of doctor-pessimal stable matcing Substitutable Existence of a Lattice + Law of Aggregate Demand Rural Hospital Teorem We now know tat a unilaterally substitutable preference as a related substitutable preference in te many-to-many setting. Moreover, all te stable matcings in te corresponding many-to-many setting wit te substitutable completions of preferences are stable under te original many-to-one setting. However, wit unilaterally substitutable preferences, te existence of a lattice structure or a doctorpessimal matcing is not guaranteed. Tis migt appear puzzling given tat tese 20

21 results old in te many-to-many setting wen te preferences for all agents ad te substitutability property. Te resolution to tis puzzle lies in te fact tat under substitutably completed (many-to-many) preferences, only a subset of stable matcings under te original (many-to-one) preferences continue to remain stable. 24 Consider te following example 25 were te unilateral substitutability condition is satisfied but not te substitutability condition. P : {x, y } {x, y} {x, y } {x, y } {x, y } {x, y } {x, y} {x, y } {x, y} {x } {y } {x } {y } P d1 : {x } {x} {x } d1 P d2 : {y } {y} {y } d2 {x} {y} Hatfield and Kojima [2010] sow tat tere is a doctor-optimal stable matcing {x, y } and two oter stable matcings, {x, y} and {x, y }, none of wic is doctorpessimal. However, under te completed preferences (using te above algoritm) we would ave {x, y } as te one and only stable matcing. It is te doctor optimal stable matcing and also te doctor pessimal stable matcing among te set of stable matcings under te completed preference. 6 Conclusion In tese closing remarks, we igligt two points tat ave not been discussed so far. First, on a tecnical note, tere could be more tan one completion of preferences wic are substitutable, and te algoritm described above arrives at just one of te possible completions. Second, te substitutably completable preferences are puzzling and are different from oter known necessary and various sufficient conditions for stability. Tis is because substitutable completability is defined implicitly and not described in terms of coices made by te ospital. Troug tis work, we provide a connection wit te unilateral substitutability property, wic is defined explicitly. 24 Te notion of stability used ere is te one for many-to-many preferences as defined in Hatfield and Kominers [2013] Definition 2. An allocation A X is stable (wit respect to X) if it is (i) Individually Rational for all f D H C f (A) = A f and (ii) Unblocked tere does not exist a nonempty blocking set Z X suc tat Z A = and for all f d(a) (A), A f C f (A Z). Also, it is under tis notion of stability tat te set of stable matcings in many-to-many preferences ave a lattice structure. 25 From page 1717 of Hatfield and Kojima [2010] 21

22 Hatfield and Kominers [2014] use tecniques to arrive at a completion for specific preferences, namely, slot-based preferences and task-based preferences. We provide an algoritm for new sub-domains of te substitutably completable preferences. In doing so, we furter clarify te connection between known concepts. A generalized algoritm for reacing a completion of preferences remains elusive as does a general caracterization of te substitutably completable preferences. Te idden structure in various conditions wic guarantee te existence of stability, wic is yet to be uncovered, will close te lacuna in our understanding. We present a small step in tis direction and ope tat furter researc will provide us te necessary and sufficient condition for te existence of stable matcings. 7 Appendix Claim 1 Te maximal subset of Ỹ wit only unrejected contracts wit a doctor, say d 1 is uniquely defined. Proof Consider two sets Ŷ1 and Ŷ2 bot being te maximal subsets as defined above in definition 3.1. We also ave Ŷ1 Ŷ2 and tey can not ordered by set inclusion. Define Ŷ = Ŷ1 Ŷ2. From te definition above we know tat Ŷ1 \ [Ŷ1] d1 = Ỹ \Ỹd 1 = Ŷ2\[Ŷ2] d1 = Ŷ \Ŷd 1. Moreover Ŷ also satisfies te second requirement wic would mean tat neiter Ŷ1 nor Ŷ2 are maximal. To prove a contradiction suppose not. Tere exists a Y Ŷ suc tat d 1 d(y ) but d 1 / d[c (Y ]. Consider any contract wit doctor d 1 in suc a Y and let us call it z. It is clear tat z Ŷ1 or z Ŷ2. Consider te set Y 1 = Y \ [Y ] d1 {z}. By IRC, we ave d 1 / d[c (Y 1 )]. However, since Y \ [Y ] d1 Ỹ \ Ỹd 1 we would ave Y 1 Ŷ1 or Y 1 Ŷ2. Tis would violate te second requirement for te above definition for eiter Ŷ1 or Ŷ2 and tis would contradict our assumption. Claim 2 Tere exists a preference relation P i over te subsets of X suc tat C i defined in step i in te completion algoritm 3.1 corresponds to P i. Proof We provide a proof by induction. Tis statement is trivially true by assumption for i = 0. Assume tat it olds for i 1 for i 1. We prove tat it olds for i constructively. i.1 Define ˆP P i 1 and Ĉ( ) Ci 1 ( ) for te ospital corresponding to te preference relation P i 1. Also define ˆX 2 X \ { }. i.2 If ˆX = ten P i ˆP and Exit tis step i in te completion algoritm 3.1 and go to next step i + 1. Else coose Ỹ ˆX and go to step i.3. 22

23 i.3 Redefine ˆX ˆX \ {Ỹ }. i.4 If d(z i ) / d[ĉ(ỹ )] ten go to step i.2. Else go to step i.5. i.5 Find te required maximal subset Ŷ for Ỹ and define Y = Ĉ(Ỹ ) Ŷd(z i ) and go to step i.6. i.6 If Ĉ(Ỹ ) = Y ten go to step i.2. Else go to step i.7. i.7 Define a preference relation ˆP by moving Y so tat it is (just) better tan Ĉ(Ỹ ), i.e. Ỹ Y and X we sould ave te following. Ỹ Y under ˆP Ỹ Ĉ(Ỹ ) under ˆP (7) Clearly Ĉ(Ỹ ) = Y under ˆP. Go to next step i.8. i.8 Define ˆP ˆP and Ĉ( ) corresponds to te new preference relation ˆP. Go to step i.2 Te above construction takes eac non-empty subset of X one at a time and modifies te preference relationsip iteratively. In a given step i were te preferences are being modified to fix te substitutability violation concerning te doctor d(z i ) te following statements follow immediately from te algoritm: Preference is modified only for sets were a d(z i ) contract is in te cosen set. Wen preference is modified for suc sets Ỹ all d(z i) contracts in Ŷ are added to te new cosen set by moving tis set as being better tan te old cosen set. Once te set is moved in te preference relation to be better tan te old cosen set ten in te subsequent iterations of te algoritm above it continues to be better tan te old cosen set, wic in turn was better tan all te feasible and infeasible subsets of te set, in a given step. Tus at te end of te algoritm above, we ave a preference relation P i and a coice function C i ( ) wic agrees wit te description in step i of te completion algoritm. Proof of Proposition 3.1 We first prove tat at step i, wit a substitutability violation (x i, z i, Y i ), tere exist no substitutability violation ( x i, z i, Ỹi) under P i suc 23

24 tat d( z i ) = d(z i ). Consider suc a substitutability violation involving te doctor of te recalled contract under P i 1. Using te same tecnique as in proof of Lemma 3.2, using part I we can claim tat ẑ i C i 1 (Ỹi { z i }) suc tat d(ẑ i ) = d( z i ) = d(z i ). Proceeding along te same lines of part II, we would establis tat for te maximal subset Ỹ i of Ỹi, z i Ỹ i and ence is added in step i. Te substitutability violation no longer exists after step i. Tus ( x i, z i, Ỹi) is not a substitutability violation under P i. Tus at eac step all te violations involving a given doctor are fixed. By Lemma 3.1, we know tat no new violations are created and ence tere will be no new violations involving te doctor d(z i ) in te subsequent steps. Tis proves tat te algoritm above identifies up to D violations and completes in D steps. References Oran Aygün and Tayfun Sönmez. Matcing wit contracts: Comment. American Economic Review, 103: , 2013a. Oran Aygün and Tayfun Sönmez. Matcing wit contracts: Critical role of irrelevance of rejected contracts. mimeo, Boston College, 2013b. URL ttps: //www2.bc.edu/~sonmezt/aygunsonmez2012a.pdf. Vincent P. Crawford and Elsie Marie Knoer. Job matcing wit eterogeneous firms and workers. Econometrica, 49(2):pp , ISSN URL ttp:// Federico Ecenique. Contracts versus salaries in matcing. American Economic Review, 102(1): , doi: /aer URL ttp://www. aeaweb.org/articles.pp?doi= /aer David Gale and Lloyd Sapley. College admissions and te stability of marriage. American Matematical Montly, 69:9 15, Jon W. Hatfield and Fuito Kojima. Matcing wit contracts: Comment. American Economic Review, 98: , Jon W. Hatfield and Fuito Kojima. Substitutes and stability for matcing wit contracts. Journal of Economic Teory, 145(5): , Jon W. Hatfield and Scott D. Kominers. Matcing in networks wit bilateral contracts. American Economic Journal: Microeconomics, 4(1): ,

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