Fairness and efficiency for probabilistic allocations with participation constraints
aa r X i v : . [ ec on . T H ] M a y FAIRNESS AND EFFICIENCY FOR ALLOCATIONS WITHPARTICIPATION CONSTRAINTS
FEDERICO ECHENIQUE, ANTONIO MIRALLES, AND JUN ZHANG
Abstract.
We propose a notion of fairness for allocation problems in whichdifferent agents may have different reservation utilities, stemming from differentoutside options, or property rights. Fairness is usually understood as the absenceof envy, but this can be incompatible with reservation utilities. It is possiblethat Alice’s envy of Bob’s assignment cannot be remedied without violating Bob’sparticipation constraint. Instead, we seek to rule out justified envy , defined as envyfor which a remedy would not violate any agent’s participation constraint. Weshow that fairness, meaning the absence of justified envy, can be achieved togetherwith efficiency and individual rationality. We introduce a competitive equilibriumapproach with price-dependent incomes obtaining the desired properties.(Echenique)
Division of the Humanities and Social Sciences, California Instituteof Technology (Miralles)
Department of Economics, Universit`a degli Studi di Messina, Universi-tat Aut`onoma de Barcelona and Barcelona Graduate School of Economics. (Zhang)
Institute for Social and Economic Research, Nanjing Audit University,Nanjing, China.
Date : May 2020.We thank Eric Budish, Fuhito Kojima, Andy McLennan, Herv´e Moulin, and Tayfun S¨onmezfor comments. Echenique thanks the National Science Foundation for its support through thegrants SES-1558757 and CNS-1518941. Miralles acknowledges financial support from the SpanishMinistry of Economy and Competitiveness (ECO2017-83534-P,) the Catalan Government (2017SGR 711,) and the Severo Ochoa Programme (SEV-2015-0563). Zhang thanks the financial supportfrom the National Natural Science Foundation of China (Grant Introduction
We investigate the meaning of fairness in allocation environments with participa-tion constraints and constrained consumption spaces. A special case is the randomallocation problem in which agents have unit demand. Without participation con-straints, we may regard all agents equally, and the absence of envy is a naturalnotion of fairness. In our model, different agents may have different reservation util-ities, stemming from outside options or property rights. Participation constraintsensure that agents get at least their reservation utilities. Absence of envy may beincompatible with agents’ participation constraints. In such environments, whatdoes it mean to treat agents fairly?It is well known that allocations satisfying both efficiency and envy-freeness exist(Varian, 1974; Hylland and Zeckhauser, 1979). In a model with participation con-straints, the challenge is to make efficient and envy-free allocations compatible withagents’ individual rationality. Our contribution is threefold. Our first contributionis to propose a notion of fairness that combines envy and individual rationality. Weprove (Theorem 1) the existence of fair, efficient, and individually rational alloca-tions. Our second contribution is to show that these fair and efficient outcomes can,under certain conditions, be viewed as market outcomes (Theorem 2), as in Varianand Hylland-Zeckhauser. Our third contribution (Theorem 3) is to accommodatequantitative constraints, such as those in course allocations (e.g all students musttake at least two math courses), or controlled school choice (e.g a school seeks certaindiversity objectives).We understand fairness as the absence of justified envy, or as “ruling out envythat can be remedied within agents’ individual rationality constraints.” We do notwant to say that an outcome is unfair if its unfairness can be traced to differencesin agents’ reservation utilities. Concretely, Alice envies Bob at an allocation x if shewould rather have Bob’s assignment in x than hers. To decide whether this envyis justified, we consider the possibility of swapping the assignments between Aliceand Bob, since swapping is an obvious remedy for Alice’s envy. We say that Alice’senvy is justified if Bob could have received Alice’s assignment without violating hisparticipation constraint, and unjustified if Alice’s assignment would put Bob belowhis reservation utility. Our fairness notion is similar to the concepts introduced by Yılmaz (2010) andAthanassoglou and Sethuraman (2011) for object allocation problems with ordinal preferences.See Section 2 for a discussion.
AIRNESS AND EFFICIENCY 3
Our notion of envy presumes that the obvious remedy for Alice’s envy towardsBob is for them to switch assignments. Clearly, if Alice wants to bring the matter tocourt, the most natural and plausible remedy she could offer is for the two of themto switch assignments. One might devise more complicated remedies, with a fullerreallocation that would seek to eliminate Alice’s envy, but these would necessarilybe complicated and require Alice’s complaint to rely on multiple agents. That said,our methods do accommodate more general remedies (Theorem 4).Importantly, our notion of fairness is compatible with efficiency. We show that,under some conditions, our solution can be achieved as a market outcome. The ideaseeks to generalize Varian’s and Hylland and Zeckhauser’s competitive equilibriumfrom equal incomes. The obvious solution would be to endogenize incomes. To thisend, we construct price-dependent income functions. We have to be careful since,as shown by Hylland and Zeckhauser (1979), when incomes depend on prices, aWalrasian equilibrium might not exist. Our careful construction of income functionsensures individual rationality and fairness. This construction could be regarded asa minimal deviation from equal incomes that sustains individual rationality and nosatiated agent overspending. If Alice envies Bob, then Bob’s maximum achievableutility is his reservation utility (Lemma 4). Besides, if Alice has less money than Boband she does not envy him, then she has just enough money to reach satiation. Weprovide an informal description of the income-function construction in Section 4.2.We organize the paper as follows. We discuss related literature in Section 2,present our model and fairness notion in Section 3, and present main theorems inSection 4. We extend our theorems to allocation environments with constraints inSection 5, and show that our theorems can account for more general remedies forenvy in Section 6. In Section 7 we apply our result to school choice.2.
Related literature
Efficiency and fairness can be achieved in models without reservation utilities.Examples are the solutions of Varian (1974), Hylland and Zeckhauser (1979) andBogomolnaia and Moulin (2001). Our problem is complicated, both conceptuallyand technically, by individual rationality constraints. Conceptually, the meaningof fairness among unequal agents is not obvious, while technically, implementationthrough market equilibrium may not be possible in economies with price-dependentincomes (see Hylland and Zeckhauser (1979)). Part of our contribution is to support
ECHENIQUE, MIRALLES, AND ZHANG fair, efficient and individually rational (IR) outcomes as competitive pseudo-marketequilibria, as in Hylland-Zeckhauser.Our notion of no justified envy is analogous to similar notions developed by Yılmaz(2010) and Athanassoglou and Sethuraman (2011). They assume that agents haveordinal preferences instead of cardinal utilities, and say that agent i justifiably enviesagent j if i does not regard her allocation as first-order stochastically dominating j ’s,while any object with positive probability in her allocation is acceptable to j . Yilmazconsiders the house allocation with existing tenants model in which some agentshave deterministic endowments. He focuses on extending the probabilistic serialrule (Bogomolnaia and Moulin, 2001). Athanassoglou and Sethuraman consider thefractional endowment environment. Their purpose is to extend Yilmaz’s mechanismand fairness notion. We work with cardinal utility, focus on market equilibriuminstead of probabilistic serial, and use very different techniques. But we share someconceptual similarities with them that extend beyond the similarity in the definitionof justified envy. These authors suggest a cake-eating algorithm that starts with allagents eating at the same speed, but when an agent is at risk of violating her IRconstraint, only this agent has the right to eat, until she reaches her reservationutility or drops out of the algorithm. So only when IR binds for some agent isshe allowed to eat at higher speed than the others. In our competitive equilibriummethod (see Theorem 2), our income functions seek to achieve similar ideas. Fairnesspushes us towards equal incomes, but IR forces us to accept some inequality.Schmeidler and Vind (1972) consider a model where IR constraints arise due tothe presence of endowments. Starting from an initial endowment, they study fairnet trades : trades leading to a Pareto optimal allocation in which no agent enviesthe trades made by others. Our model differs from theirs for two reasons. First,our model is primarily designed to address constrained consumption spaces, as inthe random allocation problem. Fair net trades may not be feasible in such en-vironments, leading to a weak notion of fairness. Specifically, under unit demandconstraints there is no reason that one agent’s net trade is feasible to any otheragent. Second, while reservation utilities in our model can arise due to the pres-ence of endowments, they may also stem from other sources. Suppose Bob is endowed with 1 / / / − / >
1, violates her unit demand constraint. Schmeidler and Vindt’s notion was nevermeant to be used in (what we term) random allocation problems.
AIRNESS AND EFFICIENCY 5
In the fractional endowment model, Yu and Zhang (2019) propose algorithms thatgeneralize Top Trading Cycle to organize endowment exchange. Their algorithmsfind ordinally efficient, fair and individually rational allocations. We differ not onlyin the use of cardinal utilities but also in the absence of preexisting endowments.Balbuzanov and Kotowski (2019) explore the role of endowments for discrete al-location problems. Different from us, they interpret endowments as the rights toexclude others, and propose a new cooperative game solution concept. Althoughthey allow for public ownership, or collective ownership by subgroups, endowmentsin their model are deterministic. As a result, their results are unrelated to ours.Our results are applicable to school choice when we wish to use endowmentsinstead of priorities to control children’s rights towards schools. It is in particularapplicable to controlled school choice. School choice was first introduced as an appli-cation of resource allocation models by Abdulkadiro˘glu and S¨onmez (2003). In thestandard model of school choice, fairness and efficiency are generally incompatible. Alot of the school choice literature has been devoted to the resulting trade-off. In oursolution, the trade-off is resolved. Hamada, Hsu, Kurata, Suzuki, Ueda, and Yokoo(2017) is the only paper we are aware of that emphasizes endowments in schoolchoice. They assume that each child owns one seat of some school as endow-ment. Their goal is to design strategy-proof allocation mechanisms to meet thedistributional constraint in the market and IR constraint of each child. Since theyconsider deterministic endowments and ordinal preferences, and their fairness no-tions are based on priorities, their results are unrelated to ours. The constraintswe analyze have been discussed extensively in the literature on controlled schoolchoice: see Ehlers (2010), Kojima (2012), Hafalir, Yenmez, and Yildirim (2013),Ehlers, Hafalir, Yenmez, and Yildirim (2014), and Echenique and Yenmez (2015);and the literature on distributional constraints (motivated by geographic distribu-tional considerations): Kamada and Kojima (2015), Kamada and Kojima (2017),and Kamada and Kojima (2019), among others. Our approach of eliminating jus-tified envy when it does not conflict with constraints is common to those papers.For example, Kamada and Kojima consider matchings where no blocking pair thatwould not violate distributional constraints are present.In separate work (Echenique, Miralles, and Zhang, 2019), we give a direct Wal-rasian approach to allocation problems with constraints. Key is the idea of settinga price for each constraint. We properly embed constraint prices into each agent’sbudget constraint.
ECHENIQUE, MIRALLES, AND ZHANG The model
Notation and preliminary definitions.
Each agent i has an associated max-imum overall demand c i ∈ R ++ . We define the c i -simplex in R n as { x ∈ R n + : P nj =1 x j = c i } , denoted by ∆ n ( c i ) ⊆ R n . The set { x ∈ R n + : P nj =1 x j ≤ c i } isdenoted by ∆ n − ( c i ). When n is understood, we simply use the notation ∆( c i ) and∆ − ( c i ). We shall use the shortened notation C i = ∆ − ( c i ) for agent i ’s consumptionspace . When c i = 1 for all i , we say that agents have unit demand, and that we arefacing a random allocation problem . For c i large enough for all i , the allocation prob-lem is indistinguishable from a standard exchange economy, where the consumptionspace is R n + .We adopt the notational conventions used in convex analysis: Denote by R ∗ = R ∪ {−∞} the extended real numbers. A function u : R n → R ∗ has domain C = { x ∈ R n : u ( x ) > −∞} . A function u : R n → R is • quasi-concave if { z ∈ R n : u ( z ) ≥ u ( x ) } is a convex set, for all x ∈ R n . • concave if, for any x, z ∈ R n , and λ ∈ (0 , λu ( z ) + (1 − λ ) u ( x ) ≤ u ( λz +(1 − λ ) x ); • linear if we can identify u ( · ) with a vector v ∈ R n , so that u ( x ) = v · x on thedomain of u . For random allocation problems, linear utility is interpreted asan expected utility function . • Lipschitz continuous with constant θ > x, z ∈ C , | u ( z ) − u ( x ) | <θ k z − x k . • satisfying the Inada property (at the axes) if, for any x in its domain, u ( x ) = u (0), unless x ≫ • has l as its favorite object if, on its domain, decreasing consumption of anyobject k = l by any amount ǫ > ǫ of object l always leads to an increase in u . For example, if u is differentiableand C = R n ++ , then l is a favorite object if ∂u ( x ) ∂x l > ∂u ( x ) ∂x k , ∀ k = l. If u is linear,identified with v ∈ R n , then v l > v k , ∀ k = l. Throughout we work with functions that have a closed and convex domain. Afunction u with domain C is said to be continuous, and monotone if it is continuous(in the relative topology on C ) at every point x ∈ C , and for any x, y ∈ C , x < y implies that u ( x ) < u ( y ). We use the conventions that are standard in convex analysis. Note that x + ( −∞ ) = −∞ , and λ · ( −∞ ) = −∞ for any scalar λ . AIRNESS AND EFFICIENCY 7
Model.
A finite set of agents are to be assigned a finite set of objects. Weassume that objects are perfectly divisible. In the random allocation problem, wewould be allocating probabilities, and preferences would be defined on the set ofprobability distributions.An allocation problem is a tuple Γ = { O, I, Q, ( c i , u i , ˜ u i ) i ∈ I } , where: • O = { , . . . , L } represents a set of objects, or goods. • I = { , . . . , N } represents a set of agents. • Q = ( q l ) l ∈ O is a capacity vector, and q l ∈ R ++ is the quantity of object l .We assume that P l ∈ O q l ≤ P i ∈ I c i (no overall excess supply.) • For each agent i , c i > C i = ∆ − ( c i ). • For each agent i , u i : R n → R ∗ is a continuous and monotone utility functionwith C i as its domain. • For each agent i , ˜ u i ∈ R is her reservation utility .We say that Γ admits a common favorite object if there is an object l that is afavorite object for every agent i .3.3. Allocations. An allocation in Γ is a vector x ∈ R LN + , which we write as x = ( x i ) Ni =1 , such that x i ∈ C i for all i ∈ I, and X i ∈ I x il = q l for all l ∈ O. Let A be the set of all allocations. Clearly, A is nonempty, compact and convex.In the random allocation problem, where P l q l = N , each agent’s assignmentis a probability distribution over O . When x il ∈ { , } for all i ∈ I and all l ∈ O , x is a deterministic allocation. The Birkhoff-von Neumann theorem (Birkhoff,1946; Von Neumann, 1953) implies that every allocation is a convex combination ofdeterministic allocations.3.4. Individual rationality and Pareto optimality.
An allocation x is accept-able to agent i if u i ( x i ) ≥ ˜ u i ; x is individually rational (IR) if it is acceptable to allagents.We assume that reservation utilities are such that an IR allocation exists. We saythat Γ admits a strictly positive IR allocation if there is an IR allocation ˜ x ∈ R LN ++ .All agents obtain strictly positive quantities of all goods in ˜ x . A sufficient condition for this is the existence of an allocation x with u i ( x i ) > ˜ u i for all i ∈ I . ECHENIQUE, MIRALLES, AND ZHANG
An allocation x is weakly Pareto optimal (wPO) if there is no allocation y suchthat u i ( y i ) > u i ( x i ) for all i . An allocation x is Pareto optimal (PO) if there is noallocation y such that u i ( y i ) ≥ u i ( x i ) for all i with at least one strict inequality.Given the bounded consumption spaces in our model, wPO is compatible withwasteful situations where one can use existing resources to make some agents strictlybetter off, but cannot make all agents strictly better off because some agents aresatiated.3.5. Fairness.
We regard agents as having the right to attain their reservationutilities. While typically reservation utilities arise from endowments, we do notconsider endowments as the only source. Guaranteed reservation utilities can arisefrom a policy that protects disavantaged groups in school choice, for example.Our notion of fairness rules out envy that cannot be justified by guaranteed reser-vation utilities. If an agent i envies another agent j at an allocation x (that is, i prefers x j to x i ), our fairness notion regards the envy as not justified if switchingtheir allocations would violate j ’s participation constraint.Formally, we say that an agent i has justified envy towards another agent j at anallocation x if u i ( x j ) > u i ( x i ) and u j ( x i ) ≥ ˜ u j , and the justified envy is said to be strong if u j ( x i ) > ˜ u j . In words, if i envies j and j could have received i ’s assignment without violating j ’s participation constraint,the envy is justified.We say that x has no (strong) justified envy (N(S)JE) if no agent has (strong)justified envy towards any other agent at x .Fairness as NJE provides defense against possible complaints. Imagine a socialplanner that proposes an IR allocation x . Suppose an agent i complains that sheenvies another agent j . An obvious remedy for i ’s complaint is a pairwise switch oftheir assignments. But if the envy is not justified, the planner’s response is that theswitch would violate j ’s right to attain her reservation utility.Of course, one may imagine complaints that could be remedied through rear-rangements more complicated than a pairwise switch. Such remedies may or maynot be realistic, but in any case our methods easily accommodate much more gen-eral remedies. Specifically, one can devise cyclic rearrangements, where arbitrarilylong sequences of agents collaborate in the satisfaction of an agent’s envy, as long AIRNESS AND EFFICIENCY 9 as the last agent’s participation constraint is not violated. Theorem 4 in Section 6.1extends our main result to cover this case.In an IR and NJE allocation x , if u i = u j and ˜ u i ≥ ˜ u j , then it must be that u i ( x i ) ≥ u j ( x j ). That is, if i and j have equal preferences and i ’s reservationutility is weakly higher than j ’s, then both agree that i ’s assignment in x is alsoweakly better than j ’s. In particular, if u i = u j and ˜ u i = ˜ u j , then it must be that u i ( x i ) = u j ( x j ). So NJE and IR imply equal treatment of equals (called symmetryby Zhou (1990)).3.6. Approximation.
Our main results will prove that there exist allocations thatsatisfy individual rationality, Pareto optimality, and no justified envy. More pre-cisely, some of our results are based on approximations of these properties: for any ε >
0, an allocation x satisfies • ε -individual rationality ( ε -IR) if u i ( x i ) ≥ ˜ u i − ε for all i ∈ I ; • ε -Pareto optimality ( ε -PO) if there is no allocation y such that u i ( y i ) >u i ( x i ) + ε for all i ∈ I . • no ε -justified envy ( ε -NJE) if there do not exist two distinct agents i, j suchthat u i ( x j ) > u i ( x i ) and u j ( x i ) > ˜ u j − ε .It is clear that ε -IR is weaker than IR, ε -PO is weaker than wPO, and ε -NJE isstronger than NJE. 4. Main results
Let Γ = { O, I, Q, ( c i , u i , ˜ u i ) i ∈ I } be an allocation problem under the assumptionsspecified above (that is, utility functions are continuous and monotone, and thereexists an IR allocation). Theorem 1.
Suppose that agents’ utility functions in Γ are concave.(1) For any ε > , there exists an allocation that is ε -individually rational, ε -Pareto optimal and has no ε -justified envy;(2) There exists an allocation that is individually rational, weak Pareto optimaland has no strong justified envy.(3) If utilities are linear, there exists an allocation that is individually rational,Pareto optimal and has no strong justified envy. Theorem 1 gives the existence of allocations with the desired properties of ef-ficiency, fairness, and individual rationality. Our next result gives a competitivemarket foundation for these allocations.
Theorem 2.
Suppose that agents’ utility functions are quasi-concave, and that atleast one the following conditions hold: • P l ∈ O q l < min i c i ( c i is sufficiently large for every i ∈ I ). • u i satisfies the Inada property and ˜ u i > u i (0) , for every i ∈ I . • There exists a common favorite object, and a strictly positive IR allocation.Then there exists continuous income functions m i : ∆ → R + and ( x, p ) = (( x i ) Ni =1 , p ) ,such that p ∈ ∆ is a price vector, and(1) P i x i = Q ( x is an allocation; or, “supply equals demand”).(2) x is individually rational, Pareto optimal and has no justified envy.(3) x i ∈ argmax { u i ( z i ) : z i ∈ C i and p · z i ≤ m i ( p ) } . Theorem 2 provides conditions under which a fair, efficient, and IR allocationexists and can be supported as a form of market equilibrium (a “pseudo-market”).Our equilibria generalize Hylland and Zeckhauser’s, or Varian’s, notion of equilib-rium with a fixed exogenous income. Here, income is not fixed. It is price dependent,and formulated through income functions m i . These are carefully calibrated to en-sure both IR and NJE.Theorem 2 allows us to improve on Theorem 1. When utilities are Lipschitz,quasi-concavity is sufficient. Corollary 1. If u i is quasi-concave and Lipschitz continuous for every i ∈ I , thereis an allocation that is individually rational, weak Pareto optimal and has no strongjustified envy. The connection between the two theorems is worth clarifying. We prove the thirdstatement of Theorem 1 using Theorem 2. The first two statements of Theorem 1have very different proofs, and can accommodate quantity constraints; See Section 5.Finally, the two theorems hold without change if we use a stronger notion of NJEthat does not rely on pairwise switch as the remedy for envy; See Section 6.1.4.1.
Remarks on Theorem 1.
Statements (1) and (2) of Theorem 1 are based onweighted utilitarian maximization. We study the problem of maximizing(1) X i ∈ I λ i u i ( x i )over all IR allocations x , for each fixed vector λ = ( λ , . . . , λ N ) of welfare weights.The trick is to find “fair” welfare weights. Ideally, one could proceed iteratively.For each λ , solve the weighted utilitarian maximization problem and check if there AIRNESS AND EFFICIENCY 11 is any justified envy. If i justifiedly envies j , then adjust λ so as to decrease λ j andincrease λ i . Yet the iterative procedure does not quite work. We use a related idea,based on the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma .The KKM lemma was used by Varian (1974) in proving the existence of Pareto-efficient allocations with no envy whatsoever. Varian does not consider participa-tion constraints, and works directly with allocations (more precisely, with the utilitypossibility frontier). Our approach using welfare weights (inspired by the Negishiapproach to equilibrium existence), is quite different. Participation constraints in-troduce some technical difficulties, which necessitates an approximation argument.The presence of ε > λ . Thus each vertex ofthe simplex is the result of putting all weight on one agent in solving (1). Eachset Λ i corresponds to the set of weights yielding an ε -Pareto optimal allocation inwhich 1) agent i does not have ε -justified envy towards any other agent and 2) ε -individually rationality holds for agent i . We show that the collection (Λ i ) i ∈ I meetsthe conditions of the KKM lemma. Any point in the intersection of (Λ i ) i ∈ I meetsthe properties in the first statement of the theorem.By taking the limit when ε →
0, we obtain the second statement of Theorem 1.Observe that we only conclude that the obtained allocation is wPO. This is irrele-vant in many allocation problems, in which wPO and PO are identical. There areenvironments, however, in which there is no such equivalence. Market design prob-lems in which the consumption spaces are bounded (say, because of unit demand)constitute one example.4.2.
Remarks on Theorem 2.
The use of competitive markets to achieve a fairand efficient allocation is inspired by Varian (1974) and Hylland and Zeckhauser(1979), and more recently by Miralles and Pycia (2014), who establish a SecondWelfare Theorem for the kind of allocation problems studied in our paper.Varian and Hylland-Zeckhauser use fixed and equal incomes for all agents. Thefirst complication in our paper is that equal incomes will, however, not respect reservation utilities. Incomes must be price-dependent and constructed to satisfyIR and ensure NJE. Our model suggests a “minimal departure” from equal incomesthat satisfies IR: We allow an agent to have above-average income only in order toobtain exactly her reservation utility. A second complication is that a competitiveequilibrium allocation with potentially satiated agents does not guarantee Pareto-optimality, unless expenses for satiated agents are minimal (Hylland and Zeckhauser,1979). For this reason, we force an agent’s income below average whenever the av-erage lies above the minimal income providing her with satiation.The main new idea in the proof of Theorem 2 lies in the construction of price-dependent incomes. The construction is done by, for each price vector, taking themedian of three magnitudes: 1) a common income level, 2) the minimum expenditureguaranteeing satiation, and 3) the minimum expenditure ensuring reservation utility.Agents’ incomes add up to the overall value of objects. An important property ofour construction is that, when i ’s income is higher than j ’s, and j is not satiated,then i ’s income must be equal to the minimum expenditure ensuring her reservationutility. This naturally establishes no justified envy: If agent j envies i , then j musthave less equilibrium income than i , so that i must find j ’s allocation unacceptable.Finally, it is clear that IR is ensured, since no agent’s income lies below the minimumexpenditure ensuring her reservation utility.Once income functions are in place, existence follows standard ideas: first show-ing the existence of quasi-equilibrium (as in e.g Mas-Colell, Whinston, Green, et al.(1995), chapter 17, appendix B), and then exploiting the conditions stated in thetheorem to bridge the gap between quasi-equilibrium and competitive equilibrium.To this end, the conditions stated in the theorem play a technical role. They serveto ensure that either all prices or incomes are strictly positive for all agents. Thefirst condition consists of virtually unbounded (above) consumption spaces. Thesecond condition considers preferences for strictly positive bundles alongside withreservation utility above the minimal level. The third condition is based on theexistence of a common favorite object, and a strictly positive IR allocation. Theresulting allocation guarantees all the desired properties in their strongest sense. When reservation utility arises from agents’ endowments, one may be tempted to use Wal-rasian incomes. These are, of course, price dependent, and ensure IR. Unfortunately, there aresimple examples of allocation problems with endowments where no Walrasian equilibria exist(Hylland and Zeckhauser, 1979). See Echenique, Miralles, and Zhang (2019) for a discussion. A corollary is that if the consumption space is R L + for all agents, then our Walrasian approachobtains an allocation that is Pareto optimal, IR and has no justified envy. AIRNESS AND EFFICIENCY 13
Similarly, Lipschitz continuity in Corollary 1 is also a technical condition, ensuringthat we can easily create an (arbitrarily) low amount of an artificial favorite good.Existence of a common favorite object is a restrictive assumption. The corollaryworks by extending an economy with the addition of an artificial common favoritegood. Lipschitz continuity is needed in order to facilitate such inclusion. We takethe limit when the supply of the artificial good tends to zero, obtaining the limit ofa sequence of Pareto optimal allocations.The addition of an artificial favorite good also lies behind the proof of State-ment (3) in Theorem 1. With linear (or expected) utility functions, the set ofPareto optimal allocations is closed in the amount of the artificial good. Hence, thelimit allocation is also Pareto optimal.5.
Efficient and fair assignment under constraints
Many allocation problems require allocations to satisfy certain quantitative con-straints. It is easy to adapt our model and results to such situations. In Sec-tion 5.1, we take a set A C ⊆ A as the primitive and interpret it as the set ofallocations that comply with some given collection of constraints. A C is closed andconvex, and we implicitly assume that the behind constraints satisfy the condition ofBudish, Che, Kojima, and Milgrom (2013), which ensures that every feasible allo-cation can be achieved as randomized deterministic feasible allocations. We discussseveral examples of explicit constraints in Section 5.2.5.1. Constrained allocations.
Given a set of feasible allocations A C ⊆ A , thedefinition of individual rationality is same as before. We assume that there existsan IR allocation in A C . The definition of efficiency extends naturally to A C . Anallocation x ∈ A C is Pareto optimal if there is no allocation y ∈ A C such that u i ( y i ) ≥ u i ( x i ) for all i ∈ I with strict inequality for some agent; x is weak Paretooptimal (wPO) if there is no allocation y ∈ A C such that u i ( y i ) > u i ( x i ) for all i ∈ I ;and x is ε -Pareto optimal ( ε -PO), for any ε >
0, if there is no allocation y ∈ A C such that u i ( y i ) > u i ( x i ) + ε for all i ∈ I .Until now, i ’s envy towards j is negated if switching their assignments violates theparticipation constraint of j . Now, additional constraints provide another reason fornegating i ’s envy: switching their assignments may not be feasible because it violatessome constraints. To formalize this idea, let x i ↔ j denote the allocation obtained byswitching the assignments of i and j in an allocation x ; that is, x ii ↔ j = x j , x ji ↔ j = x i ,and x ki ↔ j = x k for all k ∈ I \ { i, j } . An agent i has justified envy towards another agent j at an allocation x ∈ A C if u i ( x j ) > u i ( x i ) , u j ( x i ) ≥ ˜ u j and x i ↔ j ∈ A C . Under the new definition, no justified envy may no longer be compatible with effi-ciency and individual rationality. To overcome this difficulty, we classify agents intodisjoint types. Informally, think of i and j as being of equal type if the constraintsbehind A C do not distinguish between them. We identify agents’ types by checkingwhether switching their assignments in any feasible allocation is still feasible. Wethen prove that fairness among agents of equal type is compatible with efficiencyand individual rationality.Formally, we say two agents i, j are of equal type , denoted by i ∼ j , if for all x ∈ A C , x i ↔ j ∈ A C . The binary relation ∼ is reflexive and transitive. Hence itpartitions I into disjoint types. Then we say i has equal-type justified envy towards j at an allocation x ∈ A C if i has justified envy towards j , and i, j are of equal type.We say that x has no equal-type justified envy if no agent has equal-type justifiedenvy towards any other agent. No strong equal-type justified envy and no equal-type ǫ -justified envy are defined in a similar way by stating that the relevant envy isabsent in the allocation.With the above definitions, we extend the first two statements of Theorem 1 toaccommodate constraints. Theorem 3.
Suppose that agents’ utility functions are concave.(1) For any ε > , there exists an allocation that is ε -individually rational, ε -Pareto optimal and has no equal-type ε -justified envy;(2) There exists an allocation that is individually rational, weak Pareto optimaland has no strong equal-type justified envy. We say that the implicit constraints behind A C are anonymous if all agents areidentified to be of equal type. The model in Section 3 is one where constraints areanonymous since A C = A .5.2. Constraint structures.
It is often most useful to explicitly model the sourceof types and constraints. For example, types can arise from definitions of socio-economic status, or racial and ethnic classifications. Following the approach ofBudish, Che, Kojima, and Milgrom (2013), we define a general constraint structure, Suppose i ∼ j ∼ k . For all x ∈ A C , x i ↔ k = [( x i ↔ j ) j ↔ k ] i ↔ j . i ∼ j implies that x i ↔ j ∈ A C , j ∼ k implies that ( x i ↔ j ) j ↔ k ∈ A C , and i ∼ j implies that [( x i ↔ j ) j ↔ k ] i ↔ j ∈ A C . So x i ↔ k ∈ A C . AIRNESS AND EFFICIENCY 15 and then discuss some examples in which an exogenous collection of types gives riseto constrained allocations.A constraint [ H, ( q H , ¯ q H )] consists of a set H ⊆ I × O and a pair of integers( q H , ¯ q H ) with q H ≤ ¯ q H . Given a collection H of constraints, we define A H = (cid:8) x ∈ A : q H ≤ X ( i,l ) ∈ H x il ≤ ¯ q H for all [ H, ( q H , ¯ q H )] ∈ H (cid:9) as the set of feasible allocations satisfying H .The first example is controlled school choice in which the set of students I arepartitioned into disjoint subsets T , . . . , T K (which are interpreted as types), andfor each school l , the desirable number of type k students where k ∈ { , . . . , K } isbetween q l,k and ¯ q l,k . So for each school l and each type k ∈ { , . . . , K } , we have aconstraint (cid:2) T k × { l } , ( q l,k , ¯ q l,k ) (cid:3) . Theorem 3 says that there is an efficient and individually rational allocation thatachieves fairness within each type.The second example is the collection of distributional constraints studied byKamada and Kojima (2015, 2017). In that collection, every constraint is of theform (cid:2) I × O ′ , (0 , ¯ q O ′ ) (cid:3) where agents are doctors and O ′ ⊆ O is the set of hospitals in a geographic region(a city or a prefecture). The collection of constraints is anonymous because eachconstraint does not distinguish between the identities of doctors. In general, acollection H is anonymous if for every constraint [ H, ( q H , ¯ q H )] ∈ H , H is of theform I × O ′ for some O ′ ⊆ O . Theorem 3 implies that for anonymous constraints,there is an efficient and individually rational allocation that achieves fairness among any two agents.In the last example, H consists of individual constraints , which impose restric-tions on each individual’s assignment. In our model of Section 3, we have alreadyencountered the N individual constraints: [ { i } × O, (0 , c i )] for each i ∈ I . Formally, H consists of H i where each H i further consists of constraints of the form[ { i } × O ′ , ( q i,O ′ , ¯ q i,O ′ )]with O ′ ⊆ O . For example, in course allocation if a student i has to take at least onemath course but no more than three math courses, we can impose the constraint [ { i } × O ′ , (1 , O ′ is the set of math courses. Theorem 3 says that we canachieve fairness among agents of equal individual constraints.6. Discussion
Justified envy by exchange.
Our notion of NJE relies on pairwise switchas being the remedy for envy. We think of such switch as natural. But if pairwiseswitch is seen as limited, it is important to note that our result is, in fact, easilygeneralized to allow for more general remedies.Let us think of envy that can be addressed by carrying out a chain of exchanges,each agent giving up her assignment in favor of an agent who envies her, and thelast agent in the exchange being given the assignment of the first agent. If thisreallocation does not violate the last agent’s participation constraint, then the envyis justified.Formally, agent i has justified envy by exchange towards agent j at allocation x if there exists a sequence of distinct agents ( i k ) Kk =1 with • i = i and i = j ; • i k envies i k +1 , 1 ≤ k ≤ K − • and u i K ( x i ) ≥ ˜ u i K .The idea is that i could conjure a remedy for her envy towards j by proposing acoalition of agents and a reallocation of their assignments, such that all are madebetter off, with the possible exception of one agent whose participation constraintis not violated. We define strong justified envy by exchange and ε -justified envy byexchange similarly as before. We prove that our main theorems hold without changeunder this extended fairness notion. Theorem 4.
Theorem 1 and Theorem 2 hold without change if no ( ε -/strong) jus-tified envy is replaced by no ( ε -/strong) justified envy by exchange. An example of envy between agents of equal endowment in an allo-cation of no justified envy.
It is natural to think of endowments as a source forreservation utility. In this subsection, we present an example of a discrete allocationproblem in which one agent envies another agent in an allocation that is individu-ally rational, Pareto optimal, and satisfies NJE, even though the two agents haveequal endowments. The punchline is that the two agents have different preferences,so that they play very different roles in the economy. Other agents “trade” with
AIRNESS AND EFFICIENCY 17 them, and as a result one of them ends up being more useful than the other to theremaining agents. The outcome implies the presence of envy.The example also suggests that our solution may fail to be incentive compatible.We have not specified a selection mechanism, and opted not to discuss incentivesand strategy-proofness, but the example conveys some insights. One agent enviesanother even though they have equal endowments. This fact suggests that oneagent may want to pretend to be the agent that he envies. In a large economy inwhich the number of agents who report each type of preference does not changevery much after a misreport, it stands to reason that such a misreport would not beprofitable. Of course, the example falls short of proving that if we were to define afair mechanism it would not be strategy proof.
Example . Consider five agents { , , , , } and three objects { a, b, c } . There aretwo copies of objects b and c , but only one copy of object a . Agents have linearutility functions. Their von-Neumann-Morgenstern (vNM) utilities and endowmentsare described in Table 1. Observe that agents 1 and 2 have identical endowments. i u ia u ib u ic (a) Utilities i ω ia ω ib ω ic (b) Endowments i x ia x ib x ic (c) Allocation x Table 1.
Example 1Consider the allocation x in the same table. Agent 1 envies agent 2 at x because u · x = 2 < / / u · x . The envy is not justified, however, because u · x = 1 < u · ω . In fact, it is easy to see that x has no justified envy, and is individually rationaland Pareto optimal. In any PO allocation y , we cannot have y b >
0, as agent 1 andany agent j ∈ { , , } are willing to trade b for any other object. So y must bea convex combination of (1 , ,
0) and (0 , , would need to give agent 1 some shares of a , but these can only come at the expenseof agent 2. To make agent 2 better off, she would need to get more shares of a , butthese can only come at the expense of agents 3, 4 and 5. These agents could onlyexchange shares of a for shares of b , which agent 2 does not have. All agents rankobjects a and c in the same way.7. Application to school choice
School choice is the problem of allocating children to schools when we want to takeinto account children’s (or their parents’) preferences (Abdulkadiro˘glu and S¨onmez,2003). In the last 15 years, several large US school districts have implementedschool choice programs that follow economists’ recommendation and are based oneconomic theory. Practical implementation of school choice programs presents uswith a number of lessons and challenges.The first lesson is that school choice should be guided by fairness, or lack ofjustified envy. When given the choice of implementing either a fair or an efficient out-come, school districts have consistently chosen fairness (Abdulkadiro˘glu, Pathak, Roth, and S¨onmez,2005; Abdulkadiro˘glu, Pathak, and Roth, 2005). One reason could be that districtadministrators are concerned with litigation: If Alice envies Bob’s school, then thedistrict can invoke justified envy to argue as a defense that Bob had a higher prioritythan Alice at the school. It is also likely that district administrators, and societyas a whole, have an intrinsic preference for fairness, and the preference is strongenough to outweigh concerns over efficiency.The second lesson is that school districts want to give some children certain rights,like the right to attend a neighborhood school, or the right to go to the same schoolas an older sibling. In the current practice, such rights are achieved by givingchildren different priorities. Priorities seem simple, but they are not transparent:Priorities do not translate immediately into allocation outcomes. Alice may havea good priority in one school, but her chance of getting into the school dependson all students’ choices and priorities, not only on her priority at the school. Thisis especially true when priorities are coarse, which is common in practice. Also, Boston (Abdulkadiro˘glu and S¨onmez, 2003; Abdulkadiro˘glu, Pathak, Roth, and S¨onmez,2005), New York (Abdulkadiro˘glu, Pathak, and Roth, 2005), and Chicago (Pathak and S¨onmez,2013) are leading examples. Observe that this notion of justified envy is pairwise, as is ours. In school choice with coarse priorities, we can prove that it is computationally hard to deter-mine if a given student will be assigned a given school.
AIRNESS AND EFFICIENCY 19 the ordering of students in a priority may not reflect their “rights” at the school.Below we construct an example in which if two students switch priorities at oneschool, the student who climbs up in the priority ends with a worse outcome in thestudent-optimal stable matching.
Example . Consider three schools { a, b, c } and three children { , , } . Supposethat school priorities, and children preferences are as follows. a b c b a ac b ca c b Then the student-optimal stable matching is µ (1) = b , µ (2) = a and µ (3) = c .If 1 and 2 switch roles in the priority ranking of school a , then the student-optimalstable matching becomes µ (1) = c , µ (2) = b and µ (3) = a . So 1 becomes worse off.The trick here is that when 1 and 2 switch roles, they also change their positionsrelative to 3. The message of the example is that justified envy, a “pairwise” conceptin school choice, cannot rely on the relative position in schools’ priorities of the pairin question.The third lesson is that school districts have demonstrated a strong preference forcontrolling the racial and socio-economic composition of their schools: so-called con-trolled school choice . A common critique of existing school choice programs is thatthey have led to undesirable school compositions. For example, in Boston, schoolshave been left with too few neighborhood children, which has motivated a move awayfrom the system recommended by economists (Dur, Kominers, Pathak, and S¨onmez,2017). In New York City, the new school choice system exhibits high degrees of racialsegregation. Segregation in NYC schools is not new, but the complaint is that thenew school choice program may have made it worse, and certainly has not helped.In the words of a recent New York Times article “. . . school choice has not deliveredon a central promise: to give every student a real chance to attend a good school.Fourteen years into the system, black and Hispanic students are just as isolated insegregated high schools as they are in elementary schools — a situation that schoolchoice was supposed to ease.” The article points to a dissatisfaction with schoolcomposition, and access to the best schools. “The Broken Promises of Choice in New York City Schools”, New York Times , May 5th, 2017.
The situation in NYC has reached a point where there are talks of doing away withschool priorities, and instead instituting a lottery. In fact, Professor Eric Nadelsternat Columbia University, who served as deputy school chancellor when the new schoolchoice system was implemented, has recently proposed that children be allowed toapply to any school, and have a lottery deciding the allocations. Given the absence of a direct connection between priorities and outcomes, andthe situation in NYC, we propose the use of endowments to control children’s rights.This makes Nadelstern’s proposal compatible with school choice. We imagine thatthere is a lottery that gives an initial probabilistic allocation of children to schools.The lottery could be as simple as giving each child the same chance of attendingany school. It could also reflect different objectives in controlled school choice,such as giving each child a higher chance of attending his or her neighborhoodschool, or giving each minority child a chance (literally, a positive probability) ofattending the highest-ranked schools. The initial allocation, or endowment, providestransparent and immediate reservation utility. A child who is endowed with a seatat her neighborhood school can simply choose to attend that school. His or her rightto attend that school does not depend on other children in any way.The initial allocation is typically not the final allocation, because we want pref-erences to play a role. We assume that children use expected utilities to comparelotteries and ask them to report vNM utilities of schools. For convenience, we mayuse normalization by requiring each child to assign utility 1 to his favorite schooland assign utility 0 to the worst school. To capture the desired bounds on thecomposition of a school, we can use quantity constraints. Subject to constraints,our solution achieves all the desirable properties. The final allocation will be fair,efficient, and individually rational.Beyond school choice, our model and results can apply to other market designproblems. An example is time bank where agents exchange labor without usingtransfer (see Andersson, Csehz, Ehlers, Erlanson, et al. (2018)). In that problemevery agent demands others’ services and also provides services to others. Theservices an agent can supply are her endowments, and define her reservation utility. “Confronting Segregation in New York City Schools”, New York Times , May 15th, 2017.
AIRNESS AND EFFICIENCY 21 Proof of Theorems 1 and 3
We prove Theorem 3 in this section. The first two statements of Theorem 1 arecorollaries. We prove the third statement of Theorem 1 in Section 10 after provingTheorem 2 in the next section.For any given ε >
0, define A ∗ = { x ∈ A C : x is ε -individually rational } . It is easy to see that A ∗ is nonempty and compact.Let the N -dimensional simplex ∆ be the domain of welfare weights.For any λ ∈ ∆, define φ ( λ ) = argmax { X i ∈ I λ i u i ( x i ) − δ X i ∈ I k x i − k : ( x i ) i ∈ I ∈ A ∗ } , where is a vector of ones and δ > δ max x ∈A ∗ X i ∈ I k x i − k < ε. Since all u i are continuous and concave and P i ∈ I k x i − k is continuous and strictlyconvex, the objective function P i ∈ I λ i u i ( x i ) − δ P i ∈ I k x i − k is continuous andstrictly concave. Moreover, A ∗ is compact. Thus, φ : ∆ → A ∗ is a function (meaningit is singleton-valued), and, by the Maximum Theorem, continuous. Moreover, thechoice of δ implies that φ is ε -Pareto optimal.For any agent i , defineΛ i = { λ ∈ ∆ : ∄ j ∈ I s.t i has equal-type ε -justified envy towards j at φ ( λ ) } . The proof relies on an application of the so-called KKM Lemma (the lemma isdue to Knaster, Kuratowski and Mazurkiewicz; see Theorem 5.1 in Border (1989)).In the following two lemmas we prove that { Λ i } Ni =1 is a KKM covering of the simplex∆. This means that every Λ i is closed and that for any λ ∈ ∆ there is at least oneΛ i such that λ i > λ ∈ Λ i . Lemma 1.
For every i ∈ I , Λ i is closed.Proof. Let λ n be a sequence in Λ i such that λ n → λ ∈ ∆. Let x n = φ ( λ n ). Bycontinuity of φ , x n → x = φ ( λ ) ∈ A ∗ . Now we prove that λ ∈ Λ i , that is, i does nothave equal-type ε -justified envy towards any other agent. Suppose that there is anagent j of equal type with i such that u i ( x j ) > u i ( x i ) and u j ( x i ) > ˜ u j − ε . Since i and j are of equal type, x i ↔ j ∈ A C , and ( x n ) i ↔ j ∈ A C for every n . By continuity of u i and u j , for n large enough we have u i ( x jn ) > u i ( x in ) and u j ( x in ) > ˜ u j − ε . Thesemean that i has equal-type ε -justified envy towards j at x n , which is a contradiction.Therefore, λ ∈ Λ i and Λ i is closed. (cid:3) Lemma 2.
For every λ ∈ ∆ , λ ∈ ∪ i ∈ supp ( λ ) Λ i .Proof. Suppose, towards a contradiction, that for some λ ∈ ∆, λ / ∈ ∪ i ∈ supp ( λ ) Λ i .Let x = φ ( λ ). Then for every i ∈ supp( λ ) there exists some j of equal type with i such that u i ( x j ) > u i ( x i ) and u j ( x i ) > ˜ u j − ε .Suppose first that there exist some i and j in the aforementioned situation suchthat j / ∈ supp( λ ). Then consider the allocation y = x i ↔ j ∈ A C . y is ε -individuallyrational as x was ε -individually rational and u j ( x i ) > u j ( ω j ) − ε . Note that λ j = 0and u i ( x j ) > u i ( x i ) imply that P h ∈ I λ h u h ( x h ) < P h ∈ I λ h u h ( y h ). We also have that P h ∈ I k x h − k = P h ∈ I k y h − k , hence X h ∈ I λ h u h ( x h ) − δ X h ∈ I k x h − k < X h ∈ I λ h u h ( y h ) − δ X h ∈ I k y h − k . But it contradicts the definition of x = φ ( λ ).The above argument means that every i ∈ supp( λ ) has equal-type ε -justified envytowards some j ∈ supp( λ ). Then, since the set of agents in supp( λ ) is finite, theremust exist a subset of distinct agents { i , . . . i K } ⊆ supp( λ ) such that i has equal-type ε -justified envy towards i , i has equal-type ε -justified envy towards i , andso on until i K has equal-type ε -justified envy towards i . Then we can constructa new allocation y by letting agents in the cycle exchange their allocations. Sincethe agents in the cycle are of equal type, y must be feasible, that is, y ∈ A C . Asbefore, we have that P h ∈ I k x h − k = P h ∈ I k y h − k because y is obtained from x by permuting the assignments of agents in the cycle. Then we have X h ∈ I λ h u h ( x h ) − δ X h ∈ I k x h − k < X h ∈ I λ h u h ( y h ) − δ X h ∈ I k y h − k . As before, it is a contradiction. (cid:3)
Now we are ready to prove Theorem 3.
Proof of Theorem 3.
The proof is an application of the KKM lemma: see Theorem5.1 in Border (1989). We can consider a sequence of allocations { x ( k ) } K − k =0 with x (0) = x and x ( k ) = x i k ↔ i k +1 ( k − ≤ k ≤ K −
1. Since all agents in the cycle are of equal type, each x ( k ) ∈ A C . We let y = x ( K − AIRNESS AND EFFICIENCY 23
By Lemmas 1 and 2, { Λ i } ni =1 is a KKM covering of ∆. So there exists λ ∗ ε ∈ ∩ ni =1 Λ i .Let x ∗ ε = φ ( λ ∗ ε ). Then x ∗ ε is ε -individually rational, ε -Pareto optimal and has noequal-type ε -justified envy.Now let { ε n } be a sequence such that ε n > n and ε n →
0. Let x ∗ n be theallocation found above for each ε n . Since the sequence { x ∗ n } is bounded, it has asubsequence { x ∗ n k } that converges to some x ∗ . Since the set of feasible allocationsis closed, x ∗ is a feasible allocation. We prove that x ∗ is individually rational, weakPareto optimal and has no strong equal-type justified envy.Since u i ( x ∗ in k ) ≥ ˜ u i − ε n k for all n k and all i , in the limit u i ( x ∗ i ) ≥ ˜ u i for all i .So x ∗ is individually rational. Suppose x ∗ is not weak Pareto optimal, then thereexists a feasible allocation y such that u i ( y i ) > u i ( x ∗ i ) for all i . For big enough n k , u i ( y i ) > u i ( x ∗ in k ) + ε n k for all i , which contradicts the ε n k -Pareto optimality of x ∗ n k . Suppose some agent i has strong equal-type justified-envy towards anotheragent j in x ∗ ; that is, u i ( x ∗ j ) > u i ( x ∗ i ) and u j ( x ∗ i ) > ˜ u j . Then for big enough n k , u i ( x ∗ jn k ) > u i ( x ∗ in k ) and u j ( x ∗ in k ) > ˜ u j − ε n k . But given that i and j are of equal type,this contradicts the property of no equal-type ε n k -justified envy of x ∗ n k . (cid:3) Proof of Theorem 2
We let the L -dimensional simplex ∆ L be the domain of prices.9.1. Incomes.
The key to the theorem is to carefully construct price-dependentincome functions. For each consumer i , define i ’s expenditure function as e i ( v, p ) = inf { p · x : u i ( x ) ≥ v } , for p ∈ ∆ L and v ∈ R .Let v i = sup u i ( C i ) be the utility of agent i when she is satiated.For any scalar m ≥ p ∈ ∆ L , let µ i ( m, p ) = median( { e i (˜ u i , p ) , m, e i ( v i , p ) } ) . Consider the function ϕ ( m, p ) = X i µ i ( m, p ) − p · Q. Observe that • e i (˜ u i , p ) ≤ e i ( v i , p ). • µ i is continuous and m µ i ( m, p ) weakly monotone increasing. • ϕ is continuous and m ϕ ( m, p ) weakly monotone increasing. • ϕ ( m, p ) ≤ m ≥ P i e i (˜ u i , p ) ≤ p · Q (since an IRallocation x exists, e i (˜ u i , p ) ≤ p · x i for all i , and P i p · x i = p · Q .)We shall define m i ( p ). First, in the case that P i e i ( v i , p ) < p · Q , we let m i ( p ) = e i ( v i , p ) + N [ p · Q − P i e i ( v i , p )]. Second, in the case that P i e i ( v i , p ) ≥ p · Q , wehave that ϕ ( m, p ) ≤ m ≥ ϕ ( m, p ) ≥ m ≥ m ∗ ≥ ϕ ( m ∗ , p ) = 0.Now let m i ( p ) = µ i ( m ∗ , p ). To show that this is well defined, we need to provethat m i ( p ) is independent of the choice of m ∗ . To that end, suppose that there are m , m ∈ R + with m = m and 0 = ϕ ( m , p ) = ϕ ( m , p ). Suppose without lossof generality that m < m . Now, since each µ i is weakly monotone increasing as afunction of m we must have µ i ( m , p ) = µ i ( m , p ) for all i . Then the definition of m i ( p ) is the same regardless of whether we choose m or m .Note that, in all cases, p · Q = P i m i ( p ). Lemma 3. m i is continuous.Proof. Let p n → p ∈ ∆ L . Note that if P i e i ( v i , p ) − p · Q <
0, then for n largeenough we will have P i e i ( v i , p n ) − p n · Q <
0. Then m i ( p n ) = e i ( v i , p n ) + N [ p n · Q − P i e i ( v i , p n )] → e i ( v i , p ) + N [ p · Q − P i e i ( v i , p )] = m i ( p ), by continuity of theexpenditure function.So suppose that P i e i ( v i , p ) − p · Q ≥
0, and let m be such that ϕ ( m, p ) = 0. Weshall discuss two cases.Case1: Consider the case that P i e i ( v i , p n k ) − p n k · Q < p n k . Then P i e i ( v i , p ) − p · Q = 0. This means that if ϕ ( m, p ) = 0 then m ≥ e i ( v i , p )for all i . Hence m i ( p ) = e i ( v i , p ) for all i . But since m i ( p n k ) = e i ( v i , p n k ) + N [ p n k · Q − P i e i ( v i , p n k )], we get that m i ( p n k ) → m i ( p ).Case 2: Now turn to a subsequence p n k with P i e i ( v i , p n k ) − p n k · Q ≥
0. Thenthere is m n k with ϕ ( m n k , p n k ) = 0. We can take this sequence to be bounded:consider any further convergent subsequence m n ′ k and say that m n ′ k → m ′ . Then0 = ϕ ( m n ′ k , p n ′ k ) → ϕ ( m ′ , p ). Thus m i ( p n ′ k ) = µ i ( m n ′ k , p n ′ k ) → µ i ( m ′ , p ), as µ i iscontinuous. Since the sequence { m n k } is bounded, this implies that m i ( p n k ) → m i ( p ).Cases 1 and 2 exhaust all possible subsequences of p n . (cid:3) The role of the following lemma will be clear towards the end of the proof.
Lemma 4. If m i ( p ) < min { m j ( p ) , e i ( v i , p ) } then m j ( p ) = e j (˜ u j , p ) . AIRNESS AND EFFICIENCY 25
Proof.
Since m i ( p ) < e i ( v i , p ), we must be in the case P i e i ( v i , p ) ≥ p · Q of thedefinition of income functions. So let m ∗ ≥ ϕ ( m ∗ , p ) = 0.Since m i ( p ) = µ i ( m ∗ , p ) < e i ( v i , p ), we must have m ∗ ≤ m i ( p ). By hypothesis, m ∗ < m j ( p ). Then m j ( p ) = µ j ( m ∗ , p ) implies that m j ( p ) = e j (˜ u j , p ). (cid:3) Existence of quasi-equilibrium.
We first establish the existence of a quasiequi-librium with p ∗ = 0. The argument is similar to Gale and Mas-Colell (1975). Seealso Mas-Colell, Whinston, Green, et al. (1995) (Chapter 17, Appendix B).For any p ∈ ∆ L , let d i ( p ) be the set of vectors x i ′ ∈ C i that satisfy the followingproperties: p · x i ′ ≤ m i ( p ) u i ( x i ′ ) ≥ u i (ˆ x i ) for all ˆ x i ∈ C i with p · ˆ x i < m i ( p ) . We consider the correspondence p d i ( p ) with domain in ∆ L .Observe that ∅ = arg max x i ′ ∈ C i { u i ( x i ′ ) : p · x i ′ ≤ m i ( p ) } ⊆ d i ( x, p )). So d i takes non-empty values.Observe also that d i is convex valued. To see this, let z i , y i ∈ d i ( p ) and define x i ( α ) = αz i + (1 − α ) y i , for α ∈ [0 , x i ( α ) ∈ C i and that p · x i ( α ) ≤ m i ( p ). For any ˆ x i ∈ C i with p · ˆ x i < m i ( p ), min { u i ( z i ) , u i ( y i ) } ≥ u i (ˆ x i )and quasi-concavity of u i imply that u i ( x i ( α )) ≥ u i (ˆ x i ). Thus x i ( α ) ∈ d i ( p ).A third observation is that d i ( p ) is upper-hemicontinuous. To this end, considera sequence p n in ∆ L with p n → p ∈ ∆ L . Consider z in ∈ d i ( p n ) such that z in → z i .Clearly, z i ∈ C i and p · z i ≤ m i ( p ) as m i is continuous (Lemma 3). Moreover, forany ˆ x i ∈ C i with p · ˆ x i < m i ( p ), we have that p n · ˆ x i < m i ( p n ) for n large enough(again by Lemma 3). Thus u i ( z in ) ≥ u i (ˆ x i ) for n large enough, which by continuityof u i implies that u ( z i ) ≥ u i (ˆ x i ). Hence z i ∈ d i ( p ).For any x ∈ × i C i and p ∈ ∆ L , let¯ π ( x, p ) = argmax { p · X i x i − Q ! : p ∈ ∆ L } , and consider the correspondence ξ : × i C i × ∆ L ։ × i C i × ∆ L defined by ξ ( x , . . . , x N , p ) = ( × i d i ( p )) × ¯ π ( x, p ).By the previous observations, and the maximum theorem, ξ is in the hypothesesof Kakutani’s fixed point theorem. Let ( x ∗ , p ∗ ) be a fixed point of ξ . We argue that ( x ∗ , p ∗ ) is a Walrasian quasiequilibrium. We have that p ∗ · x i ∗ ≤ m i ( p ∗ ) for every i , by construction of ξ . By definition of m i , we have P i m i ( p ∗ ) = p ∗ · Q . Hence, p ∗ · ( P i x i ∗ − Q ) ≤ . This implies P i x i ∗ − Q ≤ π , we would have p ∗ · X i x i ∗ − Q ! = max p ′ ∈ ∆ L (cid:8) p ′ · X i x i ∗ − Q ! (cid:9) > . We show that P i x i ∗ − Q = 0. We first consider the case P i e i ( v i , p ∗ ) < p ∗ · Q .By definition of m i , all agents i have m i ( p ∗ ) > e i ( v i , p ∗ ) so they must be satiatedfollowing the definition of d i . By monotonicity of preferences, we observe P l x i ∗ l = c i ,hence X i X l x i ∗ l = X i c i ≥ X l q l where the inequality comes from the no overall excess supply assumption. Giventhat we knew P i x i ∗ − Q ≤
0, we conclude P i x i ∗ − Q = 0.We then consider the case P i e i ( v i , p ∗ ) ≥ p ∗ · Q . We claim that p ∗ · x i ∗ = m i ( p ∗ ) forevery i , since by definition of m i we have m i ( p ∗ ) ≤ e i ( v i , p ∗ ). Indeed, suppose that p ∗ · x i ∗ < m i ( p ∗ ) ≤ e i ( v i , p ∗ ). Since x i ∗ does not satiate the agent, for an arbitrarilysmall ball B around x i ∗ there is x i ′ ∈ B with u i ( x i ′ ) > u i ( x i ∗ ) and p ∗ · x i ′ < m i ( p ∗ ),contradicting x i ∗ ∈ d i ( x ∗ , p ∗ ). Observe that, as a consequence of the above,(2) p ∗ · Q = X i m i ( p ∗ ) = X i p ∗ · x i ∗ . Consequently, p ∗ · ( P i x i ∗ − Q ) = 0 . Since P i x i ∗ − Q ≤ , we obtain p ∗ l = 0 forany l with P i x ∗ il − q l < P i x i ∗ − Q = 0 by consuming the remainingunits of underdemanded objects for free.This proves that ( x ∗ , p ∗ ) is a Walrasian quasiequilibrium.9.3. Existence of equilibrium.
We prove now that ( x ∗ , p ∗ ) is a Walrasian equi-librium in the cases considered in the Theorem. In all cases we prove that, foreach agent i , either m i ( p ∗ ) > m i ( p ∗ ) > y i ∈ C i such that u i ( y i ) > u i ( x i ∗ ) and p ∗ · y i ≤ m i ( p ∗ ). Then, for λ < λy i ∈ C i , p ∗ · λy i < m i ( p ∗ )and, by continuity of preferences, u i ( λy i ) > u i ( x i ∗ ), contradicting x i ∗ ∈ d i ( p ∗ ). For AIRNESS AND EFFICIENCY 27 the remaining case m i ( p ∗ ) = 0, if 0 is the sole affordable bundle, then 0 = x ∗ i trivially is i ’s optimal choice subject to her budget constraint.In all cases, we skip the possibility of P i e i ( v i , p ∗ ) < p ∗ · Q . By definition of m i , allagents i would have m i ( p ∗ ) > e i ( v i , p ∗ ) so they would certainly be satiated followingthe definition of d i . Therefore we would trivially have an equilibrium. Note that,by skipping such a possibility, we have p ∗ · x i ∗ = m i ( p ∗ ) for all i .We first consider the case min i c i > P l q l . We show that p ∗ >>
0. Suppose, byway of contradiction, that p ∗ l = 0 for some good l . Since p ∗ · Q > P i x i ∗ − Q = 0,there must be an individual j with p ∗ · x j ∗ = m j ( p ∗ ) >
0. Take a vector δ containingzeros in all coordinates but l , where it contains ǫ >
0. Notice that c j > P l q l implies x j ∗ + δ ∈ C i for ǫ small enough. By monotonicity and continuity of preferences,and since x j ∗ + δ is also affordable, a standard argument shows that x j ∗ is nota quasiequilibrium allocation for j under prices p ∗ . We conclude that p ∗ >> i with m i ( p ∗ ) = 0, the 0 bundle is her only affordablebundle.We now consider the case when both Inada condition u i ( x i ) = u i (0) unless x i >> u i > u i (0) hold for all i . p ∗ ∈ ∆ L contains at least one strictly positive price,thus m i ( p ∗ ) ≥ e i (˜ u i , p ∗ ) > i .Lastly, we consider the case that a common favorite object l and a strictly positiveIR allocation ˜ x both exist. We argue that p ∗ l > . Suppose that p ∗ l = 0 . Since p ∗ ∈ ∆ L , there must be an object k = l with p ∗ k > . For any agent i who is consumingobject k , substituting his consumption of object k for an equal consumption of object l saves expenses and increases utility. Hence x i ∗ / ∈ d i ( p ∗ ). This contradiction showsthat p ∗ l > . Notice that, in consequence, e i ( v i , p ∗ ) > i .Our next step is to establish that m = min { m i ( p ∗ ) : 1 ≤ i ≤ I } >
0. First, if m = min { e i ( v i , p ∗ ) : 1 ≤ i ≤ I } then we are done because e i ( v i , p ∗ ) > i .Ruling out this case, there must exist i with m i ( p ∗ ) < e i ( v i , p ∗ ), which implies that P i m i ( p ∗ ) = p ∗ · Q = P i p ∗ · ˜ x i . Now, if(3) m = min { e i (˜ u i , p ∗ ) : 1 ≤ i ≤ I } , then there is h with m = m h ( p ∗ ) ≤ e i (˜ u i , p ∗ ) for all i ; which implies by the definitionof the income functions that m i ( p ∗ ) = e i (˜ u i , p ∗ ) for all i . But e i (˜ u i , p ∗ ) ≤ p ∗ · ˜ x i forall i and X i e i (˜ u i , p ∗ ) = X i m i ( p ∗ ) = p ∗ · Q = X i p ∗ · ˜ x i imply that e i (˜ u i , p ∗ ) = p ∗ · ˜ x i for all i . So m i ( p ∗ ) = p ∗ · ˜ x i for all i . Because ˜ x i >> m i ( p ∗ ) > i .Finally, if Equation (3) does not hold, then 0 ≤ min { e i (˜ u i , p ∗ ) : 1 ≤ i ≤ I } < m .So m i ( p ∗ ) > i .9.4. Properties of a competitive equilibrium allocation x ∗ . Pareto optimality.
We disregard the case P i e i ( v i , p ∗ ) < p ∗ · Q in which clearlyevery agent i is satiated since m i ( p ∗ ) > e i ( v i , p ∗ ). In the cases that follow below,any satiated agent i must have m i ( p ∗ ) = e i ( v i , p ∗ ). Suppose that y i ∈ C i and that u i ( y i ) ≥ u i ( x i ∗ ). Then we must have p ∗ · y i ≥ m i ( p ∗ ) because otherwise p ∗ · y i
To show that x ∗ is individually rational it suffices tonotice that m i ( p ∗ ) ≥ e i (˜ u i , p ∗ ) for all i .9.4.3. No justified envy.
Suppose that i envies j at x ∗ . This implies that i is notsatiated, hence m i ( p ∗ ) < e i ( v i , p ∗ ). It also implies that m i ( p ∗ ) < m j ( p ∗ ) as m i ( p ∗ )
Proof of Corollary 1.
We denote a constant of Lipschitz continuity commonto all utility functions by θ . Let y be an IR allocation, which exists by assumption.Consider an additional object e / ∈ O , and an α -extended economy, for any α ∈ (0 , Q α : q αl = (1 − α ) q l , ∀ l ∈ O, and q αe = α X i c i . AIRNESS AND EFFICIENCY 29
Preferences in this extended economy are defined to be: U i (( x l ) l ∈ O , x e ) = u i (( x l ) l ∈ O ) + θx e . Notice that under this construction, e is a common favorite good in this extendedeconomy. By Lipschitz continuity, the allocation y α with y iαl = (1 − α ) y il for l ∈ O and y iαe = αc i meets U i ( y αi ) > ˜ u i for all i ∈ I . Therefore, by continuity ofpreferences, for β > βQ α /N + (1 − β ) y α is a strictlypositive IR allocation in the extended economy.Therefore, by Theorem 2, each α -extended economy contains a Pareto-optimal,IR and NJE allocation x α . We construct a sequence ( x α ) α where α tends to zero.Wlog such sequence converges to some allocation x ∗ (if not, a subsequence does.)Such limit is an allocation in the original economy. x ∗ is weak Pareto optimal. Suppose not, then there is an allocation x ′ thatstrongly Pareto dominates x ∗ . Consider the allocation x ′ α in the α -extended econ-omy where x i ′ αl = (1 − α ) x i ′ l for l ∈ O and x i ′ e = αc i , for each i ∈ I . By continuity ofpreferences and for low enough α , we have that x ′ α strongly Pareto dominates x α .This contradicts that x α is Pareto optimal. x ∗ is IR, since U i ( x αi ) ≥ ˜ u i for all i ∈ I and all α . x ∗ has no strong justified envy. Suppose not. Then, some agent i envies someother agent j at x ∗ and u j ( x i ∗ ) > ˜ u j . For α low enough, and by continuity ofpreferences, i envies j at x α and U j ( x iα ) > ˜ u j . But this contradicts the fact that x α satisfies NJE.10.2. Proof of Theorem 1.3.
Following the above proof and noticing that linearutilities are Lipschitz continuous, we just need to show that, in this particular case, x ∗ constructed above is Pareto optimal instead of weak Pareto optimal.Suppose that x ∗ is not Pareto optimal. Let x ′ be an allocation with x ′ e = 0 thatPareto dominates x ∗ . For any ε ∈ (0 , x ′ αε = x α + ε ( x ′ − x ∗ ) , and observe that x ′ αε → x ∗ + ε ( x ′ − x ∗ ) as α → x ′ Pareto dominates x ∗ , we havethat U i ( x i ′ αε ) ≥ U i ( x iα ) for all i with at least one strict inequality. We have seen,by Theorem 2, that each x α in the sequence is Pareto optimal in its corresponding α -extended economy. Consequently, for any α and any ε ∈ (0 , x ′ αε cannot be anallocation in its corresponding α -extended economy. Now, given that X i x i ′ αε = X i x iα + ε ( X i x i ′ − X i x i ∗ )= Q α + ε ( Q − Q ) = Q α , the market clearing aspect of being an allocation is met. So for x ′ αε not to be anallocation it must be the case that, for every ε and α, there is at least one agent i such that x i ′ αε / ∈ C i . Observe that x i ′ αε / ∈ C i means that we are in one of two cases(1) P l x i ′ αεl > c i , or(2) x i ′ αεl < l (or both).Moreover, note by definition of x ′ αε that if we are in case (1) then we are in (1)for any ε ′ ≥ ε , and if we are in case (2) then we are in (2) for any ε ′ ≥ ε .Consider a sequence α t → x α t → x ∗ . Let I t be the set of agents i for whom x i ′ α t ε is in case (1)for all ε ∈ (0 , I t be the set of agents i for whom x i ′ α t ε is in case (2) for all ε ∈ (0 , ε ∈ (0 , i with x i ′ α t ε / ∈ C i for all ε ′ ≥ ε , and that the set of agents is finite, I t ∪ I t = ∅ for all t .Suppose first that I t = ∅ for infinitely many t . Again, since the set of agents isfinite, we can wlog assume that there exists a subsequence with the property that I t is invariant for all t large enough. Let I ∗ denote that invariant set. Select anagent i ∗ ∈ I ∗ . Then c i ∗ < X l x i ∗ ′ α t εl = X l x i ∗ α t l + ε ( X l x i ∗ ′ l − X l x i ∗ ∗ l )for all ε means that P l x i ∗ α t l = c i ∗ , as x i ∗ α t t ∈ C i ∗ . Since this is true for all t largeenough and x α t → x ∗ , P l x i ∗ ∗ l = c i ∗ . Now, we must have P l x i ∗ ′ l − P l x i ∗ ∗ l > P l x i ∗ ′ l > c i ∗ , contradicting that x ′ is an allocation.Suppose now that I t = ∅ for all but finitely many t . Then I t = ∅ for infinitelymany t . Again, since the set of agents is finite, we can wlog assume that there existsa subsequence with the property that I t = I ∗ = ∅ for all t large enough. Select anagent i ∗ ∈ I ∗ . Using the finiteness of the number of objects, there exists l with theproperty that for all t large enough, ∀ ε ∈ (0 , , x i ∗ α t l + ε ( x i ∗ ′ l − x i ∗ ∗ l ) = x ′ i ∗ α t εl < . AIRNESS AND EFFICIENCY 31
Given that x i ∗ t ∈ C i , this can only be true for all ε if x i ∗ α t l = 0, and x i ∗ ′ l − x i ∗ ∗ l < x α t → x ∗ means that x i ∗ ∗ l = 0, so we have x i ∗ ′ l <
0, which contradicts that x ′ is an allocation. 11. Proof of Theorem 4
Theorem 1.
To prove that the first two statements of Theorem 1 hold asbefore when NJE is extended, we omit the steps in common and highlight thedifferences from the previous proof. Let Λ i be the set of all λ ∈ ∆ at which i has no ε -justified envy by exchange towards any agent at φ ( λ ). We prove that the collectionof Λ i is still a KKM covering of ∆. Lemma 5.
For every i ∈ I , Λ i is closed.Proof. Let λ n be a sequence in Λ i such that λ n → λ ∈ ∆. Let x n = φ ( λ n ). Bycontinuity of φ , x n → x = φ ( λ ) ∈ A ∗ . Now we prove that λ ∈ Λ i , that is, i doesnot have ε -justified envy by exchange towards any other agent at φ ( λ ). Supposethat this is not the case. Then i has ε -justified envy by exchange towards someagent j , with the sequence ( i k ) Kk =1 being as in the definition of such envy. Bycontinuity of utility, and since the sequence ( i k ) Kk =1 is finite, for n large enough wehave u i k ( x i k +1 n ) > u i k ( x i k n ) for 1 ≤ k ≤ K − u i K ( x i n ) > ˜ u i K − ε . So i has ε -justified envy by exchange towards j at x n , which is a contradiction. Therefore, λ ∈ Λ i and Λ i is closed. (cid:3) Lemma 6.
For every λ ∈ ∆ , λ ∈ ∪ i ∈ supp ( λ ) Λ i .Proof. Suppose, towards a contradiction, that for some λ ∈ ∆, λ / ∈ ∪ i ∈ supp ( λ ) Λ i .Let x = φ ( λ ). Then for every i ∈ supp( λ ), there exists some j such that i has ε -justified envy by exchange towards j at x . Suppose first that there exists such j , withcorresponding sequence ( i k ) Kk =1 , in which λ i K = 0. Let y be the allocation obtainedfrom x by letting each i k get x i k +1 (1 ≤ k ≤ K −
1) and i K get x i . Clearly y is ε -IR,as x was ε -IR and u i K ( x i ) > ˜ u i K − ε . Note that λ i K = 0 and u i k ( y i k ) > u i k ( x i k ) forall 1 ≤ k ≤ K − P h ∈ I λ h u h ( x h ) < P h ∈ I λ h u h ( y h ). We also have that P h ∈ I k x h − k = P h ∈ I k y h − k , hence X h ∈ I λ h u h ( x h ) − δ X h ∈ I k x h − k < X h ∈ I λ h u h ( y h ) − δ X h ∈ I k y h − k , which contradicts the definition of x = φ ( λ ). The above argument means that every i ∈ supp( λ ) has ε -justified envy by ex-change towards some agent j , with corresponding sequence ( i k ) Kk =1 in which λ i K > i K ∈ supp( λ ). But it means that i K also has ε -justified envy by exchangetowards some agent j ′ , with corresponding sequence ( i ′ k ) Kk =1 in which λ i ′ K > λ ) is finite, there must exist a subset of agents { h , . . . h M } ⊆ supp( λ ) such that h has ε -justified envy by exchange towards someagent with h being the end of the corresponding sequence, h has ε -justified envyby exchange towards some agent with h being the end of the corresponding se-quence, and so on until h M has ε -justified envy by exchange towards some agentwith h being the end of the corresponding sequence. We write this situation as thefollowing cycle h → · · · → h → · · · → h → · · · → · · · → h M → · · · → h , where a → b means that a envies b , and h k → · · · → h k +1 is the correspondingsequence of h k ’s ε -justified envy by exchange towards some agent. Now note thatif an agent h appears more than once in the above cycle, we can shorten the cycleby skipping the agents between any two consecutive positions of h in the cycle. Sowe can, without loss of generality, focus on the cycle in which each agent appearsonce. If we carry out the exchange in the cycle as in the proof of Lemma 2, then weobtain an improvement on the objective that defines φ . This is a contradiction. (cid:3) The remaining part of the proof is same as before.11.2.
Theorem 2.
Let ( x, p ) be an equilibrium in Theorem 2. Suppose some agent i has justified envy by exchange towards some agent j , with the sequence ( i k ) Kk =1 being as in the definition of such envy. By our construction of income functions, m i ( p ) < m j ( p ) < m i ( p ) < · · · < m i K ( p ). So it must be that m i K ( p ) = e i K (˜ u i K , p ).It means that x i is not acceptable to i K , which is a contradiction. So x satisfies nojustified envy by exchange. Then the third statement of Theorem 1 can be provedas before. ReferencesAbdulkadiro˘glu, A., P. A. Pathak, and
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