Lexicographic Choice Under Variable Capacity Constraints
aa r X i v : . [ ec on . T H ] O c t Lexicographic Choice Under Variable CapacityConstraints ∗ Battal Do˘gan † Serhat Do˘gan ‡ Kemal Yıldız § October 30, 2019
Abstract
In several matching markets, in order to achieve diversity, agents’ priorities areallowed to vary across an institution’s available seats, and the institution is let tochoose agents in a lexicographic fashion based on a predetermined ordering of theseats, called a (capacity-constrained) lexicographic choice rule . We provide a charac-terization of lexicographic choice rules and a characterization of deferred acceptancemechanisms that operate based on a lexicographic choice structure under variable ca-pacity constraints. We discuss some implications for the Boston school choice systemand show that our analysis can be helpful in applications to select among plausiblechoice rules.
JEL
Classification Numbers: C78, D47, D78.Keywords: Choice rules, lexicographic choice, deferred acceptance, diversity. ∗ Battal Do˘gan gratefully acknowledges financial support from the British Academy/Leverhulme Trustthrough grant SR1819 \ † Department of Economics, University of Bristol; [email protected]. ‡ Department of Economics, Bilkent University; [email protected]. § Department of Economics, Bilkent University; [email protected]. Introduction
Many real-life resource allocation problems involve the allocation of an object that isavailable in a limited number of identical copies, called the capacity of the object. Choicerules, which are systematic ways of rationing available copies of an object when demandexceeds the capacity, are essential in the analysis of such problems. A well-known exampleis the school choice problem in which each school has a certain number of seats to beallocated among students. Although student preferences are elicited from the students,endowing each school with a choice rule is an essential part of the design process.Which choice rule to use is not always evident. The school choice literature, start-ing with the seminal study by Abdulkadiro˘glu and S¨onmez (2003), has widely focused onproblems where each school is already endowed with a priority ordering over studentsand chooses the highest priority students up to the capacity. Such a choice rule, which ismerely responsive to a given priority ordering, is called a responsive choice rule . However,when there are additional concerns such as achieving a diverse student body or affirmativeaction, which choice rule to use is non-trivial. For example in the Boston school choicesystem, although each school is still endowed with a priority ordering over students andrespecting student priorities is still a concern, schools would like to promote the neighbor-hood students as well by sometimes letting them override the priorities of students whoare not from the neighborhood. Such an objective can obviously not be achieved with aresponsive choice rule.The affirmative action policies that are in use in several school districts reveal thata natural way to achieve diversity is to allow students’ priorities to vary across a school’sseats, and to let the school choose students in a lexicographic fashion based on a pre-determined ordering of the seats. We call these rules (capacity-constrained) lexicographicchoice rules . To be more precise, a lexicographic choice rule specifies an ordering of the In Appendix D, we discuss responsive choice rules. In order to achieve a diverse student body, many school districts have been implementing affirmativeaction policies, such as in Boston, Chicago, and Jefferson County. These choice rules are simply called lexicographic choice rules in the recent market design literature.We introduce these choice rules using the capacity-constrained lexicographic choice terminology to differ-entiate them from other lexicographic choice rules without capacity constraints which have been studiedin the choice theory literature. Although we omit the “capacity-constrained” part for simplicity in mostpart of the paper, we include it in the statements of our results. In this study, we follow the axiomaticapproach and discover general principles (axioms) that characterize lexicographic choicerules under variable capacity constraints . In our baseline model, we consider a single decision maker who has a capacity con-straint, such as a school with a limited number of seats. The decision maker encounterschoice problems which consist of a choice set (a set of alternatives, such as students whodemand a seat at the school) and a capacity. A choice rule, at each possible choice prob-lem, chooses some alternatives from the choice set without exceeding the capacity. Notethat across different choice problems, we allow capacity to vary, since in applications ca-pacity may vary and the choice rule may need to be responsive to changes in capacity. One example is when the number of available seats at a school may change from year toyear. In fact, even during the same admissions year, a school may face two different choiceproblems with different capacities. In most of the existing school choice systems, such asNew York City and Boston, there is a second stage of admissions including those studentsand school seats that are unassigned at the end of the first stage. We consider the following three properties of choice rules that have already beenstudied in the axiomatic literature.
Capacity-filling: An alternative is rejected from a choice set at a capacity only if the Although lexicographic choice rules are used to achieve diversity in school choice, there are otherplausible choice rules that are also used, or can be used, to achieve diversity or affirmative action. Amongothers, Echenique and Yenmez (2015) and Ehlers et al. (2014) study some of those choice rules. Echenique and Yenmez (2015) also follow an axiomatic approach and characterize several choice rulesfor a school that wants to achieve diversity. There are earlier studies in the literature which also formulate choice rules by allowing capacity tovary. See, among others, Do˘gan and Klaus (2018), Ehlers and Klaus (2014), and Ehlers and Klaus (2016). The new school choice system in Chicago also has two stages of admissions. See Do˘gan and Yenmez(2019) for an analysis of the new system in Chicago. In the matching literature, capacity-filling is also referred to as acceptance , although the capacity-filling
Gross substitutes:
If an alternative is chosen from a choice set at a capacity, then itis also chosen from any subset of the choice set that contains the alternative, at the samecapacity.
Monotonicity:
If an alternative is chosen from a choice set at a capacity, then it isalso chosen from the same choice set at any higher capacity.We introduce a new property called the irrelevance of accepted alternatives . Theirrelevance of accepted alternatives requires that, if the set of rejected alternatives is thesame for two choice sets at the same capacity, then at any higher capacity, the set ofaccepted alternatives that were formerly rejected should be the same for the two choicesets. In other words, in case of an increase in the capacity, the irrelevance of acceptedalternatives requires that the new alternatives that will be chosen (if any) should notdepend on the already accepted alternatives. In Theorem 1, we show that a choice rulesatisfies capacity-filling , gross substitutes , monotonicity , and the irrelevance of acceptedalternatives if and only if it is lexicographic : there exists a list of priority orderings overpotential alternatives such that at each choice problem, the set of chosen alternatives isobtainable by choosing, first, the highest ranked alternative according to the first priorityordering, then choosing the highest ranked alternative among the remaining alternativesaccording to the second priority ordering, and proceeding similarly until the capacity isfull or no alternative is left.Besides providing a first axiomatic foundation for lexicographic choice rules undervariable capacity constraints, we also analyze the market design implications of lexico-graphic choice rules. In Section 4, we consider the variable-capacity object allocationmodel where there is more than one object (such as many schools) and agents have pref-erences over objects (such as students having preferences over schools). In that model,Ehlers and Klaus (2016) characterize deferred acceptance mechanisms where each objecthas a choice rule that satisfies capacity-filling , gross substitutes , and monotonicity . Mo-tivated by the irrelevance of accepted alternatives for choice rules, we introduce a new terminology has been increasingly popular in the recent literature. Kojima and Manea (2010) consider a setup where the capacity of each school is fixed, and characterizedeferred acceptance mechanisms where each school has a choice rule that satisfies capacity-filling and grosssubstitutes . the irrelevance of satisfied demand . Consideran arbitrary problem and the allocation chosen by the mechanism at that problem. Sup-pose that the capacity of an object is increased. Now, some of the agents who prefer thatobject to their assignments at the initial allocation may receive the object due to the capac-ity increase. The irrelevance of satisfied demand requires that the set of agents who receivethe object due to the capacity increase does not depend on the set of agents who initiallyreceive the object. We show that there is no mechanism which satisfies the irrelevance ofsatisfied demand together with some other desirable properties studied in Ehlers and Klaus(2016) (Proposition 3). In particular, lexicographic deferred acceptance mechanisms, whichare deferred acceptance mechanisms that operate based on a lexicographic choice struc-ture, violate the irrelevance of satisfied demand , which stands in contrast to lexicographicchoice rules satisfying the irrelevance of accepted alternatives . However, we show thata weaker version of the irrelevance of satisfied demand –which requires the same at anyproblem where there is only one available object–characterizes lexicographic deferred ac-ceptance mechanisms together with the desirable properties studied in Ehlers and Klaus(2016) (Proposition 4).Kominers and S¨onmez (2016) study lexicographic deferred acceptance mechanisms ina more general matching with contracts framework (Hatfield and Milgrom, 2005). In someapplications, the choice rule of an institution is subject to a feasibility constraint , in thesense that some alternatives cannot be chosen together with some other alternatives. Thematching with contracts model due to Hatfield and Milgrom (2005) introduced a generalframework that incorporates such feasibility constraints into the matching problem. Al-though for the school choice application, where such feasibility constraints are not binding,the lexicographic choice rules in Kominers and S¨onmez (2016) fall into our baseline model,in case of binding feasibility constraints, their lexicographic choice rules are not covered inour baseline analysis. In Section 3.1, we show that our baseline model and our baselineproperties can be extended to a setup with feasibility constraints, highlighting the distin-guishing properties of capacity-constrained lexicographic choice rules, including the onesdiscussed in Kominers and S¨onmez (2016), in a more general setup.Boston school district is one of the school districts that uses capacity-constrained For instance, the lexicographic choice rules in their setup may violate “substitutability”, which is ageneralization of gross substitutes to the matching with contracts setup (Hatfield and Milgrom, 2005). Dur et al. (2013) and Dur et al. (2018) analyse how the order of the priority orderings inthe choice rule of a school may cause additional bias for or against the neighbourhoodstudents. In Section 5, we consider a class of capacity-wise lexicographic choice rulesdiscussed in Dur et al. (2013) that are relevant for the design of the Boston school choicesystem and show that our analysis enables us to single out one rule from four plausiblecandidates.The paper is organized as follows. In Section 2, we review the related literature. InSection 3, we introduce and characterize lexicographic choice rules, show that our baselinemodel and our baseline properties can be extended to a setup with feasibility constraints,and also provide a characterization of responsive choice rules. In Section 4, we highlightan implication of our choice theoretical analysis for the resource allocation framework: weprovide a characterization of deferred acceptance mechanisms that operate based on alexicographic choice structure. In Section 5, we discuss some implications for the Bostonschool choice system. In Section 6, we conclude by discussing the main features of ouranalysis.
Several studies investigate choice rules that satisfy path independence (Plott, 1973), whichrequires that if the choice set is “split up” into smaller sets, and if the choices from thesmaller sets are collected and a choice is made from the collection, the final result shouldbe the same as the choice from the original choice set. Since capacity-filling together See Dur et al. (2018) for a detailed discussion of Boston’s school choice mechanism. Dur et al. (2013) is an earlier version of Dur et al. (2018). gross substitutes imply path independence , lexicographic choice rules are examplesof path independent choice rules. Aizerman and Malishevski (1981) show that for eachpath independent choice rule, there exists a list of priority orderings such that the choicefrom each choice set is the union of the highest priority alternatives in the priority order-ings. Among others, Plott (1973), Moulin (1985), and Johnson and Dean (2001) studythe structure of path independent choice rules. Path independent choice rules guaranteethe existence of stable matchings in the matching context. Chambers and Yenmez (2017)study path independence in the matching context and its connection to stable matchings.Although the structure of path independent choice rules have been extensively studied,the structure of lexicographic choice rules and what properties distinguish them from otherpath independent choice rules have not been well-understood. Houy and Tadenuma (2009)consider two classes of choice rules which are both based on “lexicographic procedures”,yet different than the ones we consider here. Similar to our setup, choice rules that theyconsider operate based on a list of binary relations. Yet, their model does not includecapacity constraints and the lexicographic procedures that operationalize the lists aredifferent. The only study that considers lexicographic choice rules that we study from anaxiomatic perspective is Chambers and Yenmez (2018a). They show that lexicographicchoice rules satisfy capacity-filling and path independence , and they also show that thereare path independent choice rules that are not lexicographic, but they do not provide acharacterization of lexicographic choice rules.Our analysis of the Boston school choice system is related to Dur et al. (2018) andthe working paper version Dur et al. (2013). Dur et al. (2013) compare alternative choicerules for schools in the Boston school district (one of which is the one used in the Bostonschool district) in terms of how much they are biased for or against the neighbourhoodstudents. We consider these alternative choice rules from a different perspective. InSection 5, we show that, although these choice rules are all based on a “lexicographicprocedure” at each capacity, only one of them satisfies all the characterizing properties in This is also noted in Remark 1 of Do˘gan and Klaus (2018), and it follows from Lemma 1 ofEhlers and Klaus (2016) together with Corollary 2 of Aizerman and Malishevski (1981). In the words of Aizerman and Malishevski (1981), each path independent choice rule is generated bysome mechanism of collected extremal choice. Houy and Tadenuma (2009) do not start with any assumptions on the list of binary relations. Theyseparately discuss under which assumptions on the list of binary relations, the resulting choice rules satisfycertain properties. (capacity-constrained) lexicographicchoice rule . The common feature of Dur et al. (2018) and our Section 5 is that we bothconsider lexicographic choice procedures in the context of school choice in Boston. Themain difference is that, although the choice rules that Dur et al. (2018) consider havedirect counterparts in a variable capacity context, their analysis pertains to the fixedcapacity case. In particular, given a fixed school capacity, Dur et al. (2018) analyze howdifferent lexicographic choice procedures perform. On the other hand, variable capacities,and properties related to variable capacities, are at the heart of our study. We show that,one of our variable capacity properties,
CWARP , is satisfied by only one of the four choicerules discussed in Dur et al. (2013).
Let A be a nonempty finite set of n alternatives and let A denote the set of all nonempty subsets of A . A (capacity-constrained) choice problem is a pair ( S, q ) ∈ A × { , . . . , n } ofa choice set S and a capacity q . A (capacity-constrained) choice rule C : A×{ , . . . , n } →A associates with each problem ( S, q ) ∈ A × { , . . . , n } , a set of choices C ( S, q ) ⊆ S suchthat | C ( S, q ) | ≤ q . Given a choice rule C , we denote the set of rejected alternatives at aproblem ( S, q ) by R ( S, q ) = S \ C ( S, q ).A priority ordering ≻ is a complete, transitive, and anti-symmetric binary relationover A . A priority profile π = ( ≻ , . . . , ≻ n ) is an ordered list of n priority orderings.Let Π denote the set of all priority profiles.A choice rule C is (capacity-constrained) lexicographic for a priority profile ( ≻ , . . . , ≻ n ) ∈ Π if for each (
S, q ) ∈ A × { , . . . , n } , C ( S, q ) is obtained by choosingthe highest ≻ -priority alternative in S , then choosing the highest ≻ -priority alternativeamong the remaining alternatives, and so on until q alternatives are chosen or no alter-native is left. A choice rule is (capacity-constrained) lexicographic if there exists apriority profile for which the choice rule is lexicographic. Remark . Note that, if a choice rule is lexicographic for a priority profile π = ( ≻ , . . . , ≻ n ),then it is lexicographic for any other priority profile that is obtained from π by replacing ≻ n with an arbitrary priority ordering. In that sense, the last priority ordering is redundant.8e consider four properties of choice rules. The following three properties are alreadyknown in the literature. Capacity-filling:
An alternative is rejected from a choice set at a capacity only if thecapacity is full. Formally, for each (
S, q ) ∈ A × { , . . . , n } , | C ( S, q ) | = min {| S | , q } . Gross substitutes: If an alternative is chosen from a choice set at a capacity, then itis also chosen from any subset of the choice set that contains the alternative, at the samecapacity. Formally, for each (
S, q ) ∈ A × { , . . . , n } and each pair a, b ∈ S such that a = b ,if a ∈ C ( S, q ) , then a ∈ C ( S \{ b } , q ) . Monotonicity:
If an alternative is chosen from a choice set at a capacity, then it isalso chosen from the same choice set at any higher capacity. Formally, for each (
S, q ) ∈A × { , . . . , n − } , C ( S, q ) ⊆ C ( S, q + 1) . We now introduce a new property called the irrelevance of accepted alternatives . Con-sider a problem and the set of rejected alternatives for that problem. Suppose that thecapacity increases. The property requires that which alternatives among the currentlyrejected alternatives will be chosen (if any) should not depend on the currently acceptedalternatives. In other words, if the set of rejected alternatives are the same for two choicesets (note that the set of accepted alternatives may be different), then at any higher ca-pacity, the set of initially rejected alternatives that become accepted should be the samefor the two choice sets.
Irrelevance of accepted alternatives:
For each
S, S ′ ∈ A and each q ∈ { , . . . , n − } ,if R ( S, q ) = R ( S ′ , q ) , then C ( S, q + 1) ∩ R ( S, q ) = C ( S ′ , q + 1) ∩ R ( S ′ , q ) . Gross substitutes was first introduced in the choice literature by Chernoff (1954). It has been studiedin the choice literature under different names such as
Chernoff ’s axiom , Sen’s α , or contraction consistency .In the matching literature, it was first studied and referred to as gross substitutes in Kelso and Crawford(1982) ( substitutability is also a commonly used name in the matching literature). We follow the termi-nology of Kelso and Crawford (1982).
9e also introduce another property called capacity-wise weak axiom of revealed pref-erence which will be helpful in our analysis. Consider the following capacity-wise revealedpreference relation. An alternative a ∈ A is revealed to be preferred to an alternative b ∈ A at a capacity q > q − a and b areboth rejected and a is chosen over b when the capacity is q . That is, a is revealed to bepreferred to b at q if there exists S ∈ A such that a, b / ∈ C ( S, q − a ∈ C ( S, q ), and b ∈ R ( S, q ). Capacity-wise weak axiom of revealed preference requires, for each capacity,the revealed preference relation to be asymmetric.
Capacity-wise weak axiom of revealed preference (CWARP):
For each capacity q > a, b ∈ A , if a is revealed to be preferred to b at q , then b is not revealedto be preferred to a at q . CWARP is a counterpart of the well-known weak axiom of revealed preference (WARP)in the standard revealed preference framework (Samuelson, 1938), where there is no capac-ity parameter. In the standard framework, an alternative is said to be revealed preferredto another alternative if there is a choice set at which the former alternative is chosen overthe latter.
WARP requires the revealed preference relation to be asymmetric, which in asense requires consistency of the choice behavior in responding to changes in the choice set.In our framework, the preference is revealed not only through the choice at a choice set,but also through a change in the capacity. Therefore, what should be the counterpart ofthe “revealed preference relation” is not entirely clear. We propose the following definition.An alternative is revealed to be preferred to another at a capacity if there is a choice setin which the former alternative is chosen over the latter at that capacity, although if thecapacity were one less, none of the alternatives would have been chosen. Put differently, ifnone of the two alternatives are chosen in a choice set at a given capacity, but one of themis chosen when capacity increases by one, this means the chosen alternative is revealedto be preferred to the unchosen one.
CWARP requires the revealed preference relation tobe asymmetric. Hence,
CWARP requires consistency of the choice behavior in respondingto changes in the choice set together with changes in the capacity. Additionally, one caninterpret
CWARP as a “no complementarities” condition, in the sense that
CWARP re-quires the new alternative to be chosen due to the capacity increase be independent of thealternatives that have already been chosen. For example, if two alternatives are comple-ments, then the choice of each one of these alternatives may depend on whether the other10ne has already been chosen or not.
CWARP rules out this type of choice behavior.
Remark . The following is an alternative definition of
CWARP , which is formulated inline with the common formulations of WARP-type revealed preference relations in theliterature.
An alternative definition of CWARP:
For each capacity q >
1, each pair
S, T ∈ A andeach pair a, b ∈ S ∩ T such that [ C ( S, q − ∪ C ( T, q − ∩ { a, b } = ∅ ,if a ∈ C ( S, q ) and b ∈ C ( T, q ) \ C ( S, q ) , then a ∈ C ( T, q ) . Lemma 1.
If a choice rule satisfies capacity-filling, monotonicity, and CWARP, then italso satisfies the irrelevance of accepted alternatives.Proof.
Let C be a choice rule. Suppose that C satisfies capacity-filling and monotonicity ,but violates the irrelevance of accepted alternatives . By violation of the irrelevance ofaccepted alternatives , there are S, S ′ ∈ A and q ∈ { , . . . , n − } such that R ( S, q ) = R ( S ′ , q ), but C ( S, q + 1) ∩ R ( S, q ) = C ( S ′ , q + 1) ∩ R ( S ′ , q ) . By monotonicity , R ( S, q + 1) ⊆ R ( S, q ) and R ( S ′ , q + 1) ⊆ R ( S ′ , q ). By capacity-filling , | R ( S, q + 1) | = | R ( S ′ , q + 1) | . Then, there exist a, b ∈ R ( S, q ) = R ( S ′ , q ) such that a ∈ C ( S, q + 1), b / ∈ C ( S, q + 1), b ∈ C ( S ′ , q + 1), and a / ∈ C ( S ′ , q + 1). But then, a is revealed preferred to b and vice versa,implying that C violates CWARP .In Appendix A, we show that each of the three properties capacity-filling , mono-tonicity , and CWARP is necessary for the implication in Lemma 1, that is, we provideexamples of choice rules which violate exactly one of the three properties and also violate the irrelevance of accepted alternatives .The following example shows that there exists a choice rule that satisfies capacity-filling , monotonicity , and the irrelevance of accepted alternatives , but violates CWARP . Example 1.
Let A = { a, b, c, d, e } . Let ≻ and ≻ ′ be defined as a ≻ b ≻ c ≻ d ≻ e and a ≻ ′ c ≻ ′ b ≻ ′ d ≻ ′ e . Let the choice rule C be defined as follows. For each problem ( S, q ) , if d ∈ S , then C ( S, q ) chooses the highest ≻ -priority alternatives from S until q alternatives are chosen or no alternative is left; if d / ∈ S , then C ( S, q ) chooses the That is, C ( S, q ) coincides with the choice rule that is “responsive” for ≻ . We discuss responsive choicerules in Section D. ighest ≻ ′ -priority alternatives from S until q alternatives are chosen or no alternative isleft. Note that C clearly satisfies capacity-filling and monotonicity. To see that C alsosatisfies the irrelevance of accepted alternatives, let S, S ′ ∈ A and q ∈ { , . . . , n − } besuch that R ( S, q ) = R ( S ′ , q ) . If d ∈ S ∩ S ′ or d ∈ A \ ( S ∪ S ′ ) , then C ( S, q + 1) ∩ R ( S, q ) = C ( S ′ , q + 1) ∩ R ( S ′ , q ) . So suppose, without loss of generality, that d ∈ S \ S ′ . Since R ( S, q ) = R ( S ′ , q ) , we have d ∈ C ( S, q ) . But then, either R ( S, q ) = ∅ or R ( S, q ) = { e } .In either case, we have C ( S, q +1) ∩ R ( S, q ) = C ( S ′ , q +1) ∩ R ( S ′ , q ) . To see that C violatesCWARP, note that C ( { a, b, c, d } ,
1) = { a } and C ( { a, b, c, d } ,
2) = { a, b } , implying that b isrevealed preferred to c at q = 2 . Also, C ( { a, b, c, e } ,
1) = { a } and C ( { a, b, c, e } ,
2) = { a, c } ,implying that c is revealed preferred to b at q = 2 . Theorem 1.
A choice rule is (capacity-constrained) lexicographic if and only if it satisfiescapacity-filling, gross substitutes, monotonicity, and the irrelevance of accepted alterna-tives. Proof.
Let C be lexicographic for ( ≻ , . . . , ≻ n ) ∈ Π. Clearly, C satisfies capacity-filling and monotonicity , and it is already known from the literature that C satisfies gross substitutes (Chambers and Yenmez, 2018a). To see that it satisfies CWARP , let a, b ∈ A and q ∈{ , . . . , n } be such that a is revealed preferred to b at q . Then, there is S ∈ A such that a, b ∈ R ( S, q − a ∈ C ( S, q ), and b ∈ R ( S, q ). But then, a ≻ q b . If also b is revealedpreferred to a at q , then by similar arguments we have b ≻ q a , contradicting that ≻ q is antisymmetric. Thus, the revealed preference relation is asymmetric and C satisfies CWARP . By Lemma 1, C also satisfies the irrelevance of accepted alternatives .Let C be a choice rule satisfying capacity-filling , gross substitutes , monotonicity , and the irrelevance of accepted alternatives . We first construct a priority profile ( ≻ , . . . , ≻ n ) ∈ Π and then show that C is lexicographic for that priority profile. For each i, j ∈ { , . . . , n } ,let a ij denote the j ’th ranked alternative in ≻ i (for instance, a i is the highest ≻ i -priorityalternative).To construct ≻ , first set { a } = C ( A, j ∈ { , . . . , n } , set { a j } = C ( A \{ a , . . . , a j − } , ≻ , consider C ( A, capacity-filling , | C ( A, | = 2.Since a ∈ C ( A, monotonicity , a ∈ C ( A, { a } = C ( A, \ { a } . Foreach j ∈ { , . . . , n − } , set { a j } = C ( A \ { a , a , . . . , a j − } , \ { a } . Set a n = a . Independence of the characterizing properties is shown in Appendix B. i ∈{ , . . . , n } , first set { a i } = C ( A, i ) \ { a , a , . . . , a ( i − } (Note that by monotonic-ity , { a , a , . . . , a ( i − } ⊆ C ( A, i ) and by capacity-filling , | C ( A, i ) | = i ). For each j ∈ { , . . . , n − i +1 } , set { a ij } = C ( A \{ a i , a i , . . . , a i ( j − } , i ) \{ a , a , . . . , a ( i − } . Notethat there are i − ≻ i , which are { a i ( n − i +2) , . . . , a in } . For each j ∈ { n − i + 2 , . . . , n } , set a ij = a ( j + i − n − (which assigns the alternatives a , . . . , a ( i − to the rankings a i ( n − i +2) , . . . , a in , respectively).Now, let ( S, q ) ∈ A × { , . . . , n } . Let b denote the highest ≻ -priority alternative in S , b denote the highest ≻ -priority alternative among the remaining alternatives, and soon up to b min {| S | ,q } . We show that C ( S, q ) = { b , . . . , b min {| S | ,q } } . If min {| S | , q } = | S | , thenby capacity-filling , C ( S, q ) = { b , . . . , b | S | } . Suppose that | S | > q .The rest of the proof is by induction: we first show that b ∈ C ( S, q ); then, foran arbitrary i ∈ { , . . . , q } , assuming that b , . . . , b i − ∈ C ( S, q ), we show that b i ∈ C ( S, q ). Let b = a j for some j ∈ { , . . . , n } . By the construction of ≻ , b ∈ C ( A \{ a , . . . , a j − } , gross substitutes and monotonicity , b ∈ C ( S, q ).Let i ∈ { , . . . , q } . Assuming that b , . . . , b i − ∈ C ( S, q ), we show that b i ∈ C ( S, q ).Let S ′ be the choice set obtained from S by replacing b with a (note that nothingchanges if b = a ), replacing b with a , . . . , and replacing b i − with a ( i − . That is, S ′ =( S \ { b , . . . , b i − } ) ∪ { a , . . . , a ( i − } . Let q ′ = i −
1. Note that { b , . . . , b i − } = C ( S, q ′ ),because otherwise, by capacity-filling , there is a ∈ S such that a ∈ C ( S, q ′ ) and a / ∈ C ( S, q ),which is a violation of monotonicity . Also, by the construction of the priority profile andby gross substitutes , { a , . . . , a ( i − } = C ( S ′ , q ′ ). Note that R ( S, q ′ ) = R ( S ′ , q ′ ). By monotonicity and the irrelevance of accepted alternatives , we have R ( S, q ) = R ( S ′ , q ).Since b i ∈ C ( S ′ , q ) by the construction of the priority profile and by gross substitutes , wealso have b i ∈ C ( S, q ). Corollary 1.
A choice rule is (capacity-constrained) lexicographic if and only if it satisfiescapacity-filling, gross substitutes, monotonicity, and CWARP.Proof.
A lexicographic choice rule satisfies capacity-filling , gross substitutes , and mono-tonicity by Theorem 1. Also note that in the proof Theorem 1, we showed that a lexi-cographic choice rule satisfies CWARP as well. To see the other direction, note that byLemma 1, capacity-filling , monotonicity , and CWARP imply the irrelevance of accepted lternatives and the rest follows by Theorem 1.There is never a unique priority profile for which a given choice rule is lexicographic.However, if C is lexicographic for two different priority profiles ( ≻ , . . . , ≻ n ) and ( ≻ ′ , . . . , ≻ ′ n ), then for each pair of alternatives a, b ∈ A , if a ≻ t b and b ≻ ′ t a for some t ∈ { , . . . , n } , then a or b must be chosen from any choice set (particularly from A ) atany capacity q < t . That is, a or b is chosen irrespective of their relative ranking at the t ’th priority ordering.To state this observation formally, for each priority ordering ≻ i on A and for eachchoice set S ∈ A , let ≻ i | S stand for the restriction of ≻ i to S . Let A = A , and for each t ∈ { , . . . , n } , let A t = A \ C ( A, t − S ∈ A and each priorityordering ≻ i , let max ( S, ≻ i ) be the top-ranked alternative in S according to ≻ i . Proposition 1.
If a choice rule C is (capacity-constrained) lexicographic for a priorityprofile ( ≻ , . . . , ≻ n ) , then C is lexicographic for another priority profile ( ≻ ′ , . . . , ≻ ′ n ) ifand only if ≻ = ≻ ′ and for each t ∈ { , . . . , n } , ≻ t | A t = ≻ ′ t | A t .Proof. (If part) Let choice rule C be lexicographic for a priority profile ( ≻ , . . . , ≻ n ).Suppose ( ≻ ′ , . . . , ≻ ′ n ) is such that ≻ = ≻ ′ and for each t ∈ { , . . . , n } , ≻ t | A t = ≻ ′ t | A t .Now, for each S ∈ A and t ∈ { , . . . , n } , if t = 1, then since ≻ = ≻ ′ , the conclusion isimmediate. Then, by proceeding inductively, for each 1 < t ≤ | S | , since C is lexicographicfor ( ≻ , . . . , ≻ n ), max ( S \ C ( S, t − , ≻ t ) = C ( S, t ) \ C ( S, t − S \ C ( S, t − ⊂ A t and ≻ t | A t = ≻ ′ t | A t , we get max ( S \ C ( S, t − , ≻ ′ t ) = C ( S, t ) \ C ( S, t − C is lexicographic for ( ≻ ′ , . . . , ≻ ′ n ).( Only if part ) For each t ∈ { , . . . , n } , let A t stand for the collection of all nonemptysubsets of A t with at least t elements. Then, define the choice function c t : A t → A t such that for each choice set S ∈ A t , c t ( S ) = C ( S, t ) \ C ( S, t − C satisfies gross substitutes , c t also satisfies gross substitutes . It follows that there is a unique priorityordering ≻ ∗ t such that c t ( S ) = max { S \ C ( S, t − , ≻ ∗ t } . Therefore, if C is lexicographicfor some ( ≻ , . . . , ≻ n ), then for each t ∈ { , . . . , n } , ≻ t | A t = ≻ ∗ t .14 .1 Lexicographic Choice Under Feasibility Constraints In some applications, the choice rule of an institution is subject to a feasibility constraint.For example, a firm may encounter a choice set which includes signing the same workerunder different terms, such as different salaries as modeled in Kelso and Crawford (1982),and it may not be possible to choose the same worker under several terms even whenthere is enough capacity (for instance, it is not possible to choose the same worker un-der different salaries). The matching with contracts model due to Hatfield and Milgrom(2005) introduced a general framework that incorporates such feasibility constraints intothe matching problem, which led to several new applications of matching theory such ascadet-branch matching by S¨onmez and Switzer (2013) and S¨onmez (2013), and matchingwith slot-specific priorities by Kominers and S¨onmez (2016). In this section, we will showthat our baseline model and our baseline properties can be extended to a setup with fea-sibility constraints, highlighting the distinguishing properties of lexicographic choice rulesin a more general setup. As in the baseline model, let A be a nonempty finite set of n alternatives and let A denote the set of all nonempty subsets of A . In addition, let F ⊆ A be a nonempty set of feasible sets. We assume that F is downward closed in the sensethat for each S ∈ F and each S ′ ⊆ S , S ′ ∈ F . We also assume that each singleton isfeasible, i.e. for each a ∈ A , { a } ∈ F . A (feasibility-constrained) choice rule C : A × { , . . . , n } → F associates with eachproblem ( S, q ) ∈ A×{ , . . . , n } , a nonempty set of choices C ( S, q ) ⊆ S which is feasible, i.e. C ( S, q ) ∈ F , and respects the capacity constraint, i.e. | C ( S, q ) | ≤ q . Given a choice rule C , we denote the set of rejected alternatives at a problem ( S, q ) by R ( S, q ) = S \ C ( S, q ).Our new framework encompasses the matching with contracts framework in the fol-lowing way. Suppose that each alternative is a contract consisting of a pair: an agent anda contractual term. Suppose that a choice set is feasible if it includes, for each agent, atmost one contract including that agent. It is easy to see that F is downward closed andit includes the singletons.A feasibility-constrained choice rule C is (capacity-constrained) lexicographic if In a matching with contracts model with distributional constraints, Goto et al. (2017) introduce theconcept of a “hereditary” distributional constraint, which implies that F is downward closed. Note that, given downward closedness, this is equivalent to requiring that each alternative belongs toat least one feasible set. ≻ , . . . , ≻ n ) ∈ Π such that for each (
S, q ) ∈ A × { , . . . , n } , C ( S, q ) is obtained by choosing the highest ≻ -priority alternative in S , then choosing thehighest ≻ -priority alternative among the remaining alternatives that induces a feasible settogether with the previously chosen alternative, and so on as long as there is a remainingalternative until finally choosing the highest ≻ q -priority alternative among the remainingalternatives that induces a feasible set together with the previously chosen alternatives. F -capacity-filling: An alternative is rejected from a choice set at a capacity only if thecapacity is full or it is infeasible to choose the alternative. Formally, for each (
S, q ) ∈A × { , . . . , n } and a ∈ S , if a / ∈ C ( S, q ), then either | C ( S, q ) | = q or C ( S, q ) ∪ { a } / ∈ F .Let us adopt the convention that for each S ∈ A , C ( S,
0) = ∅ . Now, for each capacity q ∈ { , . . . , n } , a is revealed to be F -preferred to b at q , denoted by a R F q b , if thereexists S ∈ A such that a, b / ∈ C ( S, q − a ∈ C ( S, q ) but b / ∈ C ( S, q ), although C ( S, q − ∪ { b } ∈ F . We introduce the following property which requires, for eachcapacity, the revealed preference relation be acyclic. Capacity-wise strong axiom of revealed preference (CSARP):
For each capacity q ∈ { , . . . , n } , R F q is acyclic. Proposition 2.
A feasibility-constrained choice rule is (capacity-constrained) lexicographicif and only if it satisfies F -capacity-filling, monotonicity, and the capacity-wise strong ax-iom of revealed preference.Proof. ( Only if part: ) Let C be a feasibility-constrained choice rule that is lexicographicfor ( ≻ , . . . , ≻ n ). Using similar arguments as in the proof of Theorem 1, one can easilyverify that C satisfies F -capacity-filling and monotonicity.To see that C satisfies CSARP, note that for each capacity q ∈ { , . . . , n } and a, b ∈ A ,if a R F q b , then we must have a ≻ q b . Since ≻ q is transitive, R F q is acyclic.( If part: ) Let C be a feasibility-constrained choice rule that satisfies F -capacity-filling,monotonicity, and CSARP. It follows from CSARP that for each q ∈ { , . . . , n } , R F q isacyclic. Now, for each capacity q ∈ { , . . . , n } , let ≻ q be any completion of the transitive Formally, let a be the highest ≻ -priority alternative in S . Let S ′ = { b ∈ S \ { a } : { a, b } ∈ F} . Then,the highest ≻ -priority alternative among the remaining alternatives that induces a feasible set togetherwith the previously chosen alternative is max { S ′ , ≻ } . R F q . Next, we show that C is lexicographic for ( ≻ , . . . , ≻ n ). To see this, weapply induction on capacity q . Before proceeding, let us introduce some notation. Foreach S, T ∈ A such that T ⊂ S , let F ( S | T ) be the set of alternatives in S \ T that inducea feasible set together with the alternatives in T , i.e. F ( S | T ) = { a ∈ S \ T : T ∪ { a } ∈ F } .First, we show that for each S ∈ A , C ( S,
1) = max ( S, ≻ ). By contradiction supposethat although a = max ( S, ≻ ), we have C ( S,
1) = b , where a = b . Since C ( S,
1) = b and a ∈ S , it follows that b R F a , which contradicts that a = max ( S, ≻ ). Next, assumethat for some q ∈ { , . . . , n } , we have for each S ∈ A and q ′ < q , C ( S, q ′ ) coincideswith the lexicographic choice for ( ≻ , . . . , ≻ q − ). Now, we show that for each S ∈ A , C ( S, q ) \ C ( S, q −
1) = max ( F ( S | C ( S,q − ) , ≻ q ). First, let a = max ( F ( S | C ( S,q − ) , ≻ q ). Itfollows that a / ∈ C ( S, q −
1) and C ( S, q − ∪ { a } ∈ F . By contradiction, suppose that a / ∈ C ( S, q ). Since C ( S, q − ∪ { a } ∈ F , it follows from F -capacity-filling that thereexists x ∈ C ( S, q ) \ C ( S, q −
1) such that x = a . Now, since C satisfies monotonicity, x / ∈ C ( S, q − x ∈ C ( S, q ), C ( S, q − ∪ { x } ∈ F . Therefore, we have x R F q a , but this contradicts that a = max ( F ( S | C ( S,q − ) , ≻ q ). Thus, we conclude that C is lexicographic for ( ≻ , . . . , ≻ n ). Let N denote a finite set of agents, | N | = n ≥
2. Let A be the collection of all nonemptysubsets of N . Let O denote a finite set of objects. Each agent i ∈ N has a complete,transitive, and anti-symmetric preference relation R i over O ∪ {∅} , where ∅ is the nullobject representing the option of receiving no object (or receiving an outside option).Given x, y ∈ O ∪ {∅} , x R i y means that either x = y or x = y and agent i prefers x to y .If agent i prefers x to y , we write x P i y . Let R denote the set of all preference relationsover O ∪ {∅} , and R N the set of all preference profiles R = ( R i ) i ∈ N such that for all i ∈ N , R i ∈ R .An allocation problem with capacity constraints, or simply a problem , consists ofa preference profile R ∈ R N and a capacity profile q = ( q x ) x ∈ O ∪{∅} such that for eachobject x ∈ O , q x ∈ { , , . . . , n } and q ∅ = n so that the null object has enough capacity toaccommodate all agents. Let P denote the set of all problems. Given a problem ( R, q ) ∈ P ,17n object x is available at the problem if q x > q = ( q x ) x ∈ O ∪{∅} , an allocation assigns to each agent exactlyone object in O ∪ {∅} taking capacity constraints into account. Formally, an allocation at q is a list a = ( a i ) i ∈ N such that for each i ∈ N , a i ∈ O ∪ {∅} and no object x ∈ O ∪ {∅} is assigned to more than q x agents. Let M ( q ) denote the set of all allocations at q .Given an allocation a = ( a i ) i ∈ N , a preference profile R , and an object x ∈ O ∪ {∅} ,let D x ( a, R ) = { i ∈ N : x P i a i } denote the demand for x at ( a, R ), which is the set ofagents who prefer x to their assigned object.A mechanism is a function ϕ : P → S q M ( q ) such that for each allocation problem( R, q ) ∈ P , ϕ ( R, q ) ∈ M ( q ).For mechanisms, we introduce the following property, which we call the irrelevanceof satisfied demand . Consider an arbitrary problem and the allocation chosen by themechanism at that problem. Suppose that the capacity of an object is increased. Now,some of the agents who prefer that object to their assignments at the initial allocationmay receive the object due to the capacity increase. The irrelevance of satisfied demand requires that the set of agents who receive the object due to the capacity increase doesnot depend on the set of agents who initially receive the object. In other words, for twoproblems with the same capacity, if the demands for an object are the same (note that theset of agents who receive the object at those problems may be different), then wheneverthe capacity of the object increases, the sets of agents who receive the object due to thecapacity increase should be the same for the two problems.Formally, for each x ∈ O , let 1 x be the capacity profile which has 1 unit of x andnothing else. A mechanism ϕ satisfies the irrelevance of satisfied demand if foreach pair of problems ( R, q ) and ( R ′ , q ) and each object x ∈ O , if D x ( ϕ ( R, q ) , R ) = D x ( ϕ ( R ′ , q ) , R ′ ), then D x ( ϕ ( R, q + 1 x ) , R ) = D x ( ϕ ( R ′ , q + 1 x ) , R ′ ).A (capacity-constrained) lexicographic choice structure C = ( C x ) x ∈ O asso-ciates each object x ∈ O with a lexicographic choice rule C x : A × { , . . . , n } → A . Next,we present the (capacity-constrained) lexicographic deferred acceptance algo-rithm based on C . For each problem ( R, q ) ∈ P , the algorithm runs as follows: Step 1:
Each agent applies to his favorite object in O . Each object x ∈ O such that q x > C x ( S x , q x ) where S x is the set of agents who applied18o x , and rejects all the other applicants. Each object x ∈ C such that q x = 0 rejects allapplicants. Step r ≥ : Each applicant who was rejected at step r − O . For each object x ∈ O , let S x,r be the set consisting of the agents who appliedto x at step r and the agents who were temporarily accepted by x at Step r −
1. Eachobject x ∈ O such that q x > C x ( S x,r , q x ) and rejects all theother applicants. Each object x ∈ O such that q x = 0 rejects all applicants.The algorithm terminates when each agent is accepted by an object. The allocation whereeach agent is assigned the object that he was accepted by at the end of the algorithm iscalled the C -lexicographic Deferred Acceptance allocation at ( R, q ), denoted by DA C ( R, q ). Lexicographic deferred acceptance mechanisms:
A mechanism ϕ is a lexicographicdeferred acceptance mechanism if there exists a lexicographic choice structure C such thatfor each ( R, q ) ∈ P , ϕ ( R, q ) = DA C ( R, q ).Ehlers and Klaus (2016), in their Theorem 3, characterize deferred acceptance mech-anisms based on a choice structure satisfying capacity-filling , gross substitutes , and mono-tonicity , with the following properties of mechanisms: unavailable-type-invariance (if thepositions of the unavailable types are shuffled at a profile, then the allocation should notchange); weak non-wastefulness (no agent receives the null object while he prefers an objectthat is not exhausted to the null object), resource-monotonicity (increasing the capaci-ties of some objects does not hurt any agent), truncation-invariance (if an agent truncateshis preference relation in such a way that his allotment remains acceptable under the trun-cated preference relation, then the allocation should not change), and strategy-proofness(no agent can benefit by misreporting his preferences). Next, we formally introduce theseproperties and state Theorem 3 of Ehlers and Klaus (2016). Unavailable-Type-Invariance:
Let (
R, q ) ∈ P and R ′ ∈ R N . If for each i ∈ N andeach pair of available objects x, y ∈ O ( q x > q y >
0) we have [ x R i y if and only if x R ′ i y ], then ϕ ( R, q ) = ϕ ( R ′ , q ). Weak Non-Wastefulness:
For each (
R, q ) ∈ P , each x ∈ O such that q x >
0, and each The stronger version of this property, namely non-wastefulness , requires that no agent prefers anobject that is not exhausted to his assigned object. Note that capacity-filling and non-wastefulness aresimilar in spirit, yet, capacity-filling is a property of a choice rule while non-wastefulness is a property ofa mechanism. ∈ N , if x P i ϕ i ( R, q ) and ϕ i ( R, q ) = ∅ , then |{ j ∈ N : ϕ j ( R, q ) = x }| = q x . Resource-Monotonicity:
For each R ∈ R N , and each pair of capacity profiles ( q, q ′ ), iffor each x ∈ O , q x ≤ q ′ x , then for each i ∈ N , ϕ i ( R, q ′ ) R i ϕ i ( R, q ). Truncation-Invariance:
Let (
R, q ) ∈ P and R ′ ∈ R N . If for each i ∈ N and eachpair of objects x, y ∈ O we have [ x R i y if and only if x R ′ i y ] and ϕ i ( R, q ) R ′ i ∅ , then ϕ ( R, q ) = ϕ ( R ′ , q ). Strategy-proofness:
For each (
R, q ) ∈ P , each i ∈ N , and each R ′ i ∈ R , ϕ i ( R, q ) R i ϕ i (( R ′ i , R − i ) , q ). Theorem 3 of Ehlers and Klaus(2016):
A mechanism is a deferred acceptance mech-anism based on a choice structure satisfying capacity-filling, gross substitutes, and mono-tonicity if and only if it satisfies unavailable-type-invariance, weak non-wastefulness, resource-monotonicity, truncation-invariance, and strategy-proofness.
The following impossibility result shows that the irrelevance of satisfied demand istoo strong: there is no mechanism which satisfies it together with the above desirableproperties.
Proposition 3.
Suppose that there are at least three objects, | O | ≥ . There is nomechanism which satisfies unavailable-type-invariance, weak non-wastefulness, resource-monotonicity, truncation-invariance, strategy-proofness, and the irrelevance of satisfieddemand.Proof. Suppose that there exists such a mechanism, say ϕ , which satisfies all the propertiesin the statement except for the irrelevance of satisfied demand . We will show that it mustviolate the irrelevance of satisfied demand . By Theorem 3 of Ehlers and Klaus (2016), ϕ is a deferred acceptance mechanism based on a choice structure C = ( C x ) x ∈ O whichsatisfies capacity-filling , gross substitutes , and monotonicity .Let i, j ∈ N be two distinct agents. We first claim that there exist two distinct objects a, b ∈ O such that i ∈ C a ( { i, j } , ∩ C b ( { i, j } ,
1) and j / ∈ C a ( { i, j } , ∪ C b ( { i, j } , a or b , i is chosen but j is not from { i, j } . To see this,let x, y, z ∈ O be three distinct objects. By capacity-filling , either { i } = C x ( { i, j } ,
1) or { j } = C x ( { i, j } , { i } = C x ( { i, j } , capacity-filling , either { i } = C y ( { i, j } ,
1) or { j } = C y ( { i, j } , { i } = C y ( { i, j } , capacity-filling , either { i } = C z ( { i, j } ,
1) or { j } = C z ( { i, j } , a, b ∈ O such that i ∈ C a ( { i, j } , ∩ C b ( { i, j } ,
1) and j / ∈ C a ( { i, j } , ∪ C b ( { i, j } , R and R ′ besuch that every agent other than i and j find any object unacceptable and R i , R j , R ′ i and R ′ j are as depicted below. R i R j R ′ R ′ a b a ab a b b ∅ ∅ ∅ ∅ Let q be such that q b = 1 and q x = 0 for any x ∈ O \{ b } . Let q ′ be such that q ′ a = q ′ b = 1and q ′ x = 0 for any x ∈ O \ { a, b } . Since ϕ is a deferred acceptance mechanism based on C = ( C x ) x ∈ O , D a ( ϕ ( R, q ) , R ) = D a ( ϕ ( R ′ , q ) , R ′ ) = { i, j } . However, D a ( ϕ ( R, q ′ ) , R ) = ∅ and D a ( ϕ ( R ′ , q ′ ) , R ′ ) = { j } , implying that ϕ violates the irrelevance of satisfied demand .Since the irrelevance of satisfied demand is too strong, we consider the followingweakening of it which requires that at any problem where there is only one available object ,the set of agents who receive the object due to a capacity increase does not depend on theset of agents who initially receive the object.Formally, a mechanism ϕ satisfies the weak irrelevance of satisfied demand iffor any pair of problems ( R, q ) and ( R ′ , q ) and each object x ∈ O such that for each y ∈ O \ { x } , q y = 0, D x ( ϕ ( R, q ) , R ) = D x ( ϕ ( R ′ , q ) , R ′ ) implies D x ( ϕ ( R, q + 1 x ) , R ) = D x ( ϕ ( R ′ , q + 1 x ) , R ′ ).Our next result shows that the weak irrelevance of satisfied demand together withthe above properties characterize lexicographic deferred acceptance mechanisms. Proposition 4.
A mechanism is a lexicographic deferred acceptance mechanism if and onlyif it satisfies unavailable-type-invariance, weak non-wastefulness, resource-monotonicity,truncation-invariance, strategy-proofness, and the weak irrelevance of satisfied demand. roof. The following notation will be helpful. For each x ∈ O , let R x be a preferencerelation such that x is top-ranked and ∅ is second-ranked. For each S ∈ A that is nonempty,let R xS be a preference profile such that for each i ∈ S , ( R xS ) i = R x , and for each j / ∈ S ,( R xS ) j top-ranks ∅ . For each x ∈ O and l ∈ { , . . . , n } , let l x denote the capacity profilewhere x has capacity l and every other object has capacity zero.Let ϕ be a mechanism satisfying the properties in the statement of the theorem.Let C = ( C x ) x ∈ O be defined as follows. For each x ∈ O , S ∈ A , and l ∈ { , . . . , n } , C x ( S, l ) = { i ∈ S : ϕ i ( R xS , l x ) = x } . This choice structure is the same as the oneconstructed in the proof of Theorem 3 of Ehlers and Klaus (2016).By weak non-wastefulness , C x satisfies capacity-filling . By resource-monotonicity , C x satisfies monotonicity . By Lemma 2 of Ehlers and Klaus (2016), C x satisfies gross substi-tutes . By Theorem 3 of Ehlers and Klaus (2016), ϕ is a deferred acceptance mechanismbased on C . It is easy to see that, since ϕ satisfies the irrelevance of satisfied demand , foreach x ∈ O , C x satisfies the irrelevance of accepted alternatives . Thus, C is a lexicographicchoice structure and ϕ is a lexicographic deferred acceptance mechanism.Let ϕ be a lexicographic deferred acceptance mechanism. We will show that it sat-isfies irrelevance of satisfied demand . The other properties follow from Theorem 3 ofEhlers and Klaus (2016). Let C = ( C x ) x ∈ O be a lexicographic choice structure such that ϕ = DA C . Let ( R, q ) , ( R ′ , q ) ∈ P and x ∈ O be such that for each y ∈ O \ { x } , q y = 0and let T ≡ D x ( DA C ( R, q ) , R ) = D x ( DA C ( R ′ , q ) , R ′ ). Let C x be lexicographic for thepriority profile ( ≻ , . . . , ≻ n ) ∈ Π. Let S ( R ) and S ( R ′ ) be the sets of agents who prefer x to ∅ at R and at R ′ , respectively. It is easy to see that DA C ( R, q ) = C x ( S ( R ) , q ), DA C ( R ′ , q ) = C x ( S ( R ′ ) , q ), and T = S ( R ) \ C x ( S ( R ) , q ) = S ( R ′ ) \ C x ( S ( R ′ ) , q ). Let i ∈ T be the agent who is highest ranked according to ≻ q x +1 in T . Clearly, DA C ( R, q + 1 x ) = DA C ( R, q ) ∪ { i } and DA C ( R ′ , q + 1 x ) = DA C ( R ′ , q ) ∪ { i } . Hence, D x ( DA C ( R, q + 1 x ) , R ) = D x ( DA C ( R ′ , q + 1 x ) , R ′ ) = T \ { i } . Remark . In Appendix C, we provide an example of a mechanism which satisfies all theproperties in the statement of Proposition 4 except for the irrelevance of satisfied demand ,and therefore which is not a lexicographic deferred acceptance mechanism.22
Implications for School Choice in Boston
In the Boston school choice system, there are two different priority orderings at each school:a walk-zone priority ordering , which gives priority to the school’s neighborhood studentsover all the other students, and an open priority ordering which does not give priority toany student for being a neighborhood student. The Boston school district aims to assignhalf of the seats of each school based on the walk-zone priority ordering and the other halfbased on the open priority ordering. To achieve this aim, given the capacity, each schoolchooses students in a lexicographic way according to a priority profile where half of thepriority orderings is the walk-zone priority ordering and the other half is the open priorityordering .In a recent study, Dur et al. (2013) note that two priority profiles with the same num-bers of walk-zone and open priority orderings, but with different precedence orders of thepriority orderings, may result in different choices under a lexicographic choice procedure.Starting with this observation, Dur et al. (2013) compare four different choice rules, oneof which is the one used in the Boston school district, in terms of how much they arebiased for or against the neighbourhood students. In this section, we will consider thesealternative choice rules from a different perspective. We will show that, although thesechoice rules are all based on a “lexicographic procedure” at each capacity, only one ofthem satisfies all the characterizing properties in Theorem 1, and therefore only one ofthem is actually a (capacity-constrained) lexicographic choice rule .In order to put the four choice rules in a formal context, let us consider the followingclass of choice rules which is larger than the class of lexicographic choice rules. We say thata choice rule is capacity-wise lexicographic if there exists a list of priority orderingsfor each capacity level (the number of priority orderings is the same as the capacity), andat each capacity, the rule operates based on the associated list of priority orderings in alexicographic way. For a capacity-wise lexicographic choice rule, unlike a lexicographicchoice rule, the lists for different capacity levels are not necessarily related.The capacity-wise lexicographic choice rules that can serve the Boston school district’spurpose are the choice rules for which, at each capacity, the associated list consists of onlythe walk-zone priority ordering and the open priority ordering, and the absolute differencebetween the numbers of walk-zone and open priority orderings in the list is at most one.23e formalize this property as follows.Let ≻ w and ≻ o be walk-zone and open priority orderings. We say that a capacity-wise lexicographic choice rule satisfies the Boston requirement for ( ≻ w , ≻ o ) if for eachcapacity q , the associated list of priority orderings ( ≻ , . . . , ≻ q ) is such thati. for each l ∈ { , . . . , q } , ≻ l ∈ {≻ w , ≻ o } ,ii. difference between the number of ≻ w -priorities and ≻ o -priorities is at most one, i.e. (cid:12)(cid:12)(cid:12) P qi =1 ≻ w ( ≻ i ) − P qi =1 ≻ o ( ≻ i ) (cid:12)(cid:12)(cid:12) ≤ . Now, it turns out that the following class of capacity-wise lexicographic choice rulesare the only rules satisfying our set of properties together with the Boston requirementfor ( ≻ w , ≻ o ). Proposition 5.
A capacity-wise lexicographic choice rule satisfies capacity-filling, grosssubstitutes, monotonicity, the capacity-wise weak axiom of revealed preference, and theBoston requirement for ( ≻ w , ≻ o ) if and only if it is (capacity-constrained) lexicographic fora priority profile ( ≻ , . . . , ≻ n ) such thati. for each l ∈ { , . . . , n } , ≻ l ∈ {≻ w , ≻ o } ,ii. for each l that is odd, ≻ l = ≻ w if and only if ≻ l +1 = ≻ o .Proof. By Theorem 1, a choice rule satisfying the properties must be lexicographic. Therest is straightforward.Some examples of priority profiles satisfying (i) and (ii) in the statement of Proposition5 are ( ≻ w , ≻ o , ≻ w , ≻ o , . . . ), ( ≻ o , ≻ w , ≻ o , ≻ w , . . . ), and ( ≻ w , ≻ o , ≻ o , ≻ w , ≻ w , ≻ o , ≻ w , ≻ o , . . . ).Some examples that violate (ii) are ( ≻ w , ≻ o , ≻ o , ≻ w , ≻ o , ≻ o , . . . ) and ( ≻ o , ≻ w , ≻ w , ≻ o , ≻ w , ≻ w , . . . ).Four plausible choice rules stand out from the analysis of Dur et al. (2013), one ofwhich is currently in use in Boston (Open-Walk choice rule). Dur et al. (2013) comparethe below four choice rules in terms of how much they are biased for or against theneighbourhood students. We will compare the four choice rules with respect to our set ofchoice rule properties. x ( y ) is the indicator function which has the value 1 if x = y and 0 otherwise. Walk-Open Choice Rule:
At each capacity, the first half of the priority orderings inthe list are the walk-zone priority ordering and the last half are the open priorityordering.2.
Open-Walk Choice Rule:
At each capacity, the first half of the priority orderings inthe list are the open priority ordering and the last half are the walk-zone priorityordering.3.
Rotating Choice Rule:
At each capacity, the first priority ordering in the list is thewalk-zone priority ordering, the second is the open priority ordering, the third is thewalk-zone priority ordering, and so on.4.
Compromise Choice Rule:
At each capacity, the first quarter of the priority orderingsin the list are the walk-zone priority ordering, the following half of the priorityorderings in the list are the open priority ordering, and the last quarter are againthe walk-zone priority ordering.To be precise, let us introduce the following procedures to accommodate the caseswhere the capacity is not divisible by two or four. • Walk-Open Choice Rule:
If the capacity q is an odd number, the first q +12 are thewalk-zone priority ordering. • Open-Walk Choice Rule:
If the capacity q is an odd number, the first q +12 are theopen priority ordering. • Compromise Choice Rule:
If the capacity q is not divisible by four, let q = q ′ + k for some q ′ that is divisible by 4 and some k ∈ { , , } . If k = 1, let the first q ′ + 1orderings be the walk-zone priority ordering, the following q ′ orderings be the openpriority ordering, and the last q ′ orderings be the walk-zone priority ordering. If k = 2, let the first q ′ + 1 orderings be the walk-zone priority ordering, the following q ′ + 1 orderings be the open priority ordering, and the last q ′ orderings be the walk-zone priority ordering. If k = 3, let the first q ′ +1 orderings be the walk-zone priorityordering, the following q ′ + 1 orderings be the open priority ordering, and the last q ′ + 1 orderings be the walk-zone priority ordering.Note that all of the above rules satisfy the Boston requirement for ( ≻ w , ≻ o ). Sinceall of the rules are capacity-wise lexicographic, they satisfy capacity-filling and gross sub-stitutes . It follows from the only if part of Proposition 5 that, among these four choicerules, only the Rotating Choice Rule satisfies capacity-filling , gross substitutes , monotonic- ty , and the irrelevance of accepted alternatives . However, it is not clear if the other threerules are not lexicographic under variable capacity constraints because they fail to satisfy monotonicity , the irrelevance of accepted alternatives or both. Next, we show that theother three rules satisfy monotonicity , but they violate the irrelevance of accepted alterna-tives . To show that these rules satisfy monotonicity , we first provide an auxiliary conditionthat is easier to verify and sufficient for monotonicity . Next we introduce this conditionand prove that it is sufficient for monotonicity .Let π = ( ≻ , . . . , ≻ q ) and π ′ = ( ≻ ′ , . . . , ≻ ′ q +1 ) be priority lists of size q and q + 1,respectively. We say that π ′ is obtained by insertion from π if there exists k ∈ { , . . . , q +1 } such that ≻ ′ l = ≻ l for each l < k , and ≻ ′ l = ≻ l − for each l > k . Note that when π ′ isobtained by insertion from π , a new priority ordering is inserted into the list of priorityorderings in π , by keeping relative order of the other priority orderings in the list the same.It is possible that the new ordering is inserted in the very beginning or in the very end ofthe list. Lemma 2.
Let C be a capacity-wise lexicographic choice rule. The choice rule C ismonotonic if for each q ∈ { , . . . , n } , the priority list for q is obtained by insertion fromthe priority list for q − .Proof. Let (
S, q ) ∈ A × { , . . . , n − } . Let π = ( ≻ , . . . , ≻ q ) be the list for capacity q .Let a ∈ C ( S, q ). Suppose that, in the lexicographic choice procedure, a is chosen at the t ’th step, i.e. a is chosen based on ≻ t .Let π ′ = ( ≻ ′ , . . . , ≻ ′ q +1 ) be the list for capacity q + 1 that is obtained by an insertionfrom π . Let k ∈ { , . . . , q + 1 } be such that ≻ ′ l = ≻ l for each l < k , and ≻ ′ l = ≻ l − for each l > k .Now, consider the problem ( S, q + 1). If t < k , clearly a is still chosen at the t ’thstep of the lexicographic choice procedure and thus a ∈ C ( S, q + 1). Suppose that t ≥ k .The rest of the proof is by induction. First, suppose that t = k . Note that at Step k ofthe choice procedure for the problem ( S, q + 1), the choice is made based on the insertedpriority ordering and at Step k + 1, the choice is made based on ≻ t . Then, a is eitherchosen at Step k , or at Step k + 1, the set of remaining alternatives is a subset of the setof remaining alternatives at Step t of the choice procedure for ( S, q ) where a is chosen, inwhich case a is still chosen. Thus, a ∈ C ( S, q + 1).26ow, suppose that t > k and each alternative that is chosen at a step t ′ < t of thechoice procedure at ( S, q ) is also chosen at (
S, q + 1). Then, a is either chosen before step t + 1 of the choice procedure for ( S, q + 1), or at Step t + 1, the set of remaining alternativesis a subset of the set of remaining alternatives at Step t of the choice procedure for ( S, q )where a is chosen, in which case a is still chosen. Thus, a ∈ C ( S, q + 1).
Proposition 6.
All of the four rules satisfy capacity-filling, gross substitutes, and mono-tonicity, but only the rotating choice rule satisfies the irrelevance of accepted alternativesand only the rotating choice rule is (capacity-constrained) lexicographic.Proof.
Each rule is capacity-wise lexicographic (lexicographic for a given capacity) andtherefore satisfies capacity-filling and gross substitutes . Moreover, it is easy to see thateach of the four choice rules satisfies the insertion property, so monotonicity follows fromLemma 2.As for the irrelevance of accepted alternatives , first consider ( ≻ , . . . , ≻ n ) ∈ Π suchthat the first priority ordering in the list is ≻ w , the second is ≻ o , the third is ≻ w , andso on. The rotating choice rule is clearly lexicographic for ( ≻ , . . . , ≻ n ). Moreover, byTheorem 1, it satisfies the irrelevance of accepted alternatives . We will show that each ofthe other three choice rules violates the irrelevance of accepted alternatives . Walk-Open Choice Rule:
Let A = { a, b, c, d, e } . Let ≻ w be defined as a ≻ w b ≻ w c ≻ w d ≻ w e and ≻ o be defined as e ≻ o b ≻ o d ≻ o c ≻ o a . Note that R ( { a, c, d, e } ,
2) = R ( { a, b, c, d } ,
2) = { c, d } . However, R ( { a, c, d, e } ,
3) = { d } and R ( { a, b, c, d } ,
3) = { c } ,and therefore C violates the irrelevance of accepted alternatives . Open-Walk Choice Rule:
Can be shown by interchanging the orderings for ≻ w and ≻ o inthe previous example. Compromise Choice Rule:
Let A = { a, b, c, d, x, y } . Let ≻ w be defined as a ≻ w b ≻ w c ≻ w d ≻ w x ≻ w y and ≻ o be defined as b ≻ o c ≻ o y ≻ o x ≻ o d . Note that R ( { a, b, c, x, y } ,
3) = R ( { a, b, d, x, y } ,
3) = { x, y } . However, R ( { a, b, c, x, y } ,
4) = { y } and R ( { a, b, d, x, y } ,
4) = { x } , and therefore C violates the irrelevance of accepted alternatives . Remark . Note that the particular procedures we introduced to accommodate the caseswhere the capacity is not divisible by two or four are not crucial for the proof of Propo-sition 6. For the other procedures (for example, for the walk-open choice rule, the extra27riority when the capacity is odd can alternatively be set to be the open priority ordering),the examples in the proof can simply be modified to show that the irrelevance of acceptedalternatives is still violated.It follows from our Proposition 6 that if the irrelevance of accepted alternatives orhaving a lexicographic representation under variable capacity constraints is deemed de-sirable, then the rotating choice rule should be selected since it is the only choice ruleamong the four plausible choice rules that satisfies the irrelevance of accepted alternatives together with capacity-filling , gross substitutes , and monotonicity .Another interpretation of our Proposition 6 is the following. First of all, note thatin Dur et al. (2013), the capacity is fixed and a choice rule is defined given a capacity.On the other hand, the capacity is allowed to vary and a choice rule has to specify whichalternatives are chosen from each choice set at each possible capacity in our approach,which is the fundamental difference between Dur et al. (2013) and our study. The factthat we allow the capacity to vary and we require a choice rule to respond to changes inthe capacity, allows us to define desirable properties of choice rules that address how itshould respond to changes in the capacity, such as monotonicity and the irrelevance ofaccepted alternatives . Proposition 6 shows that, although each of the four rules in Dur et al.(2013) operate based on a lexicographic procedure when we fix the capacity, in a variablecapacity framework only one of them satisfies the irrelevance of accepted alternatives andtherefore only one of them is capacity-constrained lexicographic under variable capacities,i.e., there exists a priority profile, which has as many priority orderings as the maximumpossible capacity, such that at each capacity, the rule operates based on a lexicographicprocedure with respect to the same priority profile . Our formulation of a choice rule and the properties that we consider take into accountthat the capacity may vary. When designing choice rules especially for resource allocationpurposes, such as in school choice, a designer may be interested in choice rules that respondto changes in capacity. In that framework, our Theorem 1 shows that capacity-filling , gross substitutes , monotonicity , and the irrelevance of accepted alternatives are altogether28atisfied only by (capacity-constrained) lexicographic choice rules, which identifies theproperties that distinguish lexicographic choice rules from other plausible choice rules.Besides providing an axiomatic foundation for lexicographic choice rules, this finding maybe helpful in applications to select among plausible choice rules, as we have illustrated inSection 5, and also to understand characterizing properties of popular resource allocationmechanisms, as we have illustrated in Section 4. References
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AppendixA Necessity of the Properties in Lemma 1
The following example shows that capacity-filling is necessary for the implication, that is,there exists a choice rule which satisfies monotonicity and
CWARP , but not the irrelevanceof accepted alternatives . Example 2.
Let A = { a, b, c } . Let ≻ be defined as a ≻ b ≻ c . Let the choice rule C bedefined as follows. For each problem ( S, q ) , if a ∈ S , then C ( S, q ) = { a } ; if a / ∈ S , then C chooses the highest ≻ -priority alternatives from S until q alternatives are chosen or noalternative is left. Note that C clearly violates capacity-filling and satisfies monotonicity nd CWARP. Now, let S = { a, c } and S ′ = { b, c } . Note that R ( S,
1) = R ( S ′ ,
1) = { c } , C ( S, ∩ R ( S,
1) = ∅ , C ( S ′ , ∩ R ( S ′ ,
1) = { c } , and therefore C ( S, ∩ R ( S, = C ( S ′ , ∩ R ( S ′ , , implying that C violates the irrelevance of accepted alternatives. The following example shows that monotonicity is necessary for the implication, thatis, there exists a choice rule which satisfies capacity-filling and
CWARP , but not theirrelevance of accepted alternatives . Example 3.
Let A = { a, b, c, d } . Let ≻ and ≻ ′ be defined as a ≻ b ≻ c ≻ d and b ≻ ′ c ≻ ′ d ≻ ′ a . Let the choice rule C be defined as follows. For each problem ( S, , C choosesthe highest ≻ -priority alternatives from S ; and for each problem ( S, q ) such that q > , C chooses the highest ≻ ′ -priority alternatives from S until q alternatives are chosen or noalternative is left. Note that C clearly satisfies capacity-filling and CWARP. C violatesmonotonicity because, for instance, a ∈ C ( { a, b, c } , but a / ∈ C ( { a, b, c } , . Now, let S = { a, c, d } and S ′ = { b, c, d } . Note that R ( S,
1) = R ( S ′ ,
1) = { c, d } , C ( S, ∩ R ( S,
1) = { c, d } , C ( S ′ , ∩ R ( S ′ ,
1) = { c } , and therefore C ( S, ∩ R ( S, = C ( S ′ , ∩ R ( S ′ , ,implying that C violates the irrelevance of accepted alternatives. The following example shows that
CWARP is necessary for the implication, that is,there exists a choice rule which satisfies capacity-filling and monotonicity , but not theirrelevance of accepted alternatives . Example 4.
Let A = { a, b, c, d } . Let ≻ and ≻ ′ be defined as a ≻ b ≻ c ≻ d and a ≻ ′ b ≻ ′ d ≻ ′ c . Let the choice rule C be defined as follows. For each problem ( S, q ) suchthat a ∈ S , C chooses the highest ≻ -priority alternatives from S until q alternatives arechosen or no alternative is left; and for each problem ( S, q ) such that a / ∈ S , C choosesthe highest ≻ ′ -priority alternatives from S until q alternatives are chosen or no alternativeis left. Note that C clearly satisfies capacity-filling and monotonicity. Now, let S = { a, c, d } and S ′ = { b, c, d } . Note that R ( S,
1) = R ( S ′ ,
1) = { c, d } , C ( S, ∩ R ( S,
1) = { c } , C ( S ′ , ∩ R ( S ′ ,
1) = { d } , and therefore C ( S, ∩ R ( S, = C ( S ′ , ∩ R ( S ′ , , implyingthat C violates the irrelevance of accepted alternatives. Independence of Properties in Theorem 1
Violating only capacity-filling:
Let A = { a, b, c } . Let ≻ be a priority ordering. Let C be the choice rule such that, for each problem ( S, q ), C ( S, q ) is a singleton consisting of the ≻ -maximal alternative in S . Note that C violates capacity-filling and clearly satisfies grosssubstitutes . Since the choice does not vary with capacity, C also satisfies monotonicity and the irrelevance of accepted alternatives . Violating only gross substitutes:
Let A = { a, b, c } . Let ≻ and ≻ ′ be defined as a ≻ b ≻ c and b ≻ ′ a ≻ ′ c . Let the choice rule C be defined as follows. For each problem( S, q ), C ( S, q ) consists of the ≻ -maximal alternative in S if q = 1 and c ∈ S ; otherwise, C ( S, q ) coincides with the choice rule that is responsive for ≻ ′ . Note that C satisfies capacity-filling .Since a ∈ C ( { a, b, c } ,
1) = { a } and a / ∈ C ( { a, b } ,
1) = { b } , C violates gross substitutes .To see that C satisfies monotonicity , suppose that there exists a set S and an alternative x ∈ S such that x ∈ C ( S,
1) and x / ∈ C ( S, x / ∈ C ( S,
2) implies that x = c and S = { a, b, c } . But then, x / ∈ C ( S,
1) = { a } , a contradiction. To see that C satisfies CWARP , note that the revealed preference relation at q = 2 consists of a unique pair: b is revealed preferred to c . Then, by Lemma 1, C also satisfies the irrelevance of acceptedalternatives . Violating only monotonicity:
Let A = { a, b, c } . Let ≻ be defined as a ≻ b ≻ c . Letthe choice rule C be defined as follows. For each problem ( S, q ), C ( S, q ) consists of the ≻ -maximal alternative in S if q = 1; C ( S,
2) = S if | S | = 2; and C ( { a, b, c } ,
2) = { b, c } .Note that C satisfies capacity-filling .Since a ∈ C ( { a, b, c } ,
1) and a / ∈ C ( { a, b, c } , C violates monotonicity. For q = 1, C satisfies gross substitutes , since C maximizes ≻ ; for q ∈ { , } , C clearly satisfies grosssubstitutes . Since there are no two different problems with the same capacity and the sameset of rejected alternatives, C satisfies the irrelevance of accepted alternatives . Violating only CWARP:
Note that three of the four rules that we discuss in Section 5satisfy all the properties but the irrelevance of accepted alternatives .33
Importance of the Irrelevance of Satisfied Demandin Proposition 4
We provide an example of a mechanism which satisfies all the properties in the statementof Proposition 4 except for the irrelevance of satisfied demand , and therefore which isnot a lexicographic deferred acceptance mechanism. The mechanism in the example is adeferred acceptance mechanism based on a choice structure such that the choice rule ofeach object is a walk-open choice rule. The example uses some arguments from the proofof Proposition 6, where it was shown that the walk-open choice rule violates
CWARP . Example 5.
Let N = { a, b, c, d, e } and let O be a finite set of objects. Let ≻ w be definedas a ≻ w b ≻ w c ≻ w d ≻ w e and ≻ o be defined as e ≻ o b ≻ o d ≻ o c ≻ o a . Let ( C x ) x ∈ O bethe choice structure such that for each object x ∈ O , C x is the walk-open choice rule basedon ( ≻ w , ≻ o ) . Let ϕ be the deferred acceptance mechanism based on the choice structure ( C x ) x ∈ O .Since for each x ∈ O , C x satisfies capacity-filling, gross substitutes, and monotonicity,by Theorem of Ehlers and Klaus (2016), ϕ satisfies unavailable-type-invariance, weaknon-wastefulness, resource-monotonicity, truncation-invariance, and strategy-proofness.Let x ∈ O . Let q be such that q x = 2 and for each y ∈ O \ { x } , q y = 0 . Let R be suchthat x is preferred to ∅ for all the agents except for b . Note that D x ( ϕ ( R, q ) , R ) = { c, d } since C x ( { a, c, d, e } ,
2) = { a, e } . Let R ′ be such that x is preferred to ∅ for all the agentsexcept for e . Note that D x ( ϕ ( R ′ , q ) , R ′ ) = { c, d } since C x ( { a, b, c, d } ,
2) = { a, b } . Thus, D x ( ϕ ( R, q ) , R ) = D x ( ϕ ( R ′ , q ) , R ′ ) .Now, we have D x ( ϕ ( R, q + 1 x ) , R ) = { d } since C x ( { a, c, d, e, } ) = { a, c, e } and D x ( ϕ ( R ′ , q + 1 x ) , R ′ ) = { c } since C x ( { a, b, c, d, } ) = { a, b, d } . (Note that when q = 3 , C x is lexicographic for ( ≻ w , ≻ w , ≻ o ) .) Hence, ϕ violates the irrelevance of satisfied demand. Responsive Choice
A well-known example of a lexicographic choice rule is a “responsive” choice rule, which islexicographic for a priority profile where all the priority orderings are the same. Formally, achoice rule C is responsive for a priority ordering ≻ if for each ( S, q ) ∈ A × { , . . . , n } , C ( S, q ) is obtained by choosing the highest ≻ -priority alternatives in S until q alternativesare chosen or no alternative is left. Note that C is responsive for ≻ if and only if it islexicographic for the priority profile ( ≻ , . . . , ≻ ).Chambers and Yenmez (2018b) characterize “responsive” choice rules, but in the con-text of “classical” choice problems which do not explicitly refer to a variable capacityparameter. Formally, a classical choice rule is a function C : A → A such that for each S ∈ A , C ( S ) ⊆ S . A classical choice rule is responsive if there exists a priority ordering ≻ and a capacity q ∈ { , . . . , n } such that for each S ∈ A , C ( S ) is obtained by choosingthe highest ≻ -priority alternatives until the capacity q is reached or no alternative is left.Chambers and Yenmez (2018b) show that a classical choice rule satisfies capacity-filling and the weaker axiom of revealed preference (WrARP) if and only if it is responsive. WrARP requires that for each pair a, b ∈ A and S, S ′ ∈ A such that a, b ∈ S ∩ S ′ ,if a ∈ C ( S ) and b ∈ C ( S ′ ) \ C ( S ) , then a ∈ C ( S ′ ) . To see what Chambers and Yenmez (2018b) implies in our variable capacity setup,consider the following extension of
WrARP to our setup.
Weaker axiom of revealed preference (WrARP):
For each
S, S ′ ∈ A , q ∈ { , . . . , n } ,and each pair a, b ∈ S ∩ S ′ ,if a ∈ C ( S, q ) and b ∈ C ( S ′ , q ) \ C ( S, q ) , then a ∈ C ( S ′ , q ) . Responsive choice rules have been studied particularly in the two-sided matching context(Roth and Sotomayor, 1990). A classical choice rule satisfies capacity-filling if there exists a capacity such that at each choiceproblem, an alternative is rejected only if the capacity is reached. Chambers and Yenmez (2018b) also provide a characterization of choice rules that are responsive fora known capacity (namely q -responsive choice rules). WrARP was introduced by Jamison and Lau (1973) and also studied by Ehlers and Sprumont (2008).
Proposition 7.
A choice rule satisfies capacity-filling and the weaker axiom of revealedpreference if and only if for each q ∈ { , . . . , n } , there is a priority ordering ≻ q such thatfor each S ∈ A , C ( S, q ) is obtained by choosing the highest ≻ q -priority alternatives untilthe capacity q is reached or no alternative is left. Proposition 7 states that capacity-filling and
WrARP characterizes “capacity-wiseresponsive” choice rules, which are responsive for each capacity, but the associated priorityorderings for different capacities may be different. Yet, a characterization of responsivechoice rules in our setup does not directly follow from Chambers and Yenmez (2018b).We show that, the following extension of
WrARP , together with capacity-filling , char-acterizes responsive choice rules in our variable-capacity setup. The property, called the capacity-wise weaker axiom of revealed preference (CWrARP) , requires that if an alterna-tive a is chosen and b is not chosen at a problem where they are both available, then atany problem where they are both available, a is chosen whenever b is chosen. Capacity-wise weaker axiom of revealed preference (CWrARP):
For each
S, S ′ ∈A , q, q ′ ∈ { , . . . , n } , and each pair a, b ∈ S ∩ S ′ ,if a ∈ C ( S, q ) and b ∈ C ( S ′ , q ′ ) \ C ( S, q ) , then a ∈ C ( S ′ , q ′ ) . Theorem 2.
A choice rule is responsive if and only if it satisfies capacity-filling and thecapacity-wise weaker axiom of revealed preference.Proof.
It is clear that a responsive choice rule satisfies capacity-filling and
CWrARP . Let C be a choice rule satisfying capacity-filling and CWrARP . Clearly,
CWrARP implies
WrARP , and therefore by Proposition 7, for each q ∈ { , . . . , n } , there is a priority ordering ≻ q such that for each S ∈ A , C ( S, q ) is obtained by choosing the highest ≻ q -priorityalternatives until the capacity q is reached or no alternative is left.Let ( S, q ) ∈ A × { , . . . , n } . If | S | ≤ q , then by capacity-filling , C ( S, q ) = S . Supposethat | S | > q . First note that C ( S, q − ⊆ C ( S, q ), since otherwise, by capacity-filling ,there is a pair a, b ∈ S such that a ∈ C ( S, q − \ C ( S, q ) and b ∈ C ( S, q ) \ C ( S, q − CWrARP . Now, consider any pair a, b ∈ R ( S, q −
1) such that a ∈ C ( S, q ) and36 / ∈ C ( S, q ). By
CWrARP , for any S ′ ∈ A , b is not chosen over a at ( S ′ , q ), implying that a has ≻ q -priority over b . But then, for each S ∈ A , C ( S, q ) is obtained by choosing thehighest ≻ q − -priority alternatives until the capacity q is reached or no alternative is left.Since we started with an arbitrary q ∈ { , . . . , n } , C is a choice rule that is responsive to ≻1