Constrained Pseudo-market Equilibrium
aa r X i v : . [ ec on . T H ] J un CONSTRAINED PSEUDO-MARKET EQUILIBRIUM
FEDERICO ECHENIQUE, ANTONIO MIRALLES, AND JUN ZHANG
Abstract.
We propose a market solution to the problem of resource allocationsubject to constraints, such as considerations of diversity or geographical distri-bution. Constraints give rise to pecuniary externalities, which are internalized viaprices. Agents pay to the extent that their purchases affect the value (at equi-librium prices) of the relevant constraints. The result is a constrained-efficientmarket equilibrium outcome. The outcome is fair whenever the constraints do notsingle out individual agents, which happens, for example with geographical dis-tribution constraints. In economies with endowments, moreover, we can addressparticipation constraints. Our equilibrium outcomes are then constrained efficientand approximately individually rational.(Echenique)
Division of the Humanities and Social Sciences, California Instituteof Technology (Miralles)
Department of Economics, Universitat Aut`onoma de Barcelona andBarcelona Graduate School of Economics and Universit`a degli Studi di Messina. (Zhang)
Institute for Social and Economic Research, Nanjing Audit University.
Date : First version: September 2019. This version: May 2020.We thank Eric Budish, Fuhito Kojima, Andy McLennan, Herv´e Moulin, and Tayfun S¨onmez forcomments. Echenique thanks the National Science Foundation for its support through the grantsSES-1558757 and CNS-1518941, and the Simons Institute at UC Berkeley for its hospitality whilepart of the paper was written. Miralles acknowledges financial support from the Spanish Ministryof Economy and Competitiveness, through the Severo Ochoa Programme for Centers of Excellencein R&D (SEV-2015-0563). Zhang acknowledges financial support from National Natural ScienceFoundation of China (Grant No. 71903093). Contents
1. Introduction 31.1. Motivation 51.2. Related literature 72. The model 72.1. Notational conventions 72.2. The economy 92.3. Constraints 92.4. Special case: discrete allocation 102.5. Pre-processing of constraints 102.6. Normative properties 122.7. Equilibrium 123. Main theorem 124. Application: Discrete object allocation under constraints 134.1. Constraint structures 134.2. No floor constraints 154.3. Floor constraints 155. A market for roommates, and other problems 195.1. Coalition formation 225.2. Combinatorial allocation 226. A market for “bads” 237. Endowment and α -slack equilibrium 247.1. The economy and equilibrium 247.2. Results 257.3. The Hylland and Zeckhauser example 267.4. A market-based fairness property 287.5. A market for time exchange 298. Related Literature 299. Proof of Theorem 1 and Theorem 2 3310. Proof of Theorem 3 3911. Proof of Proposition 5 40References 42 ONSTRAINTS 3 Introduction
We analyze the use of pseudo-markets for assignment problems with constraints .A pseudo-market is an artificial marketplace in which agents are given fixed budgetsof “funny money” that is only useful within the marketplace. Pseudo-markets wereproposed by Hylland and Zeckhauser (1979), who consider the allocation of indivis-ible objects to an equal number of unit-demand agents. Their running example isto assign workers to jobs.We generalize and expand the scope of applicability of pseudo markets. We dealwith general assignment problems under constraints. For example the assignmentof workers to jobs under regional “celing” and “floor” quotas (Kamada and Kojima,2015); upper bounds on assignments due to considerations of gender or demographicbalance (Ehlers, Hafalir, Yenmez, and Yildirim, 2014); or participation constraintsthat arise from a pre-existing allocation. Our approach handles models that hadnot been analyzed via markets before, such as the well-known roommate problemfrom matching theory, and coalition formation problems.Given a general set of constraints, and under standard continuity and convexityassumptions, pseudo-markets deliver (constrained) efficient outcomes in equilibrium.When constraints do not single out any particular agent, the equilibrium is efficientand fair; and when agents start out with property rights over initial endowments,we can achieve efficiency and approximate individual rationality.The key idea in our proposal is to price constraints. Think of two workers, Aliceand Bob, in Hylland and Zeckhauser’s (HZ) pseudo-market. Jobs are in fixed supply.If Alice buys a share in job J, there is less left for Bob. Such “pecuniary” externalitiesare handled in markets by pricing good J. In our proposal, we think of the priceof good J as the price on the supply constraint for job J: the constraint that saysthat the total demand for good J cannot exceed the available supply. HZ showthat pecuniary externalities are correctly internalized by equilibrium prices, andthat an efficient outcome results in equilibrium. In the present paper, we interpretall sorts of constraints, well beyond supply constraints, as giving rise to pecuniaryexternalities. In consequence, we use prices to align agents’ choices with an efficientoutcome. In models where there are only supply constraints, this is the message of the first welfaretheorem. In HZ’s model, however, there are also unit demand constraints and the first welfaretheorem fails.
ECHENIQUE, MIRALLES, AND ZHANG
Specifically, we start from a general collection of feasible assignments as our prim-itive. We use linear inequalities to characterize the “upper-right” boundary of theconvex hull of feasible assignments. These linear inequalities include the standardsupply constraints, but specific applications usually require additional constraints.Each constraint is then priced. When Alice purchases one unit of good J, she willhave to pay to the extent that her purchase affects other agents through the differentconstraints. For example, if there is a ceiling constraint on how many units of J cango to a group of agents, the agents in the group will pay the price of the constraintwhen they buy units of J. If those agents are also involved in other constraints, thenthe final personalized prices they face can be different. But if two agents are alwaysinvolved in the same constraints, they will face equal prices. Equal budgets thenensure that they will not envy each other.The idea may seem familiar from the role of shadow prices in optimization withconstraints, but the familiarity is deceptive. Imagine using the dual variables (orLagrange multipliers) associated with each constraint in order to decentralize anallocation that is constrained efficient. We run into two issues. One is that someconstraints may impose a lower bound on consumed quantities, which would leadto negative prices. The other is that decentralizing a constrained efficient alloca-tion would require transfers, as in the second welfare theorem. With endogenoustransfers, one cannot ensure a fair outcome. Or one that is individually rationalwhen there are endowments. Our approach, in contrast, can ensure fairness andindividual rationality because our prices constitute market equilibria. We first sim-plify the problem by working with a subset of the constraints, and ensure that theyhave positive prices (Sections 2.5 and 3). Individual rationality can be ensured whenagents have endowments (Section 7). And, as long as the constraints do not them-selves induce unfairness by treating agents differently, it is possible to obtain a fairoutcome in equilibrium. We differ from the literature in that we do not take a constraint structure as the primitivein our model. The influential work by Budish, Che, Kojima, and Milgrom (2013) and the recentdevelopment by Akbarpour and Nikzad (forthcoming) are concerned with the implementability ofrandom assignments. Our work is orthogonal to the question of implementability. We can addressany constraints that pin down a well-defined set of feasible ex-post assignments. We discuss the second welfare theorem without transfers developed by Miralles and Pycia(2020) in the related literature section (Section 8). Note that the outcome of the second welfaretheorem may induce envy, even in a textbook economy with no constraints other than the supplyconstraints.
ONSTRAINTS 5
We turn to a discussion of specific applications that motivate our approach, andwhere our results deliver new insights.1.1.
Motivation.
Matching workers to jobs.
HZ illustrated the use of pseudo-markets by wayof assigning jobs to a set of workers. Each worker is to receive at most one job, whatwe call a unit demand constraint . Each job is in unit supply, so the sum of workersassigned a particular job cannot exceed one: what we call a supply constraint . Aswe shall see, constraints that only involve an individual agent, such as unit demandconstraints, have no external effects and do not need to be priced. Constraints thatinvolve multiple agents, such as supply constraints, will be assigned a price. Whenan agent purchases a good, she has to pay to the extent that the purchase impactsthe constraints that are priced.Importantly, there is one supply constraint for each good. The price that corre-sponds to the supply constraint for good l is the familiar “price of good l .” Think ofthe constraint as capturing a pecuniary externality . When Alice purchases good l ,the supply constraint implies that there is less good l available for Bob. By pricingsupply constraints we ensure that Alice internalizes the effects that her purchaseshas on Bob. As we shall see, in problems with more complex constraint structures,we may not be able to ascribe a specific good to each specific price; but the logic ofusing prices to internalize the external effects induced by constraints extends.In the workers-to-jobs application, priced constraints affect all agents in the sameway. As a consequence, the prices are equal for all agents, and we can show theexistence of a market equilibrium outcome that is both efficient and fair.Finally, the methodology in our paper extends the HZ approach to situationswith multiple unit assignment; see Section 5.2 (Budish (2012) gives an overview ofmulti-unit assignment). Assigning doctors to hospital positions.
The problem of assigning each doctora hospital position is similar to the workers-to-jobs example, but with an importanttwist. Hospitals belong to different geographical regions, and the system seeks toensure a minimum number of doctors per region (Kamada and Kojima, 2015). Sowe have unit demand and supply constraints as before; but there is now a lowerbound —a floor constraint — on the number of doctors assigned to each region. Our In multi-unit assignment, the relevant set of constraints that need to be priced does not nec-essarily coincide with the set of items or the set of bundles. See Section 5.2 for a brief discussion.
ECHENIQUE, MIRALLES, AND ZHANG solution involves turning these constraints into upper bounds (see Section 4.3 fordetails).There are then two kinds of non-individual constraints that must be priced: supplyconstraints and the modified floor constraints. When Alice buys into a popularhospital position, she causes a pecuniary externality on Bob, who may have to takea position in a less-demanded regional hospital. The price on the correspondingmodified floor constraint ensures that she pays more for the popular hospital thanif she were only facing the supply constraint. In an equilibrium, now, prices ensurethat demand spills over into the less attractive regions so as to meet the lower boundfor each region.In the doctors-to-hospitals application, again, all agents are treated in the sameway by the priced constraints. In consequence, all agents face the same prices,and we obtain a market equilibrium that is fair and efficient. The application tohospital-doctor matching with regional constraints is discussed in Section 4.1.1.3.
Roommates.
A set of college students need to pair up as roommates. Eachstudent has a utility function defined over her possible roommates. We formulatethe model as an assignment problem, where we assign objects to agents by treatingthe set of objects as a copy of the set of agents. Each student has two roles: oneas agent seeking to match to an object, and one as object that can be matched todifferent agents. In addition to the familiar unit demand and supply constraints,we must now impose a symmetry property. If agent Alice is matched to objectBob, then agent Bob must be matched to object Alice. The symmetry constraintsinvolve more than one agent, and must therefore be priced. When Alice purchasessome of the “Bob good” she is committing Bob to consume an equal amount of the“Alice good.” In our pseudo-market, this pecuniary external effect is internalizedvia prices.Our results deliver an efficient equilibrium in the market for roommates. Ourfinding is significant because it is well known that stable matchings may not existin the model of roommates. Market equilibria capture a different notion of stability,one that is not game theoretic in nature, ensuring that agents are optimizing at theequilibrium outcome.The application to roommates is discussed in Section 5, where we also outlinehow pseudo-markets can be used in more general matching and coalition formationproblems.
ONSTRAINTS 7
Re-allocation of an existing assignment.
Consider a system that starts from apre-existing allocation, for example assigning teachers to schools (Combe, Tercieux, and Terrier,2018), offices to university faculty (Baccara, Imrohoroglu, Wilson, and Yariv, 2012),or upper-class students to dorms (Abdulkadiro˘glu and S¨onmez, 1999). We seek tore-assign the objects while ensuring each agent that they are not worse off thanunder the pre-existing allocation. The reason could be political. Re-assignmentproblems often need to overcome political economy obstacles stemming from an ex-isting assignment. By ensuring that agents are not made worse off, the politicalproblem is avoided.Our results allow agents to obtain market income from the value of their endow-ment at equilibrium prices. In consequence, we can not only achieve efficiency, butalso satisfy the participation constraints implied by the presence of endowments.It is crucial, we show, that not all income is derived from the endowment: someincome must have the same external source as in HZ.Endowments and participation constraints are discussed in Section 7.1.2.
Related literature.
Constrained resource allocation has received a lot of at-tention in recent years. The work by Kojima, Sun, and Yu (2020), Gul, Pesendorfer, and Zhang(2019) and ours seems to be the first to look at constrained allocation by wayof a market mechanism. The former two papers study the role of gross substi-tutes in a general model of discrete allocation. Despite a similar focus on mar-kets and constraints, the results in our papers are very different; see Section 8for more details. In studying constraints, we are motivated by the early work ofBudish, Che, Kojima, and Milgrom (2013), Ehlers, Hafalir, Yenmez, and Yildirim (2014)and Kamada and Kojima (2015). Our results apply to the same kind of constraintsthey consider, but we differ substantially in methodology. Aside from how we dealwith constraints, our approach to generating income from endowments is closelyrelated to, but distinct from, Mas-Colell (1992), Le (2017), and McLennan (2018).We provide a detailed discussion of the related literature in Section 8, once ourresults have been explained. We also provide a detailed comparison with other workon the pseudo-market mechanism in that section.2.
The model
Notational conventions.
For vectors x, y ∈ R n , x ≤ y means that x i ≤ y i for all i = 1 , . . . , n ; x < y means that x ≤ y and x = y ; and x ≪ y means that x i < y i for all i = 1 , . . . , n . The set of all x ∈ R n with 0 ≤ x is denoted by R n + , and ECHENIQUE, MIRALLES, AND ZHANG the set of all x ∈ R n with 0 ≪ x is denoted by R n ++ . Inner products are denoted as x · y = P i x i y i .Let X ⊆ R n be convex. A function u : X → R is • quasi-concave if for any x, z ∈ X and λ ∈ [0 , { u ( z ) , u ( x ) } ≤ u ( λz + (1 − λ ) x ) . • semi-strictly quasi-concave if it is quasi-concave, and for any x, z ∈ X and λ ∈ (0 , u ( z ) = u ( x ) implies thatmin { u ( z ) , u ( x ) } < u ( λz + (1 − λ ) x ) . • concave if, for any x, z ∈ X and λ ∈ (0 , λu ( z ) + (1 − λ ) u ( x ) ≤ u ( λz + (1 − λ ) x ) . • expected utility if there exists a vector v ∈ R n with u ( x ) = v · x for all x ∈ X . • strictly increasing if x > x ′ implies that u ( x ) > u ( x ′ ).Given a set A ⊆ R n , let co( A ) denote the convex hull of A in R n : the intersectionof all convex sets that contain A .A pair ( a, b ), with a ∈ R n and b ∈ R , defines a linear inequality a · x ≤ b . We saythat a linear inequality ( a, b ) has non-negative coefficients if a ≥ b ≥
0. Anylinear inequality ( a, b ) defines a (closed) half-space { x ∈ R n : a · x ≤ b } .A polyhedron in R n is a set that is the intersection of a finite number of closedhalf-spaces. A polytope in R n is a bounded polyhedron. Two special polytopes arethe simplex in R n : ∆ n = { x ∈ R n + : n X l =1 x l = 1 } , and the subsimplex ∆ n − = { x ∈ R n + : n X l =1 x l ≤ } . When n is understood, we use the notation ∆ and ∆ − . See Avriel, Diewert, Schaible, and Zang (2010) for a discussion of semi-strictly quasi-concavefunctions and their applications to economics.
ONSTRAINTS 9
The economy.
We first introduce a model without endowments. It is simpler,and goes a long way to capture the applications we have discussed. In Section 7 weaugment the model to include agents’ endowments.An economy is a tuple Γ = (
I, O, ( Z i , u i ) i ∈ I , ( q l ) l ∈ O ), where • I is a finite set of agents , with N = | I | ; • O is a finite set of objects , with L = | O | ; • Z i ⊆ R L + is i ’s consumption space ; • u i : Z i → R is i ’s utility function ; • q l ∈ R ++ is the amount of l ∈ O .In an economy, N = | I | is the number of agents, and L = | O | is the number ofdifferent objects . Each object l ∈ O is available in quantity q l .An assignment in Γ is a vector x = ( x i,l ) i ∈ I,l ∈ O with x i ∈ Z i , where x i,l is the amount of object l received by agent i . Let A denote the set of allassignments in Γ.In discrete allocation problems we often interpret assignments as probabilisticallocations: see Section 2.4. In this case, x i,l is the probability that agent i receivesan object l .For now we restrict attention to Z i = R L + , but agents’ consumption spaces willbe restricted further as we introduce constraints.2.3. Constraints. A constrained allocation problem is a pair (Γ , C ) in which Γ is aneconomy and C is a subset of the assignments in Γ . The assignments in C constitutethe assignments that satisfy the constraints: the feasible assignments in (Γ , C ).Observe that the set of constrained assignments is a primitive of the model. In-stead of starting from an explicit description of how assignments are constrained,we work directly with the set C of feasible assignments. Throughout the paper werequire that C be a polytope.In Section 4.1 we show how to apply our results to a model with explicit constraintstructures as the primitive.Our model applies to environments with infinitely divisible objects. In Section 7,for example, we obtain the textbook model of an exchange economy as a specialcase. Most market design applications, however, require indivisible objects. Weproceed to introduce some language that is pertinent to the indivisible case. Special case: discrete allocation.
In many market design applications, ob-jects are indivisible, and randomization over deterministic assignments is used toensure fairness. In these applications, we say that an assignment x is deterministic if every x i,l is an integer. When an assignment is not deterministic we call it arandom assignment.Constraints are often imposed as linear inequalities on deterministic assignments.For example, the usual unit-demand constraints require that P l ∈ O x i,l ≤ i ∈ I , and the supply constraints require that P i ∈ I x i,l ≤ q l for all l ∈ O . A deterministicassignment is feasible if it satisfies all such constraints. A random assignment is feasible if it belongs to the convex hull of feasible deterministic assignments. Thuswe obtain C from the set of feasible deterministic assignments. Observe that C is apolytope, as the number of feasible deterministic assignments is finite.Under unit demand and supply constraints, the Birkhoff-von Neumann theoremand its generalizations guarantee that every random assignment is a randomizationover deterministic assignments. Budish, Che, Kojima, and Milgrom (2013) general-ize this theorem by characterizing constraint structures that ensure the implemen-tation of feasible random assignments: we use their model in Section 4.1. By taking C as a primitive, we circumvent the implementation issue.2.5. Pre-processing of constraints.
Our approach involves pricing constraints,but not all constraints get a price. For example, in HZ, unit demand constraintsare not priced. Only supply constraints get a price. Here we proceed with a gen-eral constrained allocation problem (Γ , C ), and “pre-process” C so as to obtain theconstraints that have to be assigned a price.Recall that C is a polytope. Define the lower contour set of C to belcs( C ) = { x ∈ R NL + : ∃ x ′ ∈ C such that x ≤ x ′ } . Lemma 1.
There exists a finite set Ω of linear inequalities with non-negative coef-ficients such that lcs ( C ) = \ ( a,b ) ∈ Ω { x ∈ R LN + : a · x ≤ b } . Proof.
Consider D = { x ′ ∈ R NL : x ′ ≤ x for some x ∈ C} Lemma 1 is used by Balbuzanov (2019) to define a generalization of the probabilistic serialmechanism that accommodates constraints.
ONSTRAINTS 11 and note that lcs( C ) = D ∩ R NL + . Write D as C − R NL + ; thus, since C is a polytope, D is finitely generated. Then by Theorem 19.1 in Rockafellar (1970) D is polyhedral,and therefore the intersection of finitely many halfspaces. Let Ω be the set of linearinequalities ( a, b ) defining this collection of halfspaces, so for each ( a, b ) ∈ Ω we havethe halfspace { x ′ ∈ R NL : a · x ′ ≤ b } . Since for each i and l there is x ′ ∈ D witharbitrarily small x ′ i,l , we must have a ≥
0. If C = { } we may take b = 0. If there is x ∈ C with x > b ≥
0. Hence Ω defines a finite collection of linear inequalitieswith non-negative coefficients.To finish the proof, note that if z ∈ D \ lcs( C ) then z / ∈ R NL + . (cid:3) For any c = ( a, b ) ∈ Ω, define the support of c to besupp( c ) = { ( i, l ) ∈ I × O : a i,l > } . Let a i = ( a i,l ) l ∈ O be the vector of coefficients relevant to i in ( a, b ).Given that C is nonempty, there are two types of inequalities ( a, b ) ∈ Ω: thosewith b = 0 and those with b >
0. If b = 0, then for any x ∈ C we must have x i,l = 0 for all ( i, l ) ∈ supp( c ). We can, without loss of generality, assume thatthere is exactly one such inequality; because two inequalities ( a, , ( a ′ , ∈ Ω canbe substituted by ((max { a i,l , a ′ i,l } ) , b = 0 in Ωthen we can include the trivial inequality (0 ,
0) in Ω. Thus we can let ( a , ∈ Ω bethe unique inequality with b = 0. When ( a ,
0) is nontrivial, it forbids some agentsfrom consuming certain objects. We say that l is a forbidden object for agent i when a i,l > \{ ( a , } , we say ( a, b ) is an individual con-straint for agent i if for all j = i and l ∈ O , a j,l = 0. In words, ( a, b ) only restricts i ’s consumption. Let Ω i denote the set of all individual constraints for i . We use( a ,
0) and individual constraints to refine i ’s consumption space. Let X i be the setof vectors x i ∈ Z i such that x i,l = 0 if l is a forbidden object for i and x i satisfiesall of i ’s individual constraints. That is, X i = { x i ∈ Z i : a i · x i ≤ a i · x i ≤ b for all ( a, b ) ∈ Ω i } . Let Ω ∗ = Ω \ (cid:0) { ( a , } S ∪ i ∈ I Ω i (cid:1) collect all constraints that involve more than oneagent. Including, for example, any supply constraints. The elements of Ω ∗ will be“priced.” By pricing these constraints we seek to ensure that one agent’s pecuniaryexternality on others, imposed via the constraints present in C , are internalized. Normative properties.
Given a constrained allocation problem (Γ , C ), weanalyze constrained versions of efficiency and fairness: the efficiency and fairnessproperties that can be achieved subject to how assignments are constrained.A feasible assignment x ∈ C is weakly C -constrained Pareto efficient if there isno feasible assignment y ∈ C such that u i ( y i ) > u i ( x i ) for all i . And x ∈ C is C -constrained Pareto efficient if there is no feasible assignment y ∈ C such that u i ( y i ) ≥ u i ( x i ) for all i with a strict inequality for at least one agent.Fairness rules out envy among agents who are treated symmetrically by the prim-itive constraints. We say that two agents i and j are of equal type if X i = X j and,for all ( a, b ) ∈ Ω ∗ , a i = a j . An agent i envies another agent j at an assignment x if u i ( x j ) > u i ( x i ). An assignment x ∈ C is envy-free if no agent envies another agentat x , and equal-type envy-free if no agent envies another agent of equal type at x .2.7. Equilibrium.
For each constraint c = ( a, b ) ∈ Ω ∗ , we introduce a price p c .When agent i purchases x i,l , she affects other agents’ purchases through the role of a i,l in constraint c . Prices are meant to internalize such effects, just as the price ofgood l classically internalizes the effect that i has on other agents through the supplyconstraint for good l . Given a price vector p = ( p c ) c ∈ Ω ∗ ∈ R Ω ∗ , the personalizedprice vector faced by any agent i ∈ I is defined to be p i = ( p i,l ) Ll =1 such that p i,l = X ( a,b ) ∈ Ω ∗ a i,l p ( a,b ) . Remark . If agents i and j are of equal type, then p i = p j . Thus prices are onlypersonalized to the extent that constraints are personalized. We present severalapplications where all agents face the same prices.A pair ( x ∗ , p ∗ ) is a pseudo-market equilibrium for (Γ , C ) if(1) x ∗ i ∈ arg max x i ∈X i { u i ( x i ) : p ∗ i · x i ≤ } ;(2) x ∗ ∈ C ;(3) For any c = ( a, b ) ∈ Ω ∗ , a · x ∗ < b implies that p ∗ c = 0.3. Main theorem
For constrained allocation problems without endowments, our main result is:
Theorem 1.
Suppose that agents’ utility functions are continuous, quasi-concaveand strictly increasing.
ONSTRAINTS 13 • There exists a pseudo-market equilibrium ( x ∗ , p ∗ ) in which x ∗ is weakly C -constrained Pareto efficient. • If agents’ utility functions are semi-strictly quasi-concave, there exists apseudo-market equilibrium ( x ∗ , p ∗ ) in which x ∗ is C -constrained Pareto ef-ficient. • Every pseudo-market equilibrium assignment is equal-type envy-free.
Theorem 1 is implied by our more general result, Theorem 2 in Section 7. Weshould emphasize that the first welfare theorem does not hold in our model: onecan exhibit examples of Pareto inefficient pesudo-market equilibria, and even ofPareto-ranked equilibrium allocations. Crucial to Theorem 1 is the cheapest bundleproperty: ( x, p ) satisfies the cheapest-bundle property if, for each i , x i minimizesexpenditure p i · z i among all the z i ∈ X i for which u i ( z i ) = u i ( x i ). The cheapestbundle property, and its role in obtaining efficiency, was already established by HZ.We show the existence of a pseudo-market equilibrium with the cheapest-bundleproperty, which in consequence is C -constrained Pareto efficient.4. Application: Discrete object allocation under constraints
Our first application is to the problem of assigning indivisible objects, as in Sec-tion 2.4. We seek fair and efficient random assignments subject to constraints. Eachobject l ∈ O is available in fixed integer supply, and each agent demands at mostone copy of any object.We explicitly describe constraints by way of the constraint structures introducedby Budish, Che, Kojima, and Milgrom (2013). Many constraints in real-life alloca-tion problems can be described through such structures.4.1. Constraint structures. A constraint is defined by a triple ( S, q S , q S ), where S ⊂ I × O is a subset of agent-object pairs, and q S = ( q S , q S ) is a pair of non-negative integers with q S ≤ q S . The integers q S and q S are respectively called floor and ceiling quotas.An assignment x satisfies ( S, q S , q S ) if(1) q S ≤ X ( i,l ) ∈ S x i,l ≤ q S . For any i ∈ I and l ∈ O , a singleton constraint ( S, q S ) is such that S = { ( i, l ) } and q S = (0 , i can obtain at most one copy of l . For any i ∈ I ,a row constraint ( S, q S ) is such that S = { i } × O and q S = (0 , q i ) where q i ∈ N . The row constraint means that i obtains at most q i objects. Unit demand is anexample of a row constraint. For any l ∈ O , a column constraint ( S, q S ) is such that S = I × { l } and q S = (0 , q l ). This means that at most q l copies of l can be assigned.Supply constraints are examples of column constraints.A constraint structure H is a collection of constraints. The set of feasible assign-ments implied by H is defined to be C = { x ∈ R NL + : q S ≤ X ( i,l ) ∈ S x i,l ≤ q S for all ( S, q S ) ∈ H} . We assume that H contains the singleton constraints for all agent-object pairs,the row constrains for all agents and the column constraints for all objects. When H satisfies these assumptions, we say that it is allocative .When C is the convex hull of deterministic assignments satisfying the constraintsin H , we say that it is implementable . Budish, Che, Kojima, and Milgrom (2013)prove that a sufficient and necessary condition for C to be implementable for allpossible quotas is that H is a bihierarchy . Formally, a constraint structure H is a hierarchy if for every distinct S and S ′ in H , either S ⊂ S ′ , or S ′ ⊂ S , or S ∩ S ′ = ∅ . H is a bihierarchy if there exist two hierarchies H and H such that H = H ∪ H and H ∩ H = ∅ . H is the set of sub-row, row, and sup-row constraints, while H is the set of sub-column, column and sup-column constraints. When H is abihierarchy, C is the set of feasible assignments that we take as a primitive in ourmodel. Then we can apply our method directly to C . In the rest of this section, wediscuss applications with bihierarchy constraints. Note, however, that our approachis applicable to cases when H is not a bihierarchy. In such cases, one first needsto describe the convex hull of the set of deterministic assignments that satisfy theconstraints in H .We consider two applications of bihierarchy constraints. In the first application,all floor quotas are zero. Then we can directly price the constraints in H . It shouldbe clear that singleton and row constraints do not need to be priced. Column con-straints involve more than one agent, and thus generate pecuniary externalities thatmust be internalized through prices. In the second application, there are nontrivialfloor constraints. To characterize lcs( C ), we will then derive a new set of ceilingconstraints implied by the ceiling and floor constraints present in H . We discuss A constraint (
S, q S ) is a sub-row constraint if S = { i } × O ′ for some i ∈ I and O ′ ⊂ O , and itis a sup-row constraint if S = I ′ × O for some I ′ ⊂ I . Sub-column and sup-column constraints aresimilarly defined. ONSTRAINTS 15 two concrete examples to show how the new ceiling constraints are derived. Thegeneral model can be treated much like our examples.4.2.
No floor constraints.
Suppose that H is such that all floor quotas are zero.Then C = lcs( C ), and we can directly price all non-individual constraints in H : aset that we denote by H ∗ . Here individual constraints consist of singleton, sub-row,and row constraints. The set Ω ∗ is { ( S , q S ) ∈ R LN + × R + : ( S, , q S ) ∈ H ∗ } . Under our assumptions on utilities, an efficient pseudo-market equilibrium existswith prices for each constraint in H ∗ . It is interesting to discuss the fairness prop-erties of such equilibria.Two agents i and j are of equal type if X i = X j and, for all S ∈ H ∗ and l ∈ O ,( i, l ) ∈ S if and only if ( j, l ) ∈ S . We say that H is anonymous if every two agents areof equal type. If H is anonymous, every constraint in H ∗ must be a column or sup-column constraint. Under anonymous constraints, every pseudo-market equilibriumis envy-free: the strongest possible fairness property that we can obtain.An example with anonymous constraints is the Japanese medical residency matchwith regional caps studied by Kamada and Kojima (2015). Suppose that agents aredoctors and objects are hospital positions. Each constraint in H ∗ takes the form (cid:0) I × O ′ , (0 , ¯ q O ′ ) (cid:1) where O ′ ⊆ O is the set of hospitals in a geographic region (a city or a prefecture).Here ¯ q O ′ is the regional cap used to control the maximum number of doctors thatthe region O ′ can employ. A collection of such constraints is anonymous becauseeach constraint does not distinguish among the identities of individual doctors.4.3. Floor constraints.
Floor constraints are common in applications, but difficultto deal with theoretically. We discuss two examples. The first is the Japanesemedical residency match mentioned above. By introducing regional caps to restrictthe number of doctors assigned to urban hospitals, the Japanese government wantsto increase the number of doctors assigned to rural hospitals. But the government’sideal distribution of doctors can be described by a collection of constraints with both floor and ceiling quotas. (The regional caps described above can be motivatedthrough our approach to characterizing lcs( C ).) By S we denote the indicator vector of the set S . Specifically, the hospitals O are located in K disjoint regions. Accordingly, thereis a partition of hospitals O = R ∪ R ∪ · · · ∪ R K such that every R k is the set ofhospitals in a region. We refer to R k as a region. For each region R k , there is aconstraint q R k ≤ X l ∈ R k X i ∈ I x i,l ≤ q R k . Assume that there are enough hospital positions to assign each doctor a position.We can always add null hospitals when that is not the case. Also, assume that thereare enough doctors to meet all floor constraints. Below we derive the inequalitiesin Ω and show that they are anonymous. Our theorem applies to deliver a pseudo-market equilibrium that satisfies the constraints, and is efficient and envy-free.Let R = { R , R , . . . , R K } denote the set of regions. For each ℓ ∈ { , , . . . , K } ,let R ℓ be the collection of sets that are the union of ℓ distinct regions. That is, R ℓ = { R k ∪ R k ∪ · · · ∪ R k ℓ : { k , k , . . . , k ℓ } ⊂ { , , . . . , K }} . In particular, R = R .Consider the following inequalities(2) ≤ P l ∈ O x i,l ≤ i ∈ I, ≤ P i ∈ I x i,l ≤ q l for all l ∈ O, ≤ P i ∈ I,l ∈ R x i,l ≤ q R for all ℓ ∈ { , . . . , K } and R ∈ R ℓ , where q R is (re)defined according to the following procedure: • For every R ∈ R , redefine the ceiling quota to be q R = min (cid:8) q R , N − X R ′ ∈R\{ R } q R ′ (cid:9) . Note that q R ≥ q R because N ≥ P R ′ ∈R q R ′ , and q R is weakly smaller thanthe original ceiling quota. • For every R = R k ∪ R k ∈ R , define the ceiling quota to be q R = min (cid:8) q R k + q R k , N − X R ′ ∈R\{ R k ,R k } q R ′ (cid:9) . • In general, for every R = R k ∪ R k ∪ · · · ∪ R k ℓ ∈ R ℓ , define the ceiling quotato be q R = min (cid:8) q R \{ R kx } + q R kx for every x ∈ { , , . . . , ℓ } , N − X R ′ ∈R\{ R k ,...,R kℓ } q R ′ (cid:9) . We prove that lcs( C ) is characterized by the inequalities in (2). ONSTRAINTS 17
Proposition 1. lcs ( C ) = { x ∈ R NL + : x satifsies the inequalities in (2) } .Proof. We denote by A the set characterized by the inequalities in (2). It is easy tosee that A = lcs( A ). By the procedure to define q R , all elements of C satisfy (2). So C ⊂ A and thus lcs( C ) ⊂ A . To prove that A ⊂ lcs( C ), we first prove a claim. Claim.
For every ℓ ∈ { , . . . , K } , every R = R k ∪ R k ∪ · · · ∪ R k ℓ ∈ R ℓ , and every x ∈ { , . . . , ℓ } , q R ≥ q R kx + q R \{ R kx } .Proof of the claim. Base case ℓ = 2: For every R = R k ∪ R k ∈ R , if q R = q R k + q R k , then the claim holds obviously. Otherwise, q R = N − P R ′ ∈R\{ R k ,R k } q R ′ .By definition, N − P R ′ ∈R\{ R k } q R ′ ≥ q R k . So q R = N − P R ′ ∈R\{ R k ,R k } q R ′ ≥ q R k + P R ′ ∈R\{ R k } q R ′ − P R ′ ∈R\{ R k ,R k } q R ′ = q R k + q R k . Similarly, we prove that q R ≥ q R k + q R k .Induction step: Suppose the claim is true for 1 , , . . . , ℓ . Then we prove that it isalso true for ℓ + 1. For any R = R k ∪ R k ∪ · · ·∪ R k ℓ +1 ∈ R ℓ +1 , if q R = q R \{ R kx } + q R kx for some x ∈ { , , . . . , ℓ + 1 } , then it is obvious that q R ≥ q R \{ R kx } + q R kx . Bythe induction assumption, for every y = x , q R \{ R kx } ≥ q R ky + q R \{ R kx ,R ky } . So q R ≥ q R ky + q R \{ R kx ,R ky } + q R kx ≥ q R ky + q R \{ R ky } . The claim is proved.Otherwise, q R = N − P R ′ ∈R\{ R k ,...,R kℓ +1 } q R ′ . By definition, for every x , N − P R ′ ∈R\{ R k ,...,R kℓ +1 }∪{ R kx } q R ′ ≥ q R \{ R kx } . So q R ≥ q R \{ R kx } + q R kx .By induction, we are done.Define A ′ = { x ∈ A : ∄ x ′ ∈ A such that x < x ′ } . It is clear that A = lcs( A ′ ). Weprove that A ′ ⊂ C . Suppose there exists x ∈ A ′ such that x / ∈ C . Because x satisfiesall original ceiling constraints that define C , x must violate the floor constraintof some R k . That is, P i ∈ I,l ∈ R k x i,l < q R k . Then there must exist some doctor i such that P l ∈ O x i,l <
1, since otherwise P i ∈ I,l ∈ O \{ R k } x i,l = N − P i ∈ I,l ∈ R k x i,l >N − q R k ≥ q O \{ R k } , which contradicts the assumption that x ∈ A ′ ⊂ A . Because q R k ≤ P l ∈ R k q l , there must exist l ∈ R k such that P i ∈ I x i,l < q l . Now consider anew assignment x ′ such that x ′ i,l = x i,l + ǫ where 0 < ǫ < min { − P l ∈ O x i,l , q l − P i ∈ I x i,l , q R k − P i ∈ I,l ∈ R k x i,l } , and x ′ coincides with x in the other cells. So x < x ′ .Below we prove that x ′ ∈ A , which contradicts the assumption that x ∈ A ′ .Suppose towards a contradiction that x ′ / ∈ A . Let ℓ > R ∈ R ℓ with P i ∈ I,l ∈ R x ′ i,l > q R . It is clear that R k ⊂ R . By Claim, q R ≥ q R k + q R \ R k . So X i ∈ I,l ∈ R x ′ i,l > q R k + q R \ R k . Because ǫ is chosen such that P i ∈ I,l ∈ R k x ′ i,l < q R k . So X i ∈ I,l ∈ R \ R k x ′ i,l > q R \ R k . But it means that P i ∈ I,l ∈ R \ R k x i,l > q R \ R k , which contradicts x ∈ A . So x ′ ∈ A . (cid:3) Besides unit demand constraints, the other inequalities in (2) do not distinguishamong the identities of doctors. So every pseudo-market equilibrium is envy-free.The second application we discuss is controlled school choice. When implementingschool choice, a consideration for many school districts is demographic diversity. Wepresent a model in which the students I are simply classified into minorities I m andmajorities I M . Let the number of minorities be N m and the number of majoritiesbe N M . Each school l has a pair of quotas ( q ml , q ml ) for minorities, and a pair ofquotas ( q Ml , q Ml ) for majorities. So aside from supply constraints, each school l hasthe constraints q ml ≤ X i ∈ I m x i,l ≤ q ml ,q Ml ≤ X i ∈ I M x i,l ≤ q Ml . Of course, we assume that q ml + q Ml ≤ q l .The inequalities to characterize lcs( C ) can be derived similarly to how we dealtwith regional hospitals above. The only difference is that we need to take intoaccount the interaction between the quotas for the two student types within eachschool. After that, we can deal with the assignments for two types separately.Formally, consider the following inequalities(3) ≤ P l ∈ O x i,l ≤ i ∈ I, ≤ P i ∈ I x i,l ≤ q l for all l ∈ O, ≤ P i ∈ I m ,l ∈ O ′ x i,l ≤ q mO ′ for all nonempty O ′ ⊂ O, ≤ P i ∈ I M ,l ∈ O ′ x i,l ≤ q MO ′ for all nonempty O ′ ⊂ O, where q mO ′ and q MO ′ are (re)defined as follows: Our arguments extend to more than two types.
ONSTRAINTS 19 • For every l ∈ O , redefine the ceiling quotas to be q ml = min { q ml , q l − q Ml , N m − X l ′ ∈ O \{ l } q ml ′ } ,q Ml = min { q Ml , q l − q ml , N M − X l ′ ∈ O \{ l } q Ml ′ } . • For every non-singleton O ′ ⊂ O , (inductively) define the ceiling quotas to be q mO ′ = min { q mO ′ \{ l } + q ml for every l ∈ O ′ , N m − X l ′ ∈ O \ O ′ q ml ′ } ,q MO ′ = min { q MO ′ \{ l } + q Ml for every l ∈ O ′ , N M − X l ′ ∈ O \ O ′ q Ml ′ } . Similarly as before, lcs( C ) is characterized by the inequalities in (3). Proposition 2. lcs ( C ) = { x ∈ R NL + : x satifsies the inequalities in (3) } . The proof of Proposition 2 is similar to that of Proposition 1 and thus omitted.Note that besides unit demand constraints, the other inequalities in (3) do notdistinguish among the identities of the students of each type. So every pseudo-market equilibrium is envy-free among the students of each type. That is, minoritystudents will not envy other minority students, and majority students will not envyother majority students.5.
A market for roommates, and other problems
Our model accommodates very general assignment problems with constraints, in-cluding models with non-implementable constraints. We discuss coalition formationproblem as an illustration of the power of our approach.First we consider the roommate model, arguably the best-known example inmatching theory where game-theoretic stability solutions fail to exist. As a corollaryof our main theorem, we obtain the existence of efficient pseudo-market equilibriumassignments. Equilibria embody a form of stability: optimizing agents do not wantto change their behavior in the market. In this sense, our results offer a possibleway out of the non-existence of stable matchings.Consider a set of agents I that constitute the potential roommates or partners.Let O be a copy of I , so N = L , and think of i ∈ O as the alter ego of agent i ∈ I .If x is an assignment, interpret x i,j = 1 as agents i and j forming a partnership, orbecoming roommates. When i is alone without a roommate, we have x i,i = 1. In consequence, we restrict attention to assignments x where x i,j = x j,i , meaning thatthe matrix ( x i,j ) i ∈ I,j ∈ I is symmetric .We say that an assignment x is a matching if (1) x i,j ∈ { , } for all ( i, j ) ∈ I × I , (2) x is symmetric, (3) x satisfies the unit demand constraints with equality( P j x i,j = 1) and (4) x satisfies the allocation constraints with equality ( P i x i,j =1). Define C to be the convex hull of all matchings.Note that C is not equal to the set of symmetric assignments that satisfy the unitdemand and allocation constraints, dropping the integrality constraint x i,j ∈ { , } .Katz (1970) proves that the latter set is the convex hull of all matrices of the form(1 / P + P ′ ) ( P ′ is the transpose of P ) where P is a permutation matrix withno even cycles greater than 2. A celebrated result of Edmonds (1965) provides acharacterization of C , which we use in the proof of Proposition 3 below.To operationalize our approach, we need to work out the set of inequalities Ω forthe roommates problem. To this end, let F be the set of subsets F ⊆ I × I suchthat (1) for all i , ( i, i ) / ∈ F and (2) for every ( i, j ) ∈ F , ( j, i ) / ∈ F . For each F ∈ F ,let G F be the graph with vertex set I and edge set {{ i, j } : ( i, j ) ∈ F or ( j, i ) ∈ F } .Denote the cardinality of the maximum independent edge set of G F by k F . Forevery i ∈ I , let J i be the set of subsets J ⊂ ( { i } × I ) ∪ ( I × { i } ) such that ( i, i ) ∈ J and for every j = i , either ( i, j ) ∈ J or ( j, i ) ∈ J but not both. Then lcs( C ) ischaracterized by the following inequalities. Proposition 3. lcs ( C ) = (cid:18) \ ∅6 = F ∈F { x ∈ R I × I + : X ( i,j ) ∈ F x i,j ≤ k F } (cid:19) \ (cid:18) \ i ∈ I,J ∈J i { x ∈ R I × I + : X ( i ′ ,j ′ ) ∈ J x i ′ ,j ′ ≤ } (cid:19) . Proof.
Let D denote the set on the right-hand side of the proposition. We first provethat D ⊂ lcs( C ). For every x ∈ D , consider the matrix x ′ obtained by letting x ′ i,j =max { x i,j , x j,i } for all ( i, j ) ∈ I × I . Then x ′ is symmetric and x ≤ x ′ . We prove that x ′ ∈ D . For any ∅ 6 = F ∈ F , suppose to the contrary that P ( i,j ) ∈ F x ′ i,j > k F . Thenwe define F ′ ⊂ I × I such that for every ( i, j ) ∈ F , if x i,j ≥ x j,i , let ( i, j ) ∈ F ′ , andotherwise let ( j, i ) ∈ F ′ . So G F ′ and G F have the same (undirected) edge set, andthus k F = k F ′ . However, P ( i,j ) ∈ F ′ x i,j = P ( i,j ) ∈ F x ′ i,j > k F ′ , which contradicts that x ∈ D . Similarly we can prove that for every i and every J ∈ J i , P ( i ′ ,j ′ ) ∈ J x ′ i ′ ,j ′ ≤ x ′ ∈ D .Now define another matrix y by (1) for every ( i, j ) ∈ I × I with i = j , set y i,j = x ′ i,j , and (2) for every i ∈ I , y i,i = 1 − P j = i x ′ i,j . It is clear that y is ONSTRAINTS 21 symmetric and that x ′ ≤ y (as { i } × I ∈ J i ). For any F ∈ F , ( i, i ) / ∈ F ; hence P ( i,j ) ∈ F y i,j = P ( i,j ) ∈ F x ′ i,j ≤ k F . Since x ′ is symmetric and x ′ ∈ D , for every i andevery J ∈ J i , P ( i ′ ,j ′ ) ∈ J y i ′ ,j ′ = 1. So y ∈ D and it is a bistochastic matrix.Now we prove that y ∈ C . Edmonds (1965) proves that a symmetric bistochasticmatrix z belongs to C if and only if for every r ∈ N and every I ′ ⊂ I with | I ′ | = 2 r +1, P ( i,j ) ∈ F z i,j ≤ r , where F ⊂ I ′ × I ′ is such that there does not exist ( i, i ) ∈ F and forevery ( i, j ) ∈ I ′ × I ′ with i = j , either ( i, j ) ∈ F or ( j, i ) ∈ F but not both. For anysuch F , k F = r because I ′ is odd and we can form r pairs among the 2 r elementsof I ′ that can be paired. Since F ∈ F , then, y satisfies Edmonds’ inequalities andthus y ∈ C . Since x ≤ x ′ ≤ y , x ∈ lcs( C ). This means that D ⊂ lcs( C ).To prove lcs( C ) ⊂ D , consider any x ∈ C . Then x is the convex combination ofdeterministic matchings x k . For each ∅ 6 = F ∈ F and each i , there is at most one j with x ki,j = 1. By the definition of independent edge set, then P ( i,j ) ∈ F x ki,j ≤ k F . So P ( i,j ) ∈ F x i,j ≤ k F . It is clear that x satisfies the other inequalities related to every J i . So x ∈ D . Then it means that lcs( C ) ⊂ D . (cid:3) A pseudo-market equilibrium implies a random matching x ∗ (a probability distri-bution over matchings) that is Pareto efficient. Of course, x ∗ needs not be stable inthe game theoretic sense, but it corresponds to individual agents’ optimizing behav-ior, as long as these agents take prices as given. Price taking behavior is a plausibleassumption in a large centrally-run market for partnerships, like for example a mar-ket for roommates in college dormitories. A pseudo-market could be set up by thecollege, and equilibrium prices could be enforced.We finalize with a numerical example where stable matchings fail to exist, butwhere our results deliver an efficient equilibrium (a market stability of sorts). Example . Let I = { , , } and L = 3. For agent i ,consuming object l is the same as having agent l as her roomate. Suppose that theagents’ utilities are 1 2 31 0 1 22 2 0 13 1 2 0With these preferences, there are no stable matchings. However, there is a HZequilibrium. In the equilibrium, the price of the following constraint is two: x , + x , + x , ≤ , the price of the following constraint is one: x , + x , + x , ≤ , and the price of every other constraint is zero. Then, agent 1’s personalized pricevector is (0 , , , , , , / , / , / Coalition formation.
The application to roommates can be adapted to ageneral coalition-formation problem. Given a set of agents I , let O be the set of all coalitions from I ; that is, O = 2 I \{∅} . A deterministic assignment is a partitionof agents into coalitions, and can be represented by a matrix x ∈ { , } NL suchthat x i,l = 1 if and only if i joints the coalition l ∈ O . Unit demand constraintswill imply that agents are members of a single coalition. We may then let C bethe convex hull of the set of deterministic assignments. Then there exists a pseudo-market equilibrium, and the equilibrium assignment is a probability distributionover coalitions.5.2. Combinatorial allocation.
Our methods can be used to solve general combi-natorial assignment and matching problems (Budish, 2011, 2012). Here we discussallocation problems in which agents demand a bundle of objects, as in course allo-cation. In contrast with Budish (2011), our emphasis is on random allocations, sothere are no problems arising from the lack of convexity of deterministic allocations.There are obvious supply constraints, stemming from course capacities, but courseallocation may exhibit additional, and more problematic, constraints. For example,if a school regards two courses l and l ′ as complements, students must take both ofthem or neither. Then we have the constraint x i,l = x i,l ′ . If the school regards twocourses l and l ′ as substitutes, so that students have to take at most one of them,then we have the constraint x i,l · x i,l ′ = 0.The set of feasible (random) assignments in course allocation problems cannot beeasily characterized. In particular, an assignment that seems ex-ante feasible maynot be actually implementable. The bihierarchy condition is not met. For example,suppose there are three agents 1 , , a, b, c . Each object has onecopy. The set of bundles is O = { ab, ac, bc } . The following random assignment looksex-ante feasible because it satisfies unit demand constraints of agents and allocationconstraints of objects. But it is not feasible because bundles are not independentobjects. When a bundle is assigned, the other two bundles become unavailable. ONSTRAINTS 23 i ab ac bc A be the basic set of “items,” each of which has a number of copies.Let O ⊂ A be the set of bundles under consideration. A deterministic assignmentis represented by a matrix x ∈ { , } NL such that x i,l = 1 if and only if i obtainsthe bundle l ∈ O . Let C be the convex hull of the set of deterministic assignments.Starting from C , one needs to pre-process lcs( C ) and our theorem will deliver apseudo-market equilibrium with the desirable normative properties.6. A market for “bads”
So far we have assumed that objects are “goods,” in the sense that agents’ utilityfunctions are monotone increasing. In some applications, however, objects representduties, or tasks, that agents dislike. Yet another application is to waste disposal,or pollution. A certain minimum amount of such “bads” have to be allocated; thequestion is to whom, and in which quantities?The presence of bads gives rise to floor constraints, but we cannot use our previousmethods directly as all agents will choose zero consumption from their consumptionspace. We can, however, borrow an idea from the standard model of labor markets:labor supply is often described as consumption of leisure. We endow every agentwith a copy of every “bad,” and allow them to buy the options of not consuming abad. Such options become “goods,” and our previous methods apply.Specifically, for every l ∈ O , q l denotes the minimum number of copies of l thathave to be assigned. Every agent can be assigned at most one object (unit demand).If P l ∈ O q l = N , then every agent must obtain an object so that the problem becomesthe one studied by Hylland and Zeckhauser (1979). Assume then that P l ∈ O q l < N .For every x ∈ ∆ − and every i ∈ I , u i ( x ) is strictly decreasing in x : if x ′ > x , then u i ( x ′ ) < u i ( x ).We consider a dual problem ( I, ˜ O, ˜∆ − , (˜ u i ) i ∈ I , ( q ˜ l ) ˜ l ∈ ˜ O ) in which • the set of objects is ˜ O = { ˜ l } l ∈ O where every ˜ l is an artificial object dual to l ∈ O , and its supply is q ˜ l = N − q l . When an agent i consumes an amount a of ˜ l , it is understood that i consumes 1 − a of l . Because at least q l of l need to be assigned, the number of copies of ˜ l is N − q l . • The consumption space for every agent is ˜∆ − = { x ∈ R L + : x ˜ l ∈ [0 ,
1] for every l ∈ ˜ O, P ˜ l ∈ ˜ O x ˜ l ∈ [ L − , L ] } . So the amount of objects in O that i will consumeis L − P ˜ l ∈ ˜ O x ˜ l ∈ [0 , • Every agent i has the utility function ˜ u i such that for every x ∈ ˜∆ − , ˜ u i ( x ) = u i ( − x ). When u i is (semi-strictly) quasi-concave and strictly decreasing,˜ u i is (semi-strictly) quasi-concave and strictly increasing.In the dual problem, agents can consume multiple artificial objects. We imposefloor constraints on individual consumption, and can derive the inequalities to char-acterize lcs( C ) as in Section 4.3. Then Theorem 1 applies to give a desirable outcome.We omit the details.7. Endowment and α -slack equilibrium We turn to a version of our model in which objects are initially owned by agentsas endowments. Endowments are important in market design when the purpose isto re-assign resources. Often, one wants to improve on an existing allocation. It isthen important to be able to respect agents’ property rights. Moreover, there aremodels (such as time banks, briefly discussed in 7.5), in which the agents themselvesprovide the goods that are to be allocated.7.1.
The economy and equilibrium.
Now an economy is a tuple Γ = (
I, O, ( Z i , u i , ω i ) i ∈ I ),where • I is a finite set of agents ; • O is a finite set of objects ; • Z i ⊆ R L + is i ’s consumption space ; • u i : Z i → R is i ’s utility function ; • ω i ∈ Z i is i ’s endowment .The aggregate endowment is denoted by ¯ ω = P i ∈ I ω i . For every l ∈ O , ¯ ω l is theamount of l in the economy.A constrained allocation problem with endowments is a pair (Γ , C ) in which Γ isan economy and C is a set of feasible assignments such that(1) C is a polytope;(2) ω = ( ω i ) i ∈ I ∈ C ; that is, ω is feasible. Re-assignment problems give rise to political economy issues. The most basic issue is to ensurethat agents are not hurt in the re-allocation; that their property rights are respected.
ONSTRAINTS 25
A feasible assignment x ∈ C is acceptable to agent i if u i ( x i ) ≥ u i ( ω i ); x is individually rational (IR) if it is acceptable to all agents. We also define a notion ofapproximate individual rationality: for any ε > x is ε -individually rational ( ε -IR)if u i ( x i ) ≥ u i ( ω i ) − ε for all i ∈ I .Let X i and Ω ∗ be defined as before. We say two agents i and j are of equal type if ω i = ω j , X i = X j , and for all ( a, b ) ∈ Ω ∗ , a i = a j .In a textbook exchange economy, Walrasian equilibrium assumes that agents’ in-comes equal the value of their endowments at equilibrium prices. However, whenconsumption space is bounded, agents may have satiated preferences. Then Wal-rasian equilibrium may not exist; see Example 2 in Section 7.3.Our method to solve the nonexistence problem is to introduce an arbitrarily smallexogenous budget. Given any price vector p , let p i be the personalized price vectorfaced by i , as defined in Section 2.7. Then for any α ∈ [0 , i ’s budget be α + (1 − α ) p i · ω i . So i ’s income is a convex combination of the exogenous budget of 1 used in HZ (andin our model of Section 2), and the market value of i ’s endowment.For any α ∈ [0 , x ∗ , p ∗ ) is an α -slack equilibrium if(1) x ∗ i ∈ arg max x i ∈X i { u i ( x i ) : p ∗ i · x i ≤ α + (1 − α ) p ∗ i · ω i } ;(2) x ∗ ∈ C ;(3) For any c = ( a, b ) ∈ Ω ∗ , a · x ∗ < b implies that p ∗ c = 0. Remark . Textbook Walrasian equilibria are 0-slack equilibria. The pseudo-marketequilibria we have already discussed in detail are 1-slack equilibria.7.2.
Results.
We assume that for each c ∈ Ω ∗ , P ( i,l ) ∈ supp ( c ) ω i,l >
0. A sufficientcondition for this assumption is that every agent owns a positive amount of everyobject. Our next result is a generalization of Theorem 1.
Theorem 2.
Suppose that agents’ utility functions are continuous, quasi-concaveand strictly increasing. For any α ∈ (0 , : • There exists an α -slack equilibrium ( x ∗ , p ∗ ) , and x ∗ is weakly C -constrainedPareto efficient. • If agents’ utility functions are semi-strictly quasi-concave, there exists an α -slack equilibrium assignment x ∗ that is C -constrained Pareto efficient. • Every α -slack equilibrium assignment is equal-type envy-free. Theorem 2 ensures that we can choose α ∈ (0 ,
1] arbitrarily, but since prices areendogenous it is not clear that the nominal magnitude of α has any actual meaning.Our next result shows that it does. In fact, by choosing α arbitrarily small we ensurethat agents’ budgets approximate the market values of their endowments. In con-sequence, the α -slack equilibrium obtained is approximately individually rational. Theorem 3.
Suppose that agents’ utility functions are continuous, semi-strictlyquasi-concave and strictly increasing. For any ε > , there is α ∈ (0 , and an α -slack equilibrium ( x ∗ , p ∗ ) such that x ∗ is C -constrained Pareto efficient and max { u i ( y ) : y ∈ X i and p ∗ i · y ≤ p ∗ i · ω i } − u i ( x ∗ i ) < ε. In particular, x ∗ is ε -individually rational. The Hylland and Zeckhauser example.
A major application of Theorem 2is to the object allocation model under the supply and unit demand constraints.That is, X i = ∆ − for every i and x ∈ C if and only if P i x i = ¯ ω . There are exactly L inequalities in Ω ∗ , one for each object l , expressing that P i x i,l ≤ ¯ ω l . All agentsface equal personalized prices, so we write p l for the price of l .We present an example due to Hylland and Zeckhauser (1979) showing that aWalrasian equilibrium (a 0-slack equilibrium) may not exist, and show how thesymmetric Pareto efficient assignment in the example can be sustained as an α -slack equilibrium with any α ∈ (0 , Example . Given is an economy with three agents 1 , , a, b .Object a has one copy and b has two copies. Agents have the following von-NeumannMorgenstern utilities: i u i,a u i,b ω i = (1 / , /
3) for i = 1 , , Claim 1.
There is no Walrasian equilibrium in Example 2.Proof.
Suppose (towards a contradiction) that ( x, p ) is a Walrasian equilibrium.Suppose first that p b > p a = 0, then 1 and 2 would each buy one copy of a , which isa contradiction. So p a must be positive. The preferences of agents imply that 1 ONSTRAINTS 27 and 2 must each obtain a half of a . Therefore, 1 / p a + 2 / ≥ / p b , and we obtain p a ≤
4. However, if p a <
4, 1 and 2 would spend all of their budgets on A , andeach obtain more than a half of a , which is a contradiction. So it must be that1 / p a + 2 / / p a and p a = 4. But this means that at most 3 demands b and b must have excess supply, which is a contradiction.Now suppose p b = 0 and p a >
0. Then 3 must obtain one copy of b . Since p a ispositive, 1 and 2 must each obtain a half of a . However, their budget 1 / p a cannotafford such a consumption. (cid:3) Consider the assignment x defined by: i x i,a x i,b Claim 2.
For any α ∈ (0 , , the price vector p = ( α α , and the assignment x constitute an α -slack equilibrium in Example 2.Proof. For any α ∈ (0 ,
1] and i = 1 , , α + (1 − α ) p · ω i = α + (1 − α ) 2 α α = 3 α α = p · x i . With such budgets, agents 1 and 2 can only afford a 1 / a and a 1 / b , which is the best consumption for them. Agent 3 chooses a copy of b for free. (cid:3) Note that in the above α -slack equilibrium, the endogenous value of agents’ en-dowments is 2 α/ (1 + 2 α ). So the value of the exogenous part of the budget relativeto the value of the endogenous part is α (1 − α ) 2 α α → α →
0. So when α shrinks to zero, the value of the exogenous income is notnegligible. In the same spirit, the following proposition shows that the averageendogenous budget will always be below the exogenous budget of one. This meansthat the economy needs outside “money.” Proposition 4. If ( x, p ) is an α -slack equilibrium, then N N X i =1 p · ω i ≤ Proof.
Note that p · ( x i − ω i ) ≤ α (1 − p · ω i ). Sum over i to obtain:0 = p · X i x i − ¯ ω ! ≤ α ( N − p · ¯ ω ) . (cid:3) A market-based fairness property.
In the object allocation model of Sec-tion 7.3, agents face identical prices. It is possible to use our result to develop akind of fairness property in the presence of endowments. Fairness, in the sense ofabsence of envy, is generally incompatible with individual rationality. Imagine aneconomy with two objects, where both agents prefer object 1 over object 2, and allthe endowment of object 1 belongs to agent 1. Then, in any allocation, there willeither be envy, or agent 1’s individual rationality will be violated. So fairness hasto be amended to account for the presence of endowment. In the object allocation model with supply and unit demand constraints, in any α -slack equilibrium, if agent i envies agent j then it must be that j ’s endowmentis worth more than i ’s at equilibrium prices. In a sense, this means that agentscollectively value j ’s endowment more than i ’s. Our next result formalizes this idea. Proposition 5.
In the object allocation model with supply and unit demand con-straints, suppose that agents’ utility functions are concave and C . Let ( x, p ) bean α -slack equilibrium. Denote by S = { i : u i ( x i ) = max { u i ( z i ) : z i ∈ ∆ − }} the setof satiated agents, and by U = I \ S the set of others. Suppose that P i ∈ U x i ≫ .If i envies j in x ( u i ( x j ) > u i ( x i ) ), then p · ω j > p · ω i , and there exists welfareweights θ ∈ R U ++ such that if v ( t ) = sup { X i ∈ U θ i u i (˜ x i ) : (˜ x i ) ∈ ∆ U − and X i ∈ U ˜ x i ≤ ¯ ω + t ( ω i − ω j ) − X i ∈ S x i } , then ( x i ) i ∈ U solves the problem for v (0) , and v ( t ) < v (0) for all t small enough. The meaning of Proposition 5 is that if an agent i envies agent j then j ’s en-dowment is more valuable than i ’s in two senses. First, it is more valuable atequilibrium prices. Second, the higher price valuation translates into a statementabout how much agents value the endowments. In particular, j ’s endowment ismore valuable than i ’s to a coalition of players U (a coalition that includes i !). It is The paper by Echenique, Miralles, and Zhang (2020) deals exclusively with this problem, butproposes a very different solution. A function with domain D is C if it can be extended to a continuously differentiable functiondefined on an open set that contains D . ONSTRAINTS 29 more valuable to U in the sense that there are welfare weights for the members of U such that a change in agents’ endowment towards having more of i ’s endowmentand less of j ’s leads to a worse weighted utilitarian outcome. The result requiresthat P i ∈ U x i ≫ ω j we do not force someagent to consume negative quantities of some object. A market for time exchange.
In organizations such as time banks , membersexchange time and skills without using monetary transfers. A time exchange prob-lem can be described as an object allocation model with endowments. Formally, O is the set of service types. For each agent i and service l , ω i,l is the amount of l that i can provide. We could require that P l ∈ O ω i,l ≤ i and every agent’sdemand be no greater than one. Here “one” could mean one day, one week, orone month. Services can be regarded as divisible because time is divisible. But ofcourse, in real life time is often measured in integers such as hours, days, or weeks.Theorem 2 implies that we can find a market equilibrium to the problem. The valueof an agent’s endowment at equilibrium prices shows how much his endowment isvalued by all agents. When the value is higher, the agent is rewarded by a betterassignment. 8. Related Literature
Constrained resource allocation has received a lot of attention in recent years.Budish, Che, Kojima, and Milgrom (2013) identify the bihierarchy structure of con-straint blocks in the assignment matrix as the sufficient and necessary condition forimplementation. Akbarpour and Nikzad (forthcoming) extend this result by relax-ing some constraints and considering approximate implementation. We circumventthe implementation issue by taking the set of implementable assignments as theprimitive. Budish et al. allow for floor constraints in implementation but rule outthem in their applications. In their extension of the pseudo-market mechanism, theyconsider column constraints, row constraints and sub-row constraints. By incorpo-rating all row and sub-row constraints into agents’ consumption spaces, they provethe existence of equilibria much like Hylland and Zeckhauser’s. Their extension isa special case of ours. We can deal with more general constrains on both rows andcolumns, and allow for floor constraints. When there are no floor constraints, we The P i ∈ U x i ≫ ω j,l > P i ∈ U x i,l > See Andersson, Cseh, Ehlers, and Erlanson (2019) for more description of real-life time banks. directly price ceiling constraints, and when there are floor constraints, we translatefloor constraints into a different set of ceiling constraints.Ehlers, Hafalir, Yenmez, and Yildirim (2014) focus on the problem of controlledschool choice (which was introduced by Abdulkadiro˘glu and S¨onmez (2003)), wherebyschool children have to be assigned seats at different schools to satisfy some diversityobjective. Kamada and Kojima (2015) are mainly (but not exclusively) motivatedby the problem of allocating doctors to hospitals to satisfy geographic quotas. Theobjective of the quotas is to avoid an excessive concentration of doctors in urbanareas. Both papers proceed by adapting the notion of stability to capture thepresence of constraints, and to add structure to the constraints being considered.To address more general constraints, Kamada and Kojima (2019) relax stability andfocus on feasible, individually rational, and fair assignments. They demonstrate thatthe class of general upper-bound constraints on individual schools are the most per-missive constraints under which a student-optimal fair matching exists. That classrules out floor constraints. Our paper can deal with the same kinds of constraintsin the above papers, but we follow a different methodological tradition. Insteadof a two-sided game-theoretic matching model, we consider object allocation andpropose a competitive equilibrium solution. The above papers also investigate therole of incentives in their mechanisms. We expect our pseudo-market mechanismto be incentive compatible in large markets, but we choose to focus on existence,efficiency and fairness. The recent work of Balbuzanov (2019) considers a version of the probabilisticserial mechanism for object allocation subject to constraints. Like us, he works ona one-sided object allocation model, but the focus on probabilistic serial makes hisanalysis clearly distinct from ours. We borrow from this paper the idea, expressedin Lemma 1, allowing us to focus on non-negative linear inequalities.The use of markets over lottery shares to solve centralized allocation problemswas first proposed by Hylland and Zeckhauser (1979). They assume no constraintsother than unit demands and limited supply. They impose a fixed income for each Controlled school choice is also investigated by, among others, Ehlers (2010),Hafalir, Yenmez, and Yildirim (2013), Kominers and S¨onmez (2013), Westkamp (2013),Echenique and Yenmez (2015), Fragiadakis and Troyan (2017), Aygun and B´o (2017), andNguyen and Vohra (2017). See Kamada and Kojima (2017) for an overview. He, Miralles, Pycia, and Yan (2018) prove the asymptotic strategy-proofness of their pseudo-market mechanism.
ONSTRAINTS 31 agent, independent of prices. They also emphasize that equilibrium may not be ef-ficient, and introduce the “cheapest bundle” property that we employ as well in ourversion of the first welfare theorem. Many other papers have followed Hylland andZeckhauser in analyzing competitive equilibria as solutions in market design; seefor instance, Budish (2011), Ashlagi and Shi (2015), Hafalir and Miralles (2015),He, Miralles, Pycia, and Yan (2018). Miralles and Pycia (2020) establish the sec-ond welfare theorem for the market with satiated preferences and token money:every Pareto efficient assignment may be supported in a Walrasian equilibrium withproperly chosen budgets. But none of these papers consider constrained allocationproblems.Hylland and Zeckhauser make the point that an equilibrium may not exist ina model with endowments. Like us, Mas-Colell (1992), Le (2017) and McLennan(2018) also propose to avoid the non-existence issue by means of a hybrid incomebetween the exogenous budget and the endogenous Walrasian income. A versionof the hybrid model was first introduced by Mas-Colell (1992), who presents anexistence result with income that is the sum of a fixed income and a price-dependentincome. His result requires the first component to be determined endogenously aspart of the fixed point argument in the equilibrium existence result. Aside fromthe presence of constraints, our result differs from his by allowing us to obtainapproximate individual rationality with a small exogenous α . In Le’s (2017) notionof equilibrium, two identical objects may have different prices. As a consequence,there may be envy among identical agents, and it may be necessary for some agentsto purchase a more expensive copy of an object when a cheaper one is available. Envy among equals is problematic for normative reasons, and it is hard to implementsuch equilibria in a decentralized fashion. In Example 2, a Le’s equilibrium is as follows. Let p = (100 , , ) be a price vector in whichthe latter two elements are the prices of the two copies of B. Then all agents have an income of101 /
2. The unique optimal bundle for agents 1 and 2 is x i = (1 / , / , B , so x = (0 , , ω = (1 / , / , / ω =(1 / , / , / ω = (1 / , / , / p = (100 , , ) is still an equilibrium price, with x = ( , , x = ( , , x = (0 , ,
1) being the equilibrium assignment. Observethat agent 1 envies 2, despite they have the same utility and the same endowment: 1 / A and2 / B . One could interpret different prices for different copies of the same object as a novel endogenoustransfer scheme, but we are unaware of a normative defense of this idea.
McLennan (2018) presents an existence result for equilibrium with “slack” in ageneral model that allows for production and encompasses our model as a specialcase. But his notion of equilibrium with slack differs from ours in important ways.Agents in his (and our) model may be satiated, and his notion of slack controls thedistribution of transfers from satiated agents who spend less than their income tounsatiated agents. In contrast, our α parameter controls the role of endowments,allowing for α to specify the weight of equal incomes vs. (unequal) endowments. Infact, it is possible to construct an example to illustrate the difference between thetwo notions of equilibrium. In the example no agents are satiated, so the slack inMcLennan’s notion has no role to play, and his equilibrium allocations are indepen-dent of α ; in contrast, our equilibrium allocations range from equal division to theautartical consumption of endowments, as α ranges from placing all weight on theexogenous income, to placing all weight on initial endowments. Kojima, Sun, and Yu (2020) and Gul, Pesendorfer, and Zhang (2019) considermarket equilibrium in economies with gross substitutes utilities and constraints. Ko-jima et. al characterize the constraints that preserve the gross substitutes propertyof firms’ demands in a transferable utility model (Kelso and Crawford’s (1982) jobmatching model). Gross substitutes ensure equilibrium existence, and the authorsshow that the constraint structures have to take the form of “interval constraints.”Gul et. al prove the existence of equilibrium in economies with a finite numberof indivisible objects, and limited transfers or no transfers. They show that withlimited transfers or no transfers, equilibrium requires random allocations and canbe approached by the equilibrium with full transfers. They also show that equilib-rium allocations satisfying certain constraints can be constructed by building theseconstraints into utility functions or incorporating them into a production technol-ogy. Different from them, we price constraints and can accommodate more generalpreferences and constraint structures.Related to our applications, Manjunath (2016) proposes a competitive equilib-rium notion for a two-sided fractional matching market. The double-indexed pricesystem in his notion resembles our personalized price system, but he needs todeal with both sides’ preferences. As a consequence, his equilibrium exists whenthere are transfers, but only approximately exists when transfers are forbidden.Andersson, Cseh, Ehlers, and Erlanson (2019) propose a time exchange model inwhich each agent provides a distinct service and has dichotomous preferences. They We are grateful to Andy McLennan for this example, which can be found in his paper.
ONSTRAINTS 33 propose a priority mechanism to maximize the amount of exchanges among agents.Differently, in our model of time exchange an agent can provide multiple servicesand different agents can provide the same service. Agents can express richer prefer-ences and the prices in our market solution reveal on which service agents have moredemand. Bogomolnaia, Moulin, Sandomirskiy, and Yanovskaia (2017, 2019) studythe competitive equilibrium allocation of a mixed manna that contains “goods” and“bads”. They prove that an equilibrium always exists. Our model is different thantheirs in that agents have unit-demand constraints. So their existence result doesnot hold in our paper.Finally, the recent work by Root and Ahn (2020) looks at constrained allocationfrom a mechanism design perspective. They allow for very general constraint sets,and prove a characterization of group strategy proof rules.9.
Proof of Theorem 1 and Theorem 2
We first prove the theorem by assuming that all utility functions are semi-strictlyquasi-concave. We then explain the differences when utility functions are only quasi-concave. We also explain how the proof works for Theorem 1.With an abuse of notation, we write P l ∈ O p i,l x i,l as p i · x i . For each c ∈ Ω ∗ , wehave assumed that P ( i,l ) ∈ supp ( c ) ω i,l >
0. It implies that P ( i,l ) ∈ supp ( c ) a ci,l ω i,l > c = ( a c , b c )).We define a price ceiling¯ p = 2 N min c ∈ Ω ∗ P ( i,l ) ∈ supp ( c ) a ci,l ω i,l , and a price space P = [0 , ¯ p ] | Ω ∗ | .Given α ∈ (0 , p ∈ P , define v i = max { u i ( x i ) : x i ∈ X i } ,B i ( p, α ) = { x i ∈ X i : p i · x i ≤ α + (1 − α ) p i · ω i } ,d i ( p ) = argmax { u i ( x i ) : x i ∈ B i ( p, α ) } ,d i ( p ) = argmin { p · x i : x i ∈ d i ( p ) } ,V i ( p ) = max { u i ( x i ) : x i ∈ B i ( p, α ) } . Lemma 2. If V i ( p ) < v i then d i ( p ) = d i ( p ) .Proof. Let x i ∈ d i ( p ). We shall prove that p i · x i = α + (1 − α ) p i · ω i , whichmeans we are done because it implies that all bundles in d i ( p ) cost the same at prices p . Let z i ∈ X i be such that u i ( z i ) = v i > u i ( x i ). For any ε ∈ (0 , X i is convex, εz i + (1 − ε ) x i ∈ X i . By the semi-strict quasi-concavity of u i , u i ( εz i + (1 − ε ) x i ) > u i ( x i ). This means that, for any ε ∈ (0 , εp i · z i + (1 − ε ) p i · x i > α + (1 − α ) p i · ω i . But this is only possible, for arbitrarily small ε , if p i · x i ≥ α + (1 − α ) p i · ω i . Since x i ∈ B i ( p, α ), we have p i · x i = α + (1 − α ) p i · ω i . (cid:3) Lemma 3. If V i ( p ) = v i , then d i ( p ) = arg min { p i · x i : u i ( x i ) = v i and x i ∈ X i } . Proof.
Let x i ∈ d i ( p ). Then for any z i ∈ X i with p i · z i < p i · x i , we have z i ∈ B i ( p, α ).So u i ( z i ) < v i by definition of d i . Therefore, if z i ∈ argmin { p i · x i : u i ( x i ) = v i and x i ∈ X i } , then p i · z i = p i · x i , and therefore d i ( p ) ⊇ arg min { p i · x i : u i ( x i ) = v i and x i ∈ X i } . The converse set inclusion follows similarly because if x i is not in the right-hand set,there would exist z i ∈ X i with p i · z i < p i · x i and u i ( z i ) = v i , which is not possibleas such z i would be in B i ( p, α ). (cid:3) Lemma 4. d i is upper hemicontinuous.Proof. Let ( x n , p n ) → ( x, p ), with x n ∈ d i ( p n ). Suppose that there is x ′ ∈ B i ( p, α )with u i ( x ′ ) > u i ( x ). If p i · x ′ < α + (1 − α ) p i · ω i , then this strict inequalitywill be true for p n with n large enough; a contradiction, as u i is continuous. If p i · x ′ = α + (1 − α ) p i · ω i , then α > p i · x ′ >
0. Then there is λ ∈ (0 , u i ( λx ′ ) > u i ( x ), p i · ( λx ′ ) < p i · x ′ , and λx ′ ∈ X i (recall that theconstruction of X i ensures that this is the case)). The argument for the case of astrict inequality then applies. (cid:3) Remark . Lemma 4 uses crucially that α > Lemma 5. d i ( p ) is upper hemi-continuous.Proof. To prove upper hemi-continuity, we shall prove that d i has a closed graph.Let ( x ni , p n ) → ( x i , p ) with x ni ∈ d i ( p n ) for all n . ONSTRAINTS 35
First, consider the case where V i ( p ) < v i . By the maximum theorem, V i is contin-uous, so V i ( p n ) < v i for all large enough n . Then Lemma 2 implies that x i ∈ d i ( p )as d i is upper hemi-continuous.Second, consider the case where V i ( p ) = v i . We know that x i ∈ d i ( p ) as d i isupper hemi-continuous. Suppose (towards a contradiction) that x i / ∈ d i ( p ). Thenthere is y i ∈ d i ( p ) with p i · y i < p i · x i ≤ α + (1 − α ) p i · ω i . Then for all n large enough, p ni · y i < α + (1 − α ) p i · ω i . Since y i ∈ d i ( p ) and V i ( p ) = v i , u i ( y ) = v i . This means that V i ( p n ) = v i for all n large enough, as y i ∈ B i ( p n , α ). Then, by Lemma 3, x ni ∈ argmin { p ni · x i : u i ( x i ) = v i and x i ∈ X i } for all n large enough. But the correspondence p argmin { p i · x i : u i ( x i ) = v i and x ∈ X i } . is upper hemicontinous, by the maximum theorem. So x i ∈ argmin { p i · x i : u i ( x i ) = v i and x ∈ X i } , which is a contradiction. (cid:3) It is easy to see that d i ( p ) is nonempty, compact- and convex-valued. So d i ( p ) isalso nonempty, compact- and convex-valued. For every c ∈ Ω ∗ , define the aggregatedemand on c by D c ( p ) = X ( i,l ) ∈ supp ( c ) a ci,l d i,l ( p ) = ∪{ a · x : x ∈ × i d i ( p ) } . Define the aggregate demand correspondence by D ( p ) = ( D c ( p )) c ∈ Ω ∗ , and the excess demand correspondence by z ( p ) = D ( p ) − { b } , where b = ( b c ) c ∈ Ω ∗ .Consider the correspondence ϕ : P → P defined by ϕ c ( p ) = { min { max { , z c + p c } , ¯ p } : z ∈ z ( p ) } for all c ∈ Ω ∗ . D ( p ), and therefore z ( p ), are upper hemicontinuous, convex-valued, and compact-valued. Thus, ϕ is upper hemi-continuous, convex-valued and compact-valued. ByKakutani’s fixed point theorem, there exists p ∗ ∈ P with p ∗ ∈ ϕ ( p ∗ ).Note that there exists z ∗ ∈ z ( p ∗ ) such that(4) p ∗ c = min { max { , z ∗ c + p ∗ c } , ¯ p } for all c ∈ Ω ∗ . Choose x ∗ ∈ R NL + such that x ∗ i ∈ d i ( p ∗ ) for all i and a · x ∗ − b = z ∗ ( a,b ) for all( a, b ) ∈ Ω ∗ . We shall prove that ( x ∗ , p ∗ ) is an α -slack Walrasian equilibrium. Lemma 6. p ∗ · z ∗ ≥ .Proof. If p ∗ · z ∗ <
0, then there is some c ∈ Ω ∗ with p ∗ c > z ∗ c <
0. ByEquation 4, then, p ∗ c = p ∗ c + z ∗ c , which is not possible as z ∗ c < (cid:3) Lemma 7. p ∗ c < ¯ p for all c ∈ Ω ∗ .Proof. Suppose towards a contradiction that there exists c ∗ ∈ Ω ∗ for which p ∗ c ∗ = ¯ p .Then p ∗ c >
0. Now, x ∗ i ∈ B i ( p ∗ , α ) means that p ∗ i · x ∗ i ≤ α + (1 − α ) p ∗ i · ω i , which is equivalent to p ∗ i · ( x ∗ i − ω i ) ≤ α (1 − p ∗ i · ω i ) . Summing over i , we obtain that X i ∈ I p ∗ i · ( x ∗ i − ω i ) ≤ α ( N − X i ∈ I p ∗ i · ω i ) , which is equivalent to(5) X i ∈ I X l ∈ O p ∗ i,l ( x ∗ i,l − ω i,l ) ≤ α ( N − X i ∈ I X l ∈ O p ∗ i,l ω i,l )Note that X i ∈ I X l ∈ O p ∗ i,l ω i,l = X i ∈ I X l ∈ O (cid:0) X c ∈ Ω ∗ :( i,l ) ∈ supp ( c ) a ci,l p ∗ c (cid:1) ω i,l = X c ∈ Ω ∗ p ∗ c (cid:0) X ( i,l ) ∈ supp ( c ) a ci,l ω i,l (cid:1) (6)Now, by definition of ¯ p , we have that X c ∈ Ω ∗ p ∗ c (cid:0) X ( i,l ) ∈ supp ( c ) a ci,l ω i,l (cid:1) ≥ ¯ p (cid:0) X ( i,l ) ∈ supp ( c ∗ ) a c ∗ i,l ω i,l (cid:1) ≥ ¯ p min c ∈ Ω ∗ (cid:0) X ( i,l ) ∈ supp ( c ) a ci,l ω i,l (cid:1) = 2 N. ONSTRAINTS 37
Thus, using this inequality and equations (5) and (6), we obtain that(7) X i ∈ I X l ∈ O p ∗ i,l ( x ∗ i,l − ω i,l ) ≤ α ( N − X i ∈ I X l ∈ O p ∗ i,l ω i,l ) < . On the other hand, p ∗ · z ∗ = X c ∈ Ω ∗ p ∗ c (cid:0) X ( i,l ) ∈ supp ( c ) a ci,l x ∗ i,l − b c (cid:1) . ≤ X c ∈ Ω ∗ p ∗ c (cid:0) X ( i,l ) ∈ supp ( c ) a ci,l x ∗ i,l − X ( i,l ) ∈ supp ( c ) a ci,l ω i,l (cid:1) (8) = X i ∈ I X l ∈ O X { c ∈ Ω ∗ st ( i,l ) ∈ supp ( c ) } p ∗ c a ci,l ( x ∗ i,l − ω i,l )= X i ∈ I X l ∈ O p ∗ i,l ( x ∗ i,l − ω i,l )(9) < , (10)where (8) follows because for each c ∈ Ω ∗ , P ( i,l ) ∈ supp ( c ) a ci,l ω i,l ≤ b c , (9) follows as p ∗ i,l = X { c ∈ Ω ∗ st ( i,l ) ∈ supp ( c ) } p ∗ c a ci,l , and (10) follows from (7).Finally, (10) is absurd as it contradicts Lemma 6. (cid:3) Proof of Theorem 2.
We claim that ( x ∗ , p ∗ ) is an α -slack Walrasian equilibrium. If p ∗ c >
0, since p ∗ c < ¯ p , then p ∗ c = z ∗ c + p ∗ c , which implies z ∗ c = 0. If p ∗ c = 0, then z ∗ c + p ∗ c ≤
0, which implies z ∗ c ≤
0. Recall that z ∗ c = a c · x ∗ − b c . So this implies that x ∗ satisfies all inequalities in Ω ∗ . By definition of X i , x ∗ satisfies then all inequalitiesin Ω. Hence, x ∗ ∈ lcs( C ) . Moreover, if z ∗ c <
0, it must be that p ∗ c = 0, as p ∗ c > z ∗ c = 0.It remains to show that x ∗ ∈ C . Suppose to the contrary that x ∗ / ∈ C . Since x ∗ ∈ lcs( C ), there exists x ′ ∈ C such that x ∗ ≤ x ′ . Then x ∗ = x ′ , so there is( i ∗ , l ∗ ) ∈ I × O with x ∗ i ∗ ,l ∗ < x ′ i ∗ ,l ∗ . By definition of C , x ′ i ∗ ∈ Z i .Consider y i ∗ defined as y i ∗ ,l = x ∗ i ∗ ,l for all l = l ∗ , and y i ∗ ,l ∗ = x ′ i ∗ ,l ∗ . Since x ′ ∈ C and x ′ i ∗ ,l ∗ > l ∗ cannot be a forbidden object for i ∗ . Hence, x ′ i ∗ ∈ X i ∗ and therefore y i ∗ ∈ X i . Moreover, for any c = ( a, b ) ∈ Ω ∗ , if ( i ∗ , l ∗ ) ∈ supp( c ) then a · x ∗ < a · x ′ ≤ b andtherefore z ∗ c < c must not be binding at x ∗ ). Hence p ∗ c = 0. In consequence, X l ∈ O p ∗ i ∗ ,l y i ∗ ,l = X l = l ∗ p ∗ i ∗ ,l y i ∗ ,l + (cid:0) X ( a,b ) ∈ Ω ∗ p ∗ ( a,b ) a i ∗ ,l ∗ | {z } =0 (cid:1) y i ∗ ,l ∗ = X l = l ∗ p ∗ i ∗ ,l x ∗ i ∗ ,l ≤ α + (1 − α ) p ∗ i · ω i . Thus y i ∗ ∈ B i ∗ ( p ∗ , α ) and x ∗ i ∗ < y i ∗ , contradicting the strict monotonicity of u i ∗ andthat x ∗ i ∈ d i ( p ∗ ).We next prove that x ∗ is C -constrained Pareto efficient. Suppose towards a con-tradiction that x is an feasible allocation that Pareto dominates x ∗ . Given that x ∈ C , x i ∈ X i . Then, for all i ∈ I , u i ( x i ) ≥ u i ( x ∗ i ), so by definition of d i we havethat p ∗ i · x i ≥ p ∗ i · x ∗ i . And for some j ∈ I , u j ( x j ) > u j ( x ∗ j ), so by utility maximization, p ∗ j · x j > p ∗ j · x ∗ j . Thus, X i ∈ I p ∗ i · x i > X i ∈ I p ∗ i · x ∗ i . This is equivalent to X c ∈ Ω ∗ p ∗ c (cid:18) X ( i,l ) ∈ supp ( c ) a ci,l x i,l (cid:19) > X c ∈ Ω ∗ p ∗ c (cid:18) X ( i,l ) ∈ supp ( c ) a ci,l x ∗ i,l (cid:19) . So there must exist c ∈ Ω ∗ such that p ∗ c > X ( i,l ) ∈ supp ( c ) a ci,l x i,l > X ( i,l ) ∈ supp ( c ) a ci,l x ∗ i,l . However, p ∗ c > z ∗ c = 0 ( c is binding at x ∗ ), and thus x violates c andis not feasible, which is a contradiction.Equal-type envy-freeness follows the fact that agents of equal type have equalconsumption space and equal budgets, and face equal personalized prices. (cid:3) Remark . The proof uses semi-strict quasi-concavity only in the proof of upperhemicontinuity of d i . To prove existence of an equilibrium without imposing thecheapest-bundle property, observe that continuity and quasiconcavity of u i is enoughto ensure that d i is upper hemicontinuous, and convex- and compact-valued. If z ONSTRAINTS 39 is defined from d i in place of d i , the proof can be written same as above. Toprove that every α -slack equilibrium assignment x ∗ is weakly C -constrained Paretoefficient, suppose towards a contradiction that there exists a feasible assignment x such that for all i ∈ I , u i ( x i ) > u i ( x ∗ i ). By utility maximization, for all i ∈ I , p ∗ i · x i > p ∗ i · x ∗ i . Thus, X i ∈ I p ∗ i · x i > X i ∈ I p ∗ i · x ∗ i . So we obtain a contradiction as before.
Remark . The above proof can be easily adapted to proveTheorem 1. We first change the price space to be P = [0 , ¯ p ] | Ω ∗ | , where¯ p = Nb min + 1, and b min = min { b : ( a, b ) ∈ Ω ∗ } .By letting α = 1, Lemma 2 to Lemma 6 do not change.Lemma 7 becomes easier to prove. Suppose p ∗ c = ¯ p for some c ∈ Ω ∗ . Then z ∗ c + p ∗ c ≥ ¯ p implies that z ∗ c ≥
0. So c must be binding, and for every ( i, l ) ∈ supp ( c ), p ∗ i,l ≥ a i,l p ∗ c . However, this is impossible because P ( i,l ) ∈ supp ( c ) a i,l x ∗ i,l ≤ P ( i,l ) ∈ supp ( c ) a i,l p ∗ i,l ≤ P ( i,l ) ∈ supp ( c ) p ∗ c ≤ Np ∗ c < b min .Then we can prove as above that ( x ∗ , p ∗ ) is a pseudo-market equilibrium, and x ∗ is(weakly) C -constrained Pareto efficient.10. Proof of Theorem 3
Let d H denote the Hausdorff distance between two sets in R L . So, d H ( A, B ) = max { sup { inf {k x − y k : y ∈ B } : x ∈ A } , sup { inf {k x − y k : x ∈ A } : y ∈ B }} . Let B i ( p, α ) denote the budget set of agent i given a price vector p and slack α ∈ [0 , B i ( p, α ) = { x i ∈ X i : p i · x i ≤ α + (1 − α ) p i · ω i } denote the budgetline. Note that B i ( p, α ) = { x i ∈ X i : ∃ y ∈ ¯ B i ( p, α ) s.t. x ≤ y } . Lemma 8.
For any δ > , there is α > such that if p is an α -slack equi-librium price vector found in Theorem 2, then for any i , either p i · ω i < or d H ( ¯ B i ( p, α ) , ¯ B i ( p, < δ .Proof. Consider the price ¯ p defined in the proof of Theorem 2. If p is a price obtainedin Theorem 2, then p ∈ [0 , ¯ p ] | Ω ∗ | . Note that ¯ p is independent of α . Let K = sup {k x k : x ∈ X i , ≤ i ≤ N } . Now choose α ∈ (0 ,
1) such thatsup { (cid:12)(cid:12)(cid:12)(cid:12) − α + (1 − α ) p i · ω i p i · ω i (cid:12)(cid:12)(cid:12)(cid:12) K : p ∈ [0 , ¯ p ] | Ω ∗ | and p i · ω i ≥ } < δ Observe that when p i · ω i ≥ B i ( p, α ) ⊆ B i ( p, x ∈ B i ( p, α ),inf {k x − y k : y ∈ ¯ B i ( p, } = k x − x k = 0. Hence,sup { inf {k x − y k : y ∈ ¯ B i ( p, } , x ∈ ¯ B i ( p, α ) } = 0 . On the other hand, if we let x ∈ ¯ B i ( p, γx ∈ ¯ B i ( p, α ), where γ = α + (1 − α ) p i · ω i p i · ω i . Since γ ≤ γ ∈ X i .Note that k x − γx k = | − γ | k x k < δ. Thus inf {k x − y k : y ∈ ¯ B i ( p, α ) } < δ , and thereforesup { inf {k x − y k : y ∈ ¯ B i ( p, α ) } , x ∈ ¯ B i ( p, } < δ. Thus d H ( B i ( p, , B i ( p, α )) < δ . (cid:3) To prove the theorem, let δ > p ∈ [0 , ¯ p ] Ω ∗ , if d H ( B i ( p, , B i ( p, α )) <δ then | max { u i ( x ) : x ∈ B i ( p, α ) } − max { u i ( x ) : x ∈ B i ( p, }| < ε. For such δ , let α be as in Lemma 8.For any i , if p i · ω i < B i ( p, ⊆ B i ( p, α ), somax { u i ( y ) : y ∈ ∆ − and p i · y ≤ p i · ω i } − u i ( x ) < < ε. If, on the contrary, p i · ω i ≥
1, then Lemma 8 implies that d H ( B i ( p, , B i ( p, α )) < δ ,and the result follows from the definition of δ .11. Proof of Proposition 5
Our first observation establishes the relation between envy and the value of en-dowments at equilibrium prices.
Lemma 9.
Let ( x, p ) be a Walrasian equilibrium with slack α ∈ (0 , . If i envies j , then p · ( x j − x i ) > and p · ( ω j − ω i ) > . ONSTRAINTS 41
Proof.
Let i envy j , so u i ( x j ) > u i ( x i ). Then utility maximization implies that α + (1 − α ) p · ω j ≥ p · x j > α + (1 − α ) p · ω i ≥ p · x i , where the strict inequality follows because x j ∈ ∆ − . So p · ( x j − x i ) > p · ( ω j − ω i ) > (cid:3) Now consider a α -slack Walrasian equilibrium ( x, p ). Agent i ’s maximizationproblem is: max x ∈ R L + u i ( x ) + λ i ( I i − p · x ) + γ i (1 − · x )Where I i = α + (1 − α ) p · ω i , λ i is a multiplier for the budget constraint, and γ i for the P l x i,l ≤ C . The first-order conditions for the maximization problemsare then: ∂ l u i ( x i ) − λ i p l − g i = 0 if x i,l > ≤ x i,l = 0 , where ∂ l u i ( x i ) denotes the partial derivative of u i with respect to x i,l .Observe that if p · x i < α + (1 − α ) p · ω i , then the budget constraint is notbinding and λ i = 0. As a consequence, u i ( x i ) = max { u i ( z i ) : z i ∈ ∆ − } . Let S = { i ∈ [ N ] : p · x i < α + (1 − α ) p · ω i } be the set of satiated consumers. Let U = { i ∈ [ N ] : p · x i = α + (1 − α ) p · ω i } be the set of unsatiated , and observe thatwe can let λ i > i ∈ U . Consider the two stage social program:Stage 1: max ˜ y ∈ (∆ − ) S P i ∈ S u i (˜ y i )Stage 2: max ˜ y ∈ (∆ − ) U P i ∈ U λ i u i (˜ y i ) P i ∈ U ˜ y i ≤ ¯ ω − P i ∈ S x i Note that ( x i ) i ∈ S solves Stage 1, while satisfying P i ∈ S x i ≤ ¯ w , and that given( x i ) i ∈ S , ( x i ) i ∈ U solves Stage 2. That this is so follows from the fact that ( x i ) i ∈ U solves the first-order conditions for the Stage 2 problem with Lagrange multiplier p for the constraint that P i ∈ U ˜ y i ≤ ¯ ω − P i/ ∈ S x i .Now use the assumption that P i ∈ U x i ≫
0. This means that there exists ¯ t > t ∈ (0 , ¯ t ] then the set of ˜ y ∈ (∆ − ) U such that P i ∈ U ˜ y i ≤ ¯ ω + t ( ω i − ω j ) − P i/ ∈ S x i is nonempty (and, for constraint qualification, contains an elementthat satisfies all constraints with slack). Consider the problemmax ˜ y ∈ (∆ U − ) P i ∈ U λ i u i (˜ y i ) P i ∈ U ˜ y i ≤ ¯ ω + t ( ω i − ω j ) − P i ∈ S x i Note that for each t ∈ (0 , ¯ t ] there exists ( ν ( t ) , γ ( t ) , α ( t )) such that v ( t ) = sup { X i ∈ U λ i u i · ˜ y i + ν ( t ) · (¯ ω − X i ∈ S ˜ y i + t ( ω i − ω j )) − X i ∈ U ˜ y i )+ X i ∈ U γ i ( t )(1 − X l ∈ O ˜ y i,l )+ X i ∈ U α i ( t )˜ y i,l . } Here ν ( t ) is the Lagrange multiplier for the constraint that P i ∈ U ˜ y i ≤ ¯ ω − P i ∈ S x i + t ( ω i − ω j ), while γ ( t ) and α ( t ) are the Lagrange multipliers for the constraint that(˜ y i ) ∈ (∆ − ) N . Choose a selection ( ν ( t ) , γ ( t ) , α ( t )) such that ν (0) = p .Let ˜ ω = ¯ ω − P i ∈ S x i . The saddle point inequalities imply that( t ′ − t ) ν ( t ) · ( ω i − ω j ) = X i ∈ U λ i u i ( x i ( t ′ )) + ν ( t ) · (˜ ω + t ′ ( ω i − ω j ) − X i ∈ U x i ( t ′ ))+ X i ∈ U γ i ( t )(1 − X l ∈ O x i,l ( t ′ )) + X i ∈ U α i ( t ) x i,l ( t ′ ) − X i ∈ U λ i u i ( x i ( t ′ )) + ν ( t ) · (˜ ω + t ( ω i − ω j ) − X i ∈ U x i ( t ′ ))+ X i ∈ U γ i ( t )(1 − X l ∈ O x i,l ( t ′ )) + X i ∈ U α i ( t ) x i,l ( t ′ ) ! ≥ v ( t ′ ) − v ( t )Now recall that ν (0) = p . Then Lemma 9, together with the above inequality,imply that 0 > p · ( ω i − ω j ) t ′ ≥ v ( t ′ ) − v (0)for all t ′ > t ′ ≤ ¯ t . ReferencesAbdulkadiro˘glu, A., and
T. S¨onmez (1999): “House allocation with existingtenants,”
Journal of Economic Theory , 88(2), 233–260.(2003): “School Choice: A Mechanism Design Approach,”
The AmericanEconomic Review , 93(3), 729–747.
Akbarpour, M., and
A. Nikzad (forthcoming): “Approximate Random Alloca-tion Mechanisms,”
The Review of Economic Studies . Andersson, T., A. Cseh, L. Ehlers, and
A. Erlanson (2019): “Organizingtime exchanges: lessons from matching markets,” working paper . ONSTRAINTS 43
Ashlagi, I., and
P. Shi (2015): “Optimal allocation without money: An engi-neering approach,”
Management Science , 62(4), 1078–1097.
Avriel, M., W. E. Diewert, S. Schaible, and
I. Zang (2010):
Generalizedconcavity , vol. 63. Siam.
Aygun, O., and
I. B´o (2017): “College admission with multidimensional privi-leges: The Brazilian affirmative action case,”
Available at SSRN 3071751 . Baccara, M., A. Imrohoroglu, A. J. Wilson, and
L. Yariv (2012): “AField Study on Matching with Network Externalities,”
American Economic Re-view , 102(5), 1773–1804.
Balbuzanov, I. (2019): “Constrained random matching,” working paper . Bogomolnaia, A., H. Moulin, F. Sandomirskiy, and
E. Yanovskaia (2017): “Competitive division of a mixed manna,”
Econometrica , 85(6), 1847–1871. (2019): “Dividing bads under additive utilities,”
Social Choice and Welfare ,52(3), 395–417.
Budish, E. (2011): “The combinatorial assignment problem: Approximate com-petitive equilibrium from equal incomes,”
Journal of Political Economy , 119(6),1061–1103.(2012): “Matching” versus” mechanism design,”
ACM SIGecom Ex-changes , 11(2), 4–15.
Budish, E., Y.-K. Che, F. Kojima, and
P. Milgrom (2013): “Designingrandom allocation mechanisms: Theory and applications,”
American EconomicReview , 103(2), 585–623.
Combe, J., O. Tercieux, and
C. Terrier (2018): “The design of teacherassignment: theory and evidence,” Mimeo, University College London.
Echenique, F., A. Miralles, and
J. Zhang (2020): “Fairness and efficiencyfor probabilistic allocations with participation constraints,” arXiv:1908.04336.
Echenique, F., and
M. B. Yenmez (2015): “How to control controlled schoolchoice,”
The American Economic Review , 105(8), 2679–2694.
Edmonds, J. (1965): “Maximum matching and a polyhedron with 0, 1-vertices,”
Journal of research of the National Bureau of Standards B , 69(125-130), 55–56.
Ehlers, L. (2010): “Controlled School Choice,” mimeo, University of Montreal.
Ehlers, L., I. E. Hafalir, M. B. Yenmez, and
M. A. Yildirim (2014):“School choice with controlled choice constraints: Hard bounds versus softbounds,”
Journal of Economic Theory , 153, 648–683.
Fragiadakis, D., and
P. Troyan (2017): “Improving matching under harddistributional constraints,”
Theoretical Economics , 12(2), 863–908.
Gul, F., W. Pesendorfer, and
M. Zhang (2019): “Market design and Wal-rasian equilibrium,” mimeo, Princeton University.
Hafalir, I., and
A. Miralles (2015): “Welfare-maximizing assignment of agentsto hierarchical positions,”
Journal of Mathematical Economics , 61, 253–270.
Hafalir, I. E., M. B. Yenmez, and
M. A. Yildirim (2013): “Effective affir-mative action in school choice,”
Theoretical Economics , 8(2), 325–363.
He, Y., A. Miralles, M. Pycia, and
J. Yan (2018): “A pseudo-market ap-proach to allocation with priorities,”
American Economic Journal: Microeco-nomics , 10(3), 272–314.
Hylland, A., and
R. Zeckhauser (1979): “The Efficient Allocation of Individ-uals to Positions,”
Journal of Political Economy , 87(2), 293–314.
Kamada, Y., and
F. Kojima (2015): “Efficient matching under distributionalconstraints: Theory and applications,”
American Economic Review , 105(1), 67–99. (2017): “Recent developments in matching with constraints,”
AmericanEconomic Review , 107(5), 200–204.(2019): “Fair matching under constraints: Theory and applications,”Mimeo: Stanford University.
Katz, M. (1970): “On the extreme points of a certain convex polytope,”
Journalof Combinatorial Theory , 8(4), 417–423.
Kelso, A. S., and
V. P. Crawford (1982): “Job matching, coalition formation,and gross substitutes,”
Econometrica: Journal of the Econometric Society , pp.1483–1504.
Kojima, F., N. Sun, and
N. N. Yu (2020): “Job matching under constraints,”American Economic Review, forthcoming.
Kominers, S. D., and
T. S¨onmez (2013): “Designing for diversity in matching,”in EC , pp. 603–604. Le, P. (2017): “Competitive equilibrium in the random assignment problem,”
In-ternational Journal of Economic Theory , 13(4), 369–385.
Manjunath, V. (2016): “Fractional matching markets,”
Games and EconomicBehavior , 100, 321–336.
Mas-Colell, A. (1992): “Equilibrium theory with possibly satiated preferences,”in
Equilibrium and Dynamics , ed. by M. Majumdar, pp. 201–213. Springer.
ONSTRAINTS 45
McLennan, A. (2018): “Efficient disposal equilibria of pseudomarkets,” mimeo,University of Queensland.
Miralles, A., and
M. Pycia (2020): “Foundations of Pseudomarkets: WalrasianEquilibria for Discrete Resources,” working paper . Nguyen, T., and
R. Vohra (2017): “Stable Matching with Proportionality Con-straints.,” in EC , pp. 675–676. Rockafellar, R. T. (1970):
Convex analysis , vol. 28. Princeton university press.
Root, J., and
D. S. Ahn (2020): “Incentives and Efficiency in Constrained Allo-cation Mechanisms,” mimeo: UC Berkeley.
Westkamp, A. (2013): “An analysis of the German university admissions system,”