Equivalent Choice Functions and Stable Mechanisms
aa r X i v : . [ ec on . T H ] J a n Equivalent Choice Functions and StableMechanisms
Jan Christoph Schlegel ∗ Department of Economics, City, University of London, United Kingdom [email protected]
Abstract
We study conditions for the existence of stable and group-strategy-proofmechanisms in a many-to-one matching model with contracts if students’preferences are monotone in contract terms. We show that “equivalence”,properly defined, to a choice profile under which contracts are substitutesand the law of aggregate holds is a necessary and sufficient condition for theexistence of a stable and group-strategy-proof mechanism.Our result can be interpreted as a (weak) embedding result for choicefunctions under which contracts are observable substitutes and the observ-able law of aggregate demand holds.
JEL-classification:
C78, D47
Keywords:
Matching with contracts; Embedding; College admission; Sub-stitutes; Observable Substitutes; Group-Strategy-Proofness
Centralized clearing houses based on the deferred-acceptance mechanism are at theheart of many successful real-world matching markets (Roth, 1984a; Abdulkadiroglu and S¨onmez, ∗ The current paper has its origins in my job market paper entitled ”Virtual Demand andStable Mechanisms”, an extended abstract of which appeared in the Proceedings of the 2016 ACMConference on Economics and Computation (EC16). Chapter 4 of the original paper, has beenpublished as an independent working paper with the title “Group-Strategy-Proof Mechanisms forJob Matching with Continuous Transfers”. I am grateful to my adviser Bettina Klaus for manyuseful comments that greatly improved the paper and to Sangram Kadam for many insightfuldiscussions. I thank Battal Do˘gan, Federico Echenique, Ravi Jagadeesan, Flip Klijn, FuhitoKojima, Maciej Kotowski, Jordi Mass´o, participants of the 2016 Meeting of the Social Choiceand Welfare Society, the 17th ACM Conference on Economics and Computation, the 2018 ASSAmeetings, seminar participants at Bocconi University, in Marseille, in Maastricht and at CityUniversity for comments on a previous version of the paper. I gratefully acknowledge financialsupport by the Swiss National Science Foundation (SNSF) under project 100018-150086. Moreover, it is safe for the applying side of themarket to report their true preferences to the mechanism. Thus, the mechanismsuccessfully aggregates the information in the market and levels the playing fieldfor naive and sophisticated participants.In some applications, the market does not only match agents, but determinesalso the contractual details of the match. In a labor market, firms and work-ers may have some discretion on how to set the salary. In the cadet-to-branchmatch (S¨onmez and Switzer, 2013), cadets can choose between different lengths ofservice time in exchange for a higher priority in their branch of choice. In a collegeadmission problem, students can be admitted with or without a stipend. Thesemarkets can be understood as hybrids between matching markets and auctions andhave first been analyzed in the seminal paper of Kelso and Crawford (1982), withlater important extensions by Roth (1984b); Fleiner (2003); Hatfield and Milgrom(2005); Hatfield and Kojima (2009) among others. In a general model of many-to-one matching with endogenous contracting, a generalized version of the deferredacceptance mechanism can be defined, and is stable and group-strategy-proof un-der the assumption that contracts are substitutes for colleges and the law ofaggregate demand holds. In many applications of matching models with contracts, there is a naturalordering on contract terms and it is reasonable to assume that preferences aremonotone with respect to the ordering: In the cadet-to-branch matching modelof S¨onmez and Switzer (2013) the contract-term is the service time in the militaryand it is natural to assume that cadets prefer a shorter to a longer service in thesame branch. This assumption is for example made in the analysis of Jagadeesan(2016). In college admission problems (Hassidim et al., 2017; Abizada and Dur,2017) with stipends, it is natural to assume that students prefer being admittedwith a stipend to being admitted without a stipend at the same college, or moregenerally, being admitted with a higher stipend than a lower stipend at the samecollege. For the medical match, mechanisms which allow for flexible salaries have See Roth (1991) for evidence that clearing houses using unstable mechanisms tend to fail inpractice. In the following we call the applying side of the market “students” and the admitting sideof the market “colleges” motivated by the application of college admission. However, the modelequally applies to the other applications mentioned in this introduction. This means that if we shrink the choice set of a college, an equal or small number of contractswill be chosen. There is empirical evidence that monotonicity is violated for reported preferences of someparticipants in the Israeli psychology match, see Hassidim et al., 2016. But it seems likely thatthis monotonicity violations can be attributed to students failing to play the weakly dominant Monotonicity means that contractterms are totally ordered and students’ preferences are monotone with respectto this ordering. Under the assumption of monotone preferences, we show thatsubstitutability and the law of aggregate demand are not only sufficient, but alsoessentially necessary for the existence of a stable and (group)-strategy-proof mech-anism. For this purpose, we introduce the notion of an equivalent choice profile. ADA-equivalent choice profile is a choice profile such that the (student-proposing)deferred-acceptance algorithm produces the same outcome if the original choiceprofile is replaced by the equivalent choice profile. The domain of choice profilesthat are (1) DA-equivalent to a choice profile under which contract are substi-tutes for colleges and the law of aggregate demand holds, and for which (2) thedeferred-acceptance algorithm is stable, turns out to be equivalent to the domain ofchoice profiles under which contracts are observable substitutes and the observablelaw of aggregate demand holds - two notions recently introduced by Hatfield et al.(2018). Moreover, these choice profiles form a maximal (Cartesian and unitary)domain for the existence of a stable and (group)-strategy-proof mechanism.Our result have two important consequences: First, for monotone preferences,group-strategy-proofness is “for free” in the sense that the maximal domain forthe existence of a strategy-proof and stable mechanism is also maximal for theexistence of a group-strategy-proof and stable mechanism. Second obtain an em-bedding result in the sense of Jagadeesan (2016) for the class of choice functionsunder which contracts are observable substitutes and the observable law of ag-gregate demand holds. Thus, if attention is restricted to the case of monotonepreferences for students, it is, in some sense without loss of generality to workwith the model of matching with salaries of Kelso and Crawford (1982) ratherthan the full matching with contracts model.
Stable many-to-one matching mechanisms and their incentive properties have beenextensively studied (Hatfield and Kojima, 2010; Chen et al., 2016; Hirata and Kasuya,2017; Kominers and S¨onmez, 2016; Hatfield et al., 2018). Most papers focus on thepure matching model or on the matching with contracts model (Hatfield and Milgrom,2005; Roth, 1984b; Fleiner, 2003). Working with monotone preferences makes our strategy of revealing preferences truthfully, rather than that they have non-monotone preferences. Colleges’ preferences are not necessarily monotone in our analysis. In recent work, Abizada and Dur (2017) make similar observations and considera model of college admissions with stipends where complementarities in contractterms are present for colleges: In their model three contract terms { t + , t , t − } areavailable, interpreted as admission with stipend, admission without stipend butwith tuition waiver, and admission without either of the two, and the number of To illustrate this point, consider a college admission problem of the following kind: Thereare two colleges c , c and three students s , s , s . Suppose there are two kinds of contracts: Astudent can be admitted with stipend (represented by the contract term ”1”) or without stipend(represented by the contract term ”0”). The colleges have choice functions induced (in the usualway) by the following preferences { ( s , , ( s , , ( s , } ≻ c { ( s , } ≻ c { ( s , } ≻ c ∅ ≻ c . . . { ( s , } ≻ c { ( s , } ≻ c ∅ ≻ c . . . Suppose student always prefer to be admitted at a college under a stipend to being admitted atthe same college without a stipend. Going through all different cases, one can show that, for anypreferences satisfying this monotonicity assumption, a stable allocation (in the matching withcontracts sense) exists. This changes if students can report non-monotonic preferences. Considerthe following preferences:( c , ≻ s ( c , ≻ s ( c , ≻ s ( c , ≻ s ∅ ( c , ≻ s ( c , ≻ s ∅ ≻ s . . . ( c , ≻ s ( c , ≻ s ( c , ≻ s ( c , ≻ s ∅ Student s prefers to go to college c without a stipend rather than a stipend. Thus, in a stableallocation it will never be the case that s goes to college c with a stipend, because otherwise c and s could block that allocation. This in turn implies that no stable allocation exists: Theallocation that matches all three students to c without stipend is blocked by student s andcollege c . Any allocation that matches s to c without stipend is blocked by students s , s and s being admitted to college c without stipend. Any allocation that matches s to c withoutstipend is blocked by student s being admitted to college c without a stipend. Finally, allother allocations are either not individually rational or blocked by students s , s and s beingadmitted to college c without a stipend. − -contracts signed by a college constraints the number of t + -contracts it signs.Importantly, students have monotone preferences in these contract terms. Themodel is a special case of ours. In particular, the result for “Max-Min Respon-sive” preferences can be obtained as a special case of ours and for this case theirstrategy-proofness result can be strengthened to group-strategy-proofness. How-ever, Abizada and Dur (2017) also analyze pairwise-stable outcomes and this partof their analysis does not have a counter-part in our paper.Our original working paper (Schlegel, 2016) contained a version of our maximaldomain result for a model of matching with contracts where also colleges’ choicefunctions are monotone in contract-terms. Technically the two maximal domainresults are independent. However, the adaption to obtain the previous versionof the theorem from the current one are minimal. The current version of themaximality result can also be obtained with a similar proof as the one in theprevious version. For the original version of the theorem we refer the interestedreader to the original version of the working paper.While preparing the current version of the paper, Hatfield et al. (2018) releaseda new version of their working paper, where the authors also analyze preferencesrestrictions such as monotonicity and show that their analysis of observable substi-tutes and the observable law of aggregate demand extend to restricted preferencesdomains such as monotone preferences. Their additional work allows us to shortenthe proof of the “sufficiency part” of our maximal domain result considerably. Itnow suffices to show that observable substitutability and the observable law ofaggregate demand for monotone preferences (which for choice functions that aremonotone in contract terms, are equivalent to the virtual substitutability and vir-tual law of aggregate demand conditions from our original working paper) implyDA-equivalence to a profile of choice functions under which contracts are substi-tutes and the law of aggregate demand holds. We emphasize however that ourresult for monotone preferences is stronger than the new result in Hatfield et al.(2018) in the following regards: We obtain the stronger incentive property ofgroup-strategy-proofness instead of just strategy-proofness. Moreover, the ”non-manipulability” axiom is redundant under monotone preferences. Finally, theexplicit construction of an equivalent choice profile allows for a natural interpre- One has to make sure that in the “necessity part” of the maximal domain proof, the profileof unitary choice functions for the other colleges can be chosen to be induced by monotonepreferences. The main difference is that now stability of the deferred acceptance algorithm for the originalchoice profile has to be assumed on top of DA-equivalence. The mechanisms fall outside of the domain defined by Barber`a et al. (2016) on whichstrategy-proofness is equivalent to group-strategy-proofness. Thus, it is not sufficient to onlyshow strategy-proofness and invoke the result of Barber`a et al. (2016). The “non-manipulability” axiom is discussed in more detail in Remark 1.
There are two finite disjoint sets of agents, a set of colleges C and a set of students S . There is a finite set of possible contract-terms T which are totallyordered by ⊲ . Colleges can accept students under different bilateral contracts.The set of possible contracts is X ⊆ C × S × T . For a contract x ∈ X , we denoteby x C ∈ C the college involved in x , by x S ∈ S the student involved in X , and by x T ∈ T the contract term involved in x . We write x T D x ′ T whenever x T ⊲ x ′ T or x = x ′ .Each college c has a choice function Ch c : 2 X c → X c that from each set Y ⊆ X c chooses a subset of contracts. Each college can only sign one contract with anygiven student, i.e. for each x, y ∈ Ch c ( Y ) with x = y we have x S = y S . Throughoutthis paper, we assume that all considered choice functions satisfy the irrelevanceof rejected contracts (IRC) (Ayg¨un and S¨onmez, 2013), which means that forall Y ⊆ X , x ∈ X \ Y , x / ∈ Ch c ( Y ∪ { x } ) ⇒ Ch c ( Y ) = Ch c ( Y ∪ { x } ) . We also define a rejection function R c : 2 X c → X c by R c ( Y ) := Y \ Ch c ( Y ) . Wedenote the set of all choice functions for college c ∈ C that satisfy IRC, by C c .Each student s has preferences (cid:23) s over different contracts involving him, andan outside option which we denote by “ ∅ ”. We make the following assumption onstudents’ preferences:1. Preferences are strict , for x, x ′ ∈ X with x ′ S = x S , we have x = x ′ ⇒ x ≻ x S x ′ or x ′ ≻ x S x, and x ≻ x S ∅ or ∅ ≻ x S x.
2. Preferences are monotone in contract terms, for each x, x ′ ∈ X with x S = x ′ S we have x T ⊲ x ′ T ⇒ x ≻ x S x ′ .
6e denote the set of all strict and monotone preferences for student s ∈ S by R s . A market is a pair ( Ch, (cid:23) ) consisting of a choice profile Ch = ( Ch c ) c ∈ C ∈ × c ∈ C C c and a preference profile (cid:23) = ( (cid:23) s ) s ∈ S ∈ × s ∈ S R s .An allocation is a set Y ⊆ X that contains at most one contract per student.We denote the set of allocations by A . In the following it will be useful to definefor each set of contracts Y ⊆ X c with a college c the allocation Y min := { y ∈ Y : ∄ y ′ ∈ Y, y ′ S = y S , y T ⊲ y ′ T } of contracts that gives each student the worst contract among the contract in Y , and the set U ( Y ) := { x ∈ X c : x T D y T for some y ∈ Y with x S = y S } of contracts, not necessarily in Y , which are as least as good for the involvedstudent as his worst contract in Y .Finally, an allocation Y is individually rational in ( Ch, (cid:23) ) if for each c ∈ C , we have Y c = Ch c ( Y ),and for each y ∈ Y we have y ≻ y S ∅ , blocked in ( Ch, (cid:23) ) if there are c ∈ C and an allocation Z with Z c = Y c ,such that Z c = Ch c ( Y ∪ Z ) and for each z ∈ Z c we have z (cid:23) z S Y z S , stable in ( Ch, (cid:23) ) if it is individually rational and not blocked. A mechanism (for the students) is a mapping from preference profiles to allo-cations M : × s ∈ S R s → A . Mechanism M is strategy-proof if it is a weaklydominant strategy for students to report their true preferences to the mechanism,i.e. for each s ∈ S , (cid:23) − s ∈ × s ′ ∈ S \{ s } R s ′ and (cid:23) s , (cid:23) ′ s ∈ R s we have M ( (cid:23) s , (cid:23) − s ) (cid:23) s M ( (cid:23) ′ s , (cid:23) − s ) . Mechanism M is group-strategy-proof if for each S ′ ⊆ S , (cid:23) − S ′ ∈ × s ∈ S \S ′ R s and (cid:23) S ′ , (cid:23) ′ S ′ ∈ × s ∈ S ′ R s , there is a s ′ ∈ S ′ with M ( (cid:23) S ′ , (cid:23) − S ′ ) (cid:23) s ′ M ( (cid:23) ′ S ′ , (cid:23) − S ′ ) . Let Ch be a choice profile. Mechanism M is Ch -stable if for each (cid:23)∈ × s ∈ S R s allocation M ( (cid:23) ) is stable in ( Ch, (cid:23) ). Note that the worst contract for a student with a given college c from some set is the sameunder all monotone preferences. .3 Examples Several examples from applied marked design fit into our model. Later in Sec-tion 3.4, we will show how our results apply to these models.
A finite
Kelso-Crawford economy consists of a finite set of firms F , a finiteset of workers W , a finite set of salaries Σ ⊆ R ++ and a profile u i ∈ F ∪ W of utilityfunctions, where for each f ∈ F , utility function u f assigns to each W ′ ⊆ W and p ∈ Σ W ′ a utility level u f ( W ′ , p ) and for each W ′ ⊆ W , u f ( W ′ , · ) is strictlydecreasing in salaries, and for each w ∈ W , the utility function u w assigns to each( f, p ) ∈ F × Σ a utility level u w ( f, p ) and a utility level to the outside option u w ( ∅ )such that for each w ∈ W , u w ( f, · ) is strictly increasing in salaries.The model fits in our framework with C = F , S = W , T = Σ, x T ⊲ x ′ T ⇔ x T > x ′ T , X = F × W × Σ, choice functions are defined by Ch f ( Y ) = max Y ′ ⊆ Y min u f ( Y ′ ) , and preferences ( (cid:23) w ) w ∈ W are induced by utility functions ( u w ) w ∈ W . Note that the constructed market with contracts (
Ch, (cid:23) ) does not only satisfymonotonicity of students’ preferences, but also monotonicity of colleges’ choicefunction, where for c ∈ C , Ch c is monotone in contract-terms if for each Y ⊆ X C we have Ch c ( Y ) ⊆ Y min . A college admission problem with stipends of Hassidim et al. (2017), consistsof set of colleges C , a set of students S , a finite set of contract terms T ⊆ N , whereeach t ∈ T correspond to a funding level, a set of contracts X ⊆ C × S × T andpreferences ( (cid:23) i ) i ∈ C ∪ S for colleges and students. Preferences of a student s are over X s ∪ {∅} and monotone with respect to > . Preferences of colleges are responsiveaccording to quotas (( q tc ) t ∈ T ) c ∈ C with t < t ′ ⇒ q tc ≤ q t ′ c where q tc is the number ofseats available with funding level t or lower and a profile of master lists ( ≫ c ) c ∈ C which are linear orders over T ∪ {∅} such that allocations that do not violate thequota and only differ in contract terms, are ranked responsively according to themaster-list(the specifications accommodates different possibilities of choosing thecontract terms, so preferences and induced choice functions can be non-monotone,for the full details see Hassidim et al., 2017).8 .3.3 Cadet to branch matching A cadet-to-branch matching problem of S¨onmez (2013), consist of a set ofcadets I , a set of branches B , a finite set of service times t < t < . . . < t k , aprofile of order of merit lists ( ≫ b ) b ∈ B which are strict priority orderings over I ,and a profile of capacities ( q b , q b ) b ∈ B ⊆ ( Z ≥ ) B . Choice functions of branches are“bid for your career (BfYC) choice functions” which can be informally described asfollows (for a formal description see the original paper of S¨onmez, 2013): Brancheschoose the q b highest ranked cadets according to ≫ b and selects for each of themthe contract with the shortest service time available and, afterwards choose the q b next ranked cadets according to ≫ b and selects for each of them the contract withthe longest service time available. The model fits in our framework with C = B , S = I , T = { t , . . . , t k } , where x T ⊲ x ′ T ⇔ x T < x ′ T , X = C × S × T . In general, a stable allocation does not need to exist for our model. A sufficientcondition for stability is that contracts are substitutes for colleges, i.e. if a contractis chosen from some set of contracts, then this contract is also chosen from eachsubset of that set of contracts.
Substitutability (Roth, 1984b; Hatfield and Milgrom, 2005): For each Y ⊆ X and each x, z ∈ X \ Y , x ∈ Ch c ( Y ∪ { x, z } ) ⇒ x ∈ Ch c ( Y ∪ { x } ) . Not only is substitutability sufficient for the existence of a stable allocationbut it also guarantees that the set of stable allocations has a lattice structure.If contracts are substitutes for colleges, then the set of stable allocation forms alattice with respect to the preferences of students (Blair, 1988). In particular,there is a unique stable allocation that is most preferred by all students among allstable allocations. We call this allocation the student-optimal stable allocation.It can be found by the deferred acceptance (DA) algorithm that is definedas follows.1. Each student applies under his favorite acceptable and unrejected contractor stays alone, if he finds no unrejected contract acceptable.2. Each college tentatively accepts the contracts it chooses among the proposedcontracts and rejects all other contracts.3. If no applications has been rejected in the current round, the chosen contractsform the final allocation. If some applications are rejected we repeat.9e denote the outcome of the DA-algorithm in market (
Ch, (cid:23) ) by DA ( Ch, (cid:23) ).The deferred acceptance mechanism for Ch assigns to each (cid:23)∈ × s ∈ S R s theoutcome of the deferred acceptance algorithm DA ( Ch, (cid:23) ).In general, the deferred acceptance algorithm does not need to converge to astable allocation. It could be the case that a student s applies in some round tosome college c which tentatively accepts the student, but another college c ′ wantsto recall an application made by s to c ′ in an earlier round that c ′ had previouslyrejected. Substitutability rules out this possibility, since it guarantees that collegeswill never want to recall applications made in previous rounds. Later we will seethat also weaker conditions than substitutability guarantee the convergence to astable allocation.In the following, we call a choice profile Ch DA-stable if for each (cid:23)∈ × s ∈ S R s , allocation DA ( Ch, (cid:23) ) is stable.Student-optimality is related to group-strategy-proofness. Under substitutabil-ity and the following additional condition on the colleges’ choice functions the de-ferred acceptance mechanism is group-strategy-proof.
Law of Aggregate Demand (Hatfield and Milgrom, 2005): For
Y, Z ⊆ X wehave Z ⊆ Y ⇒ | Ch c ( Y ) | ≥ | Ch c ( Z ) | . The following proposition summarizes known results about side-optimal stableallocations, the invariance of the set of matched students in stable allocations (therural hospitals theorem), and group-strategy-proofness.
Proposition 1 (Kelso and Crawford, 1982; Blair, 1988; Hatfield and Milgrom,2005; Hatfield and Kojima, 2009) .
1. If contracts are substitutes for colleges, then the deferred acceptance algorithmconverges to a stable allocation that is most preferred by all students amongall stable allocations.2. If choice functions satisfy, moreover, the law of aggregate demand, then(a) the set of accepted students is the same in all stable allocations and eachcollege accepts the same number of students in each stable allocation,(b) the deferred acceptance mechanism is group-strategy-proof.
It is a natural question whether the conditions of Section 2.4 for the stability andgroup-strategy-proofness of the deferred acceptance mechanism are also necessary.10ext we provide a counter example showing that substitutability and the law ofaggregate demand are not necessary for the deferred-acceptance mechanism tobe stable and group-strategy-proof. The choice function example will have thefollowing structure: There is one college c for which contracts are not substitutes.For each college except for c , contracts are substitutes and the law of aggregatedemand holds. However, c ’s choice function can be replaced by another choicefunction, such that1. the outcome of the deferred acceptance algorithm is the same under theoriginal choice profile and the profile where c ’s choice function is replaced,2. under the replacing choice function, contracts are substitutes for c and thelaw of aggregate demand holds.The deferred acceptance mechanism is stable and group-strategy-proof both forthe original market and the market where we have replaced c ’s choice function bythe other choice function. Example . We reconsider the college c from the example in Footnote 8. Let x = ( c , s , , x ′ = ( c , s , , y = ( c , s , , y ′ = ( c , s , , z = ( c , s , , z ′ =( c , s , . Let X = { x, x ′ , y, y ′ , z, z ′ } and 1 ⊲
0. The choice function of the collegeis Ch c ( Y ) = { x, y, z } , if { x, y, z } ⊆ Y, { y } , if y ∈ Y and x / ∈ Y or z / ∈ Y, { y ′ } , if y ′ ∈ Y, y / ∈ Y, ∅ , else.Now consider the alternative choice function Ch ′ c defined by: Ch ′ c ( Y ) = { y } , if y ∈ Y, { y ′ } , if y ′ ∈ Y, y / ∈ Y, ∅ , else.Note that under Ch c contracts are not substitutes as x ∈ Ch c ( X ) = { x, y, z } but x / ∈ Ch c ( X \ { z } ) = { y } and that under Ch ′ c contracts are substitutes.Suppose colleges C \ { c } have choice functions Ch − c = ( Ch c ) c = c underwhich contracts are substitutes and the law of aggregate demand holds. Define Ch := ( Ch c , Ch − c ) and Ch ′ := ( Ch ′ c , Ch − c ). Let (cid:23)∈ × s ∈ S R s . We show thatthe deferred acceptance algorithm process in the market ( Ch, (cid:23) ) and the deferredacceptance algorithm in the market ( Ch ′ , (cid:23) ) converge to the same allocation. Ob-serve that the choice functions Ch c and Ch ′ c differ only for sets Y ⊆ X with11 x, y, z } ⊆ Y . In particular, for the deferred acceptance algorithm to differ in thetwo markets, student s must apply to c under contract y during the deferredacceptance algorithm in ( Ch, (cid:23) ). Note however that before s applies to c undercontract y , he applies to c under contract y ′ , as y ′ T ⊲ y T . But once the process ten-tatively matches s to c under contract y ′ , the college will not subsequently dropthe contract or accept additional contracts. Thus, s will never apply to c undercontract y . Hence, the deferred acceptance algorithm in the two markets convergeto the same allocation, which is the student-optimal stable allocation in ( Ch ′ , (cid:23) ).By Proposition 1, the deferred acceptance mechanism for Ch ′ is group-strategy-proof and Ch ′ -stable. It is also easy to see that, as Ch c is induced by monotonepreferences, the mechanism is Ch -stable as well. Thus, there is a Ch -stable andgroup-strategy-proof mechanism.The example motivates the following definition. DA-equivalence : Two choice profiles Ch and Ch ′ are deferred-acceptance-equivalentif for each (cid:23)∈ × s ∈ S R s we have DA ( Ch, (cid:23) ) = DA ( Ch ′ , (cid:23) ).From this definition and Proposition 1, we obtain the following result: Proposition 2. If Ch is DA-equivalent to a profile under which contracts aresubstitutes and the law of aggregate demand holds, then the deferred acceptancemechanism for Ch is group-strategy-proof. If contracts are substitutes under the equivalent choice profile, the deferredacceptance mechanism is stable for the equivalent choice profile. It is not in gen-eral true that the deferred acceptance mechanism is also stable for the originalchoice profile. The outcome can fail to be stable, if colleges would like to recallcontracts that they had rejected during a previous round of the deferred accep-tance algorithm. In Hatfield et al. (2018) propose a condition on choice functions,called observable substitutability, that rules out exactly the problem that duringthe deferred acceptance algorithm colleges would like to recall contracts. We nextintroduce this property formally and derive some useful implications.In the following, a sequence of contracts x , x , . . . , x τ is generated frommonotone preferences if for 1 ≤ t ≤ τ and each x ∈ X with x S = x tS and x T ⊲ x tT , we have x ∈ { x , . . . , x t − } . A sequence x , x , . . . , x τ is observable under Ch c if for 1 ≤ t ≤ τ − x t +1 S / ∈ [ Ch c { x , . . . , x t } ] S . Then observable A similar notion has been introduced independently by Jagadeesan (2016). However, sinceJagadeesan (2016) does not assume monotonicity on student preferences, his notion of DA-equivalence is much stronger. Without monotonicity, DA-equivalence requires that choices coin-cide on all sets of contracts which contain at most one contract per student. (see Theorem 1 ofJagadeesan, 2016). With monotonicity the choices only have to coincide at some of these sets ofcontracts.
Observable Substitutability (Hatfield et al., 2018): A choice function Ch c isobservably substitutable for monotone preferences if for each observable sequence x , x , . . . , x τ under Ch c that is generated from monotone preferences, we have R c { x , . . . x t } ⊆ R c { x , . . . , x t +1 } for each 0 < t < τ. Similarly, we can define an observable version of the law of aggregate demand.
Observable Law of Aggregate Demand (Hatfield et al., 2018): A choice func-tions Ch c satisfies the observable law of aggregate demand for monotone prefer-ences if for each observable sequence under Ch c that is generated from monotonepreferences, we have | Ch c { x , . . . x t }| ≤ | Ch c { x , . . . , x t +1 }| for each 0 < t < τ. Since we exclusively deal with monotone preferences, from now one we drop theterm ”for monotone preferences.” However, we emphasize that observable sub-stitutability for monotone preferences is a weaker notion than observable sub-stitutability for general preferences. In the following, we call a set of contracts Y ⊆ X c observable under Ch c , if there is a sequence of contracts x , . . . , x | Y | that is observable under Ch c and generated from monotone preferences such that Y = { x , . . . , x | Y | } . We call ˜ Y ⊆ Y ⊆ X c a maximal observable subset of Y under Ch c , if there is no observable set Y ′ under Ch c with ˜ Y ( Y ′ ⊆ Y . Itis straightforward to see that substitutability (the law of aggregate demand) forobservable sequences (generated by monotone preferences) implies substitutability(the law of aggregate demand) for observable sets. We prove a stronger version ofthis result, which will be useful in the subsequent proofs. Lemma 3.
If contracts are observable substitutes under Ch c , then for each Y ⊆ X c there is a unique maximal observable subset of Y under Ch c . In this case, for Z ⊆ Y ⊆ X c , let ˜ Z ⊆ Z and ˜ Y ⊆ Y be the unique maximal observable subsetsunder Ch c . Then R c ( ˜ Z ) ⊆ R c ( ˜ Y ) . If moreover, the observable law of aggregate demand holds for Ch c , then | Ch c ( ˜ Z ) | ≤ | Ch c ( ˜ Y ) | . Proof.
First we prove the existence of a unique maximal observable subset. Theproof strategy is due to Hirata and Kasuya (2014). We use induction on the sizeof the set Y . 13 nduction Basis: If | Y | = 0, then Y = ∅ and trivially ˜ Y = ∅ = Y is the onlymaximal observable subset of Y . Induction Assumption:
For each Y ⊆ X s with | Y | ≤ n there exists a uniquemaximal observable subset of Y under Ch c . Induction Step:
Consider maximal sequences x , . . . , x T and y , . . . , y T ′ in Y that are observable under Ch c and generated from monotone preferences. Weshow that { x , . . . , x T } = { y , . . . , y T ′ } . Suppose not. Then w.l.o.g. there is a1 ≤ t ≤ T ′ with y t / ∈ { x , . . . , x T } . Choose the smallest such t and considerthe set Y ′ := Y \ { y t } . Since y t / ∈ { x , . . . , x T } , by the induction assumption { x , . . . , x T } is the unique maximal observable subset of Y ′ under Ch c . Nowconsider, the sequence y , . . . , y t − . Observe that { y , . . . , y t − } ⊆ Y ′ and thatthis sequence is generated from monotone preferences and is observable under Ch c . Extend y , . . . y t − to a maximal sequence y , . . . , y t − , ˜ y t , . . . , ˜ y ˜ T in Y ′ thatis observable under Ch c and generated from monotone preferences. By the in-duction assumption, we have { y . . . , y t − , ˜ y t , . . . , ˜ y ˜ T } = { x , . . . , x T } . Further-more we have y tS ∈ Ch c { x , . . . , x T } S , since otherwise x , . . . , x T , y t would beobservable under Ch c , and, as t was chosen to be the smallest index such that y t / ∈ { x , . . . , x T } , x , . . . , x T , y t would be generated from monotone preferences.Observe furthermore, that by observability of y , . . . , y t , we have U ( { y t } ) \ { y t } ⊆ R c ( { y , . . . , y t − } ). By observable substitutability, we have U ( { y t } ) \ { y t } ⊆ R c ( { y , . . . , y t − } ) ⊆ R c { y . . . , y t − , ˜ y t , . . . , ˜ y ˜ T } and therefore y tS / ∈ Ch c { y . . . , y t − , ˜ y t , . . . , ˜ y ˜ T } S . But this contradicts { y . . . , y t − , ˜ y t , . . . , ˜ y ˜ T } = { x , . . . , x T } . Now consider Z ⊆ Y ⊆ X c and let ˜ Z ⊆ Z and ˜ Y ⊆ Y be the unique maximalobservable subsets under Ch c . Let z , . . . , z | ˜ Z | be an observable sequence, gener-ated from monotone preferences such that ˜ Z = { z , . . . , z | ˜ Z | } . Maximally extendthe sequence to a sequence z , . . . , z | ˜ Z | , z | ˜ Z | +1 , . . . , z ˜ Y with z | ˜ Z | +1 , . . . , z ˜ Y ∈ ˜ Y thatis observable and generated from monotone preferences. Since ˜ Y is observable, wehave ˜ Y = { z , . . . , z | ˜ Z | , z | ˜ Z | +1 , . . . , z ˜ Y } . Thus observable substitutability applied tothe sequence { z , . . . , z | ˜ Z | , z | ˜ Z | +1 , . . . , z ˜ Y } implies R c ( ˜ Z ) ⊆ R c ( ˜ Y ) , and the observable law of aggregate demand applied to { z , . . . , z | ˜ Z | , z | ˜ Z | +1 , . . . , z ˜ Y } implies | Ch c ( ˜ Z ) | ≤ | Ch c ( ˜ Y ) | . Lemma 4.
Let ( Ch, (cid:23) ) be a market such that contracts are observable substitutes.1. Let Y ⊆ Y ⊆ . . . ⊆ Y T ⊆ X such that Y t is the set of proposed contractsup to round t during the deferred acceptance algorithm in ( Ch, (cid:23) ) . Then foreach c ∈ C , the sets Y c , . . . , Y Tc are observable under Ch c ,2. DA ( Ch, (cid:23) ) is stable. With the two lemmata, we obtain the following result.
Theorem 5.
For a profile of choice functions Ch = ( Ch c ) c ∈ C the following areequivalent:1. Under Ch contracts are observable substitutes and the observable law of ag-gregate demand holds.2. Ch is DA-stable and DA-equivalent to a choice profile Ch ′ under which con-tracts are substitutes and the law of aggregate demand holds.Moreover, for each c ∈ C the choice function Ch ′ c can be chosen to be monotonein contract terms and its construction only depends on Ch c and is independent of Ch − c .Proof. First we prove that 1 . ⇒ . We define for each choice function Ch c a secondchoice function Ch ′ c that we call the virtual choice function for Ch c . For each Y ⊆ X c let Y ∨ ⊆ U ( Y ) be the, by Lemma 3 well-defined, maximal subset of U ( Y )that is observable under Ch c . We define Ch ′ c : 2 X c → X c by Ch ′ c ( Y ) := (cid:8) x ∈ Y min : x S ∈ Ch c ( Y ∨ ) S (cid:9) . First note that Ch ′ c satisfies our assumptions on choice functions: By definition Ch ′ c ( Y ) ⊆ Y min ⊆ Y , and since Y min contains at most one contract per student,also Ch ′ c ( Y ) contains at most one contract per student. The IRC condition for Ch ′ c will follow from substitutability and the law of aggregate (see Ayg¨un and S¨onmez,2013) or Ch ′ c which we will establish next. The result is Proposition A.1 in Hatfield et al. (2018). Their proof is for the case of gen-eral preferences for students and with the notion of observable substitutability that applies togeneral preferences, i.e. when observable sequences are not necessarily generated from monotonepreferences. However, the proof applied verbatim also to the case of monotone preferences forstudents where observable substitutability is with regard to observable sequences generated frommonotone preferences. Z ⊆ Y ⊆ X c . Note that U ( Z ) ⊆ U ( Y ). Thus by Lemma 3, we have R c ( Z ∨ ) ⊆ R c ( Y ∨ ) and | Ch c ( Z ∨ ) | ≤ | Ch c ( Y ∨ ) | . Immediately from this, we obtainthe law of aggregate demand for Ch ′ c , as | Ch ′ c ( Z ) | = | Ch c ( Z ∨ ) | ≤ | Ch c ( Y ∨ ) | = | Ch ′ c ( Y ) | . To show that contracts are substitutes under Ch ′ c , first note that, by definition, Y \ ( Y ) min ⊆ R ′ c ( Y ). Thus it suffices to show that for x ∈ Y min ∩ R ′ c ( Z ) we have x ∈ R ′ c ( Y ) . First note that x ∈ R ′ c ( Z ) ∩ Y min implies that x ∈ Z min and therefore x S / ∈ Ch ′ c ( Z ) S = Ch s ( Z ∨ ) S . Thus U ( { x } ) ⊆ R c ( Z ∨ ) ⊆ R c ( Y ∨ ) and therefore x S / ∈ Ch ′ c ( Y ) S = Ch c ( Y ∨ ) S . Hence x ∈ R ′ c ( Y ). Thus contracts are substitutesunder Ch ′ c . Next observe that if Y is observable under Ch c , then by definition Y = U ( Y ) = Y ∨ . Moreover, by observable substitutability of Ch c and observability of Y , wehave Ch c ( Y ) ⊆ Y min . Thus if Y is observable under Ch c , then Ch ′ c ( Y ) = Ch c ( Y ) . Thus, for each (cid:23)∈ × s ∈ S R s , by the first part of Lemma 4, we have DA ( Ch ′ , (cid:23) ) = DA ( Ch, (cid:23) ), and by the second part of Lemma 4, allocation DA ( Ch, (cid:23) ) is stablein (
Ch, (cid:23) ).Note furthermore that Ch ′ c is monotone by construction, since for each Y , wehave Ch c ( Y ) ⊆ Y min .Next we show that 2 . ⇒ . Let Ch be DA-equivalent to Ch ′ such that under Ch ′ contracts are substitutes and the law of aggregate demand and such that DA ( Ch, · ) is Ch -stable. Let c ∈ C and consider a sequence x , . . . , x T that isobservable under Ch c and generated from monotone preferences. We show that Ch ′ c { x , . . . , x T } = Ch c { x , . . . , x T } . Since under Ch ′ c contracts are substitutesand the law of aggregate demand holds, this will imply observable substitutabilityand the observable law of aggregate demand under Ch c . In the following, wedenote by (cid:23) ∈ × s ∈ S R s a profile such that no contract is acceptable, and for1 ≤ t ≤ T we denote by (cid:23) t ∈ × s ∈ S R s a preference profile such that x ≻ tx S ∅ for x ∈ { x , . . . , x t } and ∅ ≻ tx S x for x / ∈ { x , . . . , x t } . First note that ∅ = Ch ′ c ( ∅ ) = DA ( Ch ′ , (cid:23) ) = DA ( Ch ′ , (cid:23) ) = Ch s ( ∅ ) . We now show that if for 0 ≤ τ < T we have Ch ′ c { x , . . . , x t } = DA ( Ch ′ , (cid:23) t ) = DA ( Ch, (cid:23) t ) = Ch c { x , . . . , x t } for each 0 ≤ t ≤ τ , then Ch ′ c { x , . . . , x τ } = DA ( Ch ′ , (cid:23) τ ) = DA ( Ch, (cid:23) τ ) = Ch c { x , . . . , x τ } . Since Ch c { x , . . . , x t } = Ch ′ c { x , . . . , x t } for 0 ≤ t < τ , observability of x , . . . , x τ under Ch c implies observability of x , . . . , x τ under Ch ′ c . Let Y be the set of16roposed contract during the deferred acceptance mechanism in ( Ch ′ , (cid:23) τ ) and notethat by Lemma 4, is observable under Ch ′ . Moreover, Y is a maximal observablesubset of the set of acceptable contracts under (cid:23) τ , since Ch ′ c ( Y ) = DA ( Ch ′ , (cid:23) τ )by (observable) substitutability of Ch ′ c . By construction, the set of acceptablecontracts under (cid:23) τ is { x , . . . , x τ } . As { x , . . . , x τ } is observable under Ch ′ c and Y is a maximal observable subset of { x , . . . , x τ } Ch ′ c , Lemma 3 applied to Ch ′ c implies Y = { x , . . . , x τ } . Thus, Ch ′ c { x , . . . x τ } = DA ( Ch ′ , (cid:23) τ ). Next we showthat Ch c ( Y ) = DA ( Ch, (cid:23) τ ). Suppose not. Then Ch c ( Y ) blocks DA ( Ch, (cid:23) τ ).But this contradicts the stability of DA ( Ch, · ). Hence by DA-equivalence Ch ′ c { x , . . . x τ } = DA ( Ch ′ , (cid:23) τ ) = DA ( Ch, (cid:23) ) = Ch c { x , . . . x τ } . An immediate consequence of our Theorem 5 is that DA-stability and DA-equivalenceto profile such that contracts are substitutes and the law of aggregate demand holdsis a necessary and sufficient condition for the existence of group-strategy-proof andstable mechanism, in the following sense: In the following a choice domain is a setof choice profiles
D ⊆ × c ∈ C C c . A choice domain D is Cartesian if D = × c ∈ C D c where D c ⊆ C c for each college c ∈ C . A choice function Ch c for college c is unitdemand if | Ch c ( Y ) | ≤ Y ⊆ X c . A choice domain is unitary if it includesall profiles of unit demand choice functions. We obtain the following result whichis a direct consequence of our Theorem 5 and Theorem 6 in Hatfield et al. (2018): Corollary 6.
The choice domain of choice profiles that are DA-stable and DA-equivalent to a profile under which contracts are substitutes and the law of aggregatedemand holds is a maximal unitary, Cartesian domain for the existence of a stableand group-strategy-proof mechanism and for the existence of a stable and strategy-proof-mechanism.Proof.
Let D be the domain of choice profiles that are DA-stable and DA-equivalentto a profile under which contracts are substitutes and the law of aggregate demandholds. Let Ch ∈ D . By DA-stability of Ch , DA ( Ch, · ) is Ch -stable and by Propo-sition 2, DA ( Ch, · ) is group-strategy-proofness. Moreover, by Theorem 5, domain D is unitary and Cartesian.To show maximality, note that by Theorem 5, under Ch contracts are observ-able substitutes and the observable law of aggregate demand holds. Theorem 6 ofHatfield et al. (2018) implies that there is no Cartesian and unitary domain D ′ with D ( D ′ such that for each Ch ∈ D ′ there exists a Ch -stable and strategy-proofmechanism. In particular, there is no such domain such that for each Ch ∈ D ′ there exists a Ch -stable and group-strategy-proof mechanism.17 emark . Hatfield et al. (2018) prove that the domain of all choice profiles underwhich contracts are observable substitutes, the observable law of aggregate demandholds and which satisfy a property called non-manipulability, is a maximal unitaryCartesian domain for the existence of a stable and strategy-proof mechanism. Non-manipulability requires that for each profile in the domain the deferred-acceptancemechanism is strategy-proof on the domain of preference profiles where only con-tracts with a given college c are acceptable. Whereas this property is in generalindependent of the other two properties, our Corollary 6 implies that this prop-erty is implied by observable substitutability and the observable law of aggregatedemand for monotone preferences. As a second consequence of Theorem 5, we obtain an embedding result. In re-cent work Jagadeesan (2016) shows that for BfYC choice profiles as introduced inSection 2.3.3 there is a DA-equivalent choice profile such that the DA-equivalentmarket can be embedded into a Kelso-Crawford economy. There are three dif-ferences between the result by Jagadeesan (2016) and earlier result by Echenique(2012) that has been extended by Kominers (2012); Schlegel (2015):1. The results apply to different domains of choice functions.2. The result of Jagadeesan (2016) requires to first construct a DA-equivalentchoice profile. The embedding is then performed for a market where theoriginal choice profile is replaced by the equivalent choice profile. So hisembedding establishes an isomorphism between the deferred acceptance al-gorithm in the original market and the salary adjustment process in theKelso-Crawford economy. In contrast to this, Echenique (2012) establishes afull isomophism between the sets of stable allocations in the original marketand the Kelso-Crawford market.3. Utility function in the Kelso-Crawford economy satisfy stronger regularityconditions in Jagadeesan (2016). In Echenique (2012), monotonicity of utilityfunctions can only be achieved for salaries corresponding to “un-dominated”contracts (see the discussion in Schlegel, 2015), whereas in Jagadeesan (2016)monotonicity can be achieved for all salaries. Moreover, the utility functionsfor firms can be chosen to be quasi-linear in salaries.Thus, the notion of isomorphism of Jagadeesan (2016) is neither weaker (becauseof 3.) nor stronger (because of 2.) than the one of Echenique (2012). In thefollowing, when we talk of a “embedding result” we mean embedding in the senseof Jagadeesan (2016). Formally, an isomorphism in the sense of Jagadeesan,18016(see Definition C.2 in his paper) between a matching market with contracts(
Ch, (cid:23) ) and a Kelso-Crawford-economy (Σ , u ) is a bijection ( f, w, σ ) : C × S × T → F × W × Σ such that1. for each x, x ′ ∈ X , if x ′ C = x C then f ( x ) = f ( x ′ ) and if x S = x ′ S then w ( x ) = w ( x ′ ),2. for each x, x ′ ∈ X we have x ≻ x S x ′ ⇔ u w ( x ) ( f ( x ) , σ ( x )) > u w ( x ) ( f ( x ) , σ ( x ))and x ≻ x S ∅ ⇔ u w ( x ) ( f ( x ) , σ ( x )) > u w ( x ) ( ∅ ) ,
3. for each c ∈ C and Y ⊆ X c we have Ch c ( Y ) = argmax Y ′ ⊆ Y min u f ( c ) ( { ( w ( x ) , σ ( x )) : x ∈ Y ′ } ) ,
4. for each ( c, s, t ) ∈ ( C × S × T ) \ X , we have u w ( c,s,t ) ( f ( c, s, t ) , σ ( c, s, t ))
Corollary 7.
For each choice profile Ch such that contracts are observable sub-stitutes and the observable law of aggregate demand holds, there exists a DA-equivalent choice profile Ch ′ , a finite set Σ ⊆ R ++ , utility functions ( u c ) c ∈ C , andfor each (cid:23)∈ × s ∈ S R s , utility functions ( u s ) s ∈ S and an isomorphism in the senseof Jagadeesan, 2016 between ( Ch ′ , (cid:23) ) and (Σ , u ) .Proof. Let Ch be a profile such that contracts are observable substitutes and theobservable law of aggregate demand holds. By Theorem 5, there exist a DA-equivalent profile Ch ′ under which contracts are substitutes and the law of aggre-gate demand holds. Moreover, Ch ′ can be chosen to be monotone in contract-terms. It can be shown that a choice function Ch ′ c satisfying IRC can be rational-ized by a strict preference relation (cid:23) ′ c over A c (see e.g.Alva, 2018). W.l.o.g. therepresentation can be chosen such that for allocations Y, Y ′ ∈ A c , if Y ⊆ Y ′ and Ch c ( Y ′ ) = Y the only allocations ranked between Y and Y ′ according to (cid:23) ′ c aresubsets of Y ′ . Note that then in particular (cid:23) ′ c can be chosen to be monotone incontracts terms, i.e. such that for allocation Y, Y ′ ∈ A c , Y = Y ′ with Y S = Y ′ S wehave that y T D y ′ T for each y ∈ Y and y ′ ∈ Y ′ with y S = y ′ S implies Y ′ (cid:23) ′ c Y : Indeednote that by monotonicity in contract-terms of Ch ′ c , we have Ch ′ c ( Y ∪ Y ′ ) ⊆ Y ′ This item is not required in Jagadeesan (2016), as he only considers situations in which thereis the same number of contracts between any college and student. Y ′ and Ch ′ c ( Y ∪ Y ′ ) are subsets of Y ′ , we either have Y ≻ ′ c Ch ′ c ( Y ∪ Y ′ ) or Y ′ ≻ ′ c Y . Since (cid:23) ′ c rationalizes Ch ′ c , the first cannot be the case, hence Y ′ ≻ ′ c Y .Now let F = C, W = S, Σ = { , . . . , | T |} . For each c ∈ C and s ∈ S let x bethe ⊲ -maximal contract in X with x C = c and x S = s and define f ( x ) = c, w ( x ) = s, σ ( x ) = T . Similarly for the ⊲ -maximal contract x ′ in X ⊆ { x } with x C = c and x S = s and define f ( x ) = c, w ( x ) = s, σ ( x ) = T − | T | contracts between c and s extend the bijection arbitrarily. For each c ∈ F = C , we can choose u c to be any monotonic utility representation such thatfor allocations Y, Y ′ ∈ A c we have Y ≻ ′ c Y ′ : ⇔ u c ( { ( w ( x ) , σ ( x )) : x ∈ Y } ) > u c ( { ( w ( x ′ ) , σ ( x ′ )) : x ′ ∈ Y ′ } ) , and for s ∈ W = S we can choose u s to be any monotonic utility representationsuch that for x, x ′ ∈ X s we have x ≻ s x ′ ⇔ u s ( f ( x ) , σ ( x )) > u s ( f ( x ) , σ ( x ))and x ≻ s ∅ ⇔ u s ( f ( x ) , σ ( x )) > u s ( ∅ ) . Remark . While we use the same notion of isomorphism as Jagadeesan (2016),we do not assume quasi-linearity of firm utility functions in the Kelso-Crawfordeconomy. In this sense our result is weaker. However, our result applies to a largerdomain of choice functions.
In a Kelso-Crawford economy where firms’ utility functions are quasi-linear insalaries, our Theorem 5, can be simplified considerably. Quasi-linearity impliesthe following invariance property of the choice function: A choice function satisfies demand-invariance if for Z ⊆ Y ⊆ X c such that Z s = Y s for s / ∈ Ch c ( Z ) S ,we have Ch c ( Y ) S = Ch c ( Z ) S . A direct consequence of this property is that theequivalent choice profile in Theorem 5, can be chosen to be the original profile.More generally this proposition holds for monotone choice functions satisfyingdemand invariance.
Proposition 8.
If a choice function is induced by quasi-linear preferences, thechoice function and the virtual choice function is the same. In particular, if a rofile Ch of choice functions induced by quasi-linear preferences is DA-equivalentto a profile Ch ′ such that contracts under Ch ′ are substitutes, then we can choose Ch ′ = Ch.
Proof.
Observe that by monotonicity of Ch c in contract terms, we have C c ( Y ) = C c ( U ( Y )). By construction, Y ∨ ⊆ U ( Y ) and Y ∨ s = U ( Y ) s for s / ∈ Ch c ( Y ∨ ) S .Thus, by demand invariance Ch c ( Y ∨ ) S = Ch c ( U ( Y )) S = Ch c ( Y ) S , and hence bymonotonicity of Ch c in contract terms, Ch ′ c ( Y ) = Ch c ( Y ) ⊆ Y min .We obtain the following corollary: Corollary 9.
For a discrete quasi-linear Kelso-Crawford economies without ties,gross substitutable valuations form a maximal cartesian domain of valuations forthe existence of a stable and (group)-strategy-proof mechanism.Proof.
By Proposition 8 and Theorem 5, for each substitutable valuation profilethere exists a stable and group-strategy-proof mechanism. For the opposite di-rection note that the proof of Theorem 1 in Hatfield et al. (2018) can be slightlymodified such that the constructed unit demand choice functions are induced byquasi-linear valuations.
Remark . In a continuous Kelso-Crawford model with quasi-linear utility forfirms, substitutable valuations do not only form a maximal domain of valuations forthe existence of a stable and strategy-proof mechanism, but also for the existencefor a stable allocation. (Gul and Stacchetti, 1999) See the example in Appendix Aof Schlegel (2018).
For a choice function from the class defined by Hassidim et al. (2017), the virtualchoice function can be defined as follows: The choice Ch c ( Y ) of college c from Y ⊆ X c is constructed iteratively. Consider the student s who is ranked topaccording to ≫ c in Y S and choose the contract with the smallest stipend with i in Y that does not make the choice violate the quota. Remove, all contracts with i from Y and iterate.Observe that, in the language of Hassidim et al. (2017), the virtual choice func-tion prioritizes merit over need in college admission. This highlights the fundamen-tal trade-off in Hassidim et al. (2017): if stability and (group)-strategy-proofnessare required the mechanism behaves necessarily, by our equivalence result, like amechanisms that assigns seats based on merit only (as encoded in the master list)and keeps stipends as low as possible independently of the applicant.21 .4.3 Cadet-to-branch matching For the cadet-to-branch matching problem, Jagadeesan (2016) has constructedDA-equivalent choice profile for the BfYC choice profiles of S¨onmez (2013).
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