aa r X i v : . [ q -f i n . E C ] J u l Multi-unit Assignment under Dichotomous Preferences
Josue Ortega
University of Essex, Colchester, UK.Center for European Economic Research (ZEW), Mannheim, Germany.
Abstract
I study the problem of allocating objects among agents without using money.Agents can receive several objects and have dichotomous preferences, mean-ing that they either consider objects to be acceptable or not. In this set-up,the egalitarian solution is more appealing than the competitive equilibriumwith equal incomes because it is Lorenz dominant, unique in utilities, andgroup strategy-proof. Moreover, it can be adapted to satisfy a new fairnessaxiom that arises naturally in this context. Both solutions are disjoint.
Keywords: multi-unit assignment, dichotomous preferences, Lorenzdominance, competitive equilibrium with equal incomes.
JEL Codes:
C78, D73.
Email address: [email protected]
This paper is based on the second chapter of my PhD thesis supervised by Herv´eMoulin, to whom I am indebted for his valuable suggestions. I have also received helpfulcomments from Anna Bogomolnaia, Yeon-Koo Che, Bram Driesen, Georg N¨oldeke, FedorSandomirskiy, Erel Segal-Halevi, Jay Sethuraman, Olivier Tercieux, and audiences at theWinter Meeting of the Econometric Society in Barcelona and the Tenth Conference onEconomic Design at York. Special thanks to Columbia University for their kind hospitality.I thank Erin and Christina Loennblad for proofreading the paper. Errors are mine alone.
October 17, 2018 . Introduction
An assignment problem is an allocation problem where scarce objects areto be allocated among several agents without using monetary transfers. As-signment problems include the allocation of senators to committees, coursesto students, or job interviews to applicants. In this paper, I study assign-ment problems in which each agent can receive more than one object, butat most one unit of each, and several identical units are available of eachobject. These are called multi-unit assignment problems. They include thethree examples previously discussed. A U.S. senator on average participatesin four committees, a student can take many courses during a semester, anda job candidate can schedule many interviews. However, senators cannothave more than one seat on each committee, students cannot take a coursetwice for credit, and applicants cannot be interviewed more than once for thesame position.For such multi-unit assignment problems, we would like to have a system-atic (probabilistic) procedure to decide fairly which agents should get whichobjects, which, at the same time, does not offer incentives to coalitions ofagents to lie about their true preferences. My contribution is to proposean egalitarian solution that achieves this purpose for multi-unit assignmentproblems in the dichotomous preference domain, in which objects are eitherconsidered acceptable or not, and in which agents are indifferent between allobjects that they find acceptable.The egalitarian solution is based on the well-known leximin principle.In the domain of dichotomous preferences, it performs better than the cele-brated competitive equilibrium with equal incomes, a solution used in similarassignment models on larger preference domains (Hylland and Zeckhauser,1979; Budish, 2011; Reny, 2017), and which has been successfully applied toallocate courses in business schools (Budish et al., 2017). By better, I meanthat, unlike the competitive equilibrium with equal incomes, the egalitariansolution is Lorenz dominant, unique in utilities, and impossible to manipulateby groups. In contrast with the single-unit assignment problem (Bogomol-naia and Moulin, 2004), both solutions are disjoint, meaning that in generalwe cannot obtain the egalitarian solution as a competitive equilibrium whenagents are endowed with equal incomes . Source: “The many roles of a Member of Congress”,
Indiana University Center onRepresentative Government . a ranking generally accepted as the unambiguousarbiter of inequality comparison ” (Foster and Ok, 1999) and is “ widely ac-cepted as embodying a set of minimal ethical judgements that should be made ”(Dutta and Ray, 1989). Given two vectors of size n , the first Lorenz domi-nates the second if, when arranged in ascending order, the sum of the first k ≤ n elements of the first is always greater than or equal to the sum ofthe k first elements of the second. A utility profile is Lorenz dominant if itLorenz dominates any other feasible utility profile. In our set-up, the factthat a utility profile is Lorenz dominant implies that it uniquely maximizesany strictly concave utility function representing agents’ preferences and is,therefore, a strong fairness property.Uniqueness of the solution (in the utility profile obtained) is also a de-sirable property, for it gives a clear recommendation of how the resourcesshould be split. A multi-valued solution leaves the schedule designer withthe complicated task of selecting a particular division among those suggestedby the solution, thus raising the possibility of justified complaints by someagents who may argue that other allocations were also recommended by thesolution that were more beneficial to them.It is equally interesting that the egalitarian solution is group strategy-proof, implying that coalitions of agents can never profit from misrepresentingtheir availability. On the contrary, the competitive solution is manipulable bygroups in this set-up, as in many others. Yet, it is remarkable that even in oursmall dichotomous preference domain, where the possibilities to misreportare very limited, the pseudo-market solution can still be manipulated bycoalitions of agents.The fact that the egalitarian solution satisfies these three desirable prop-erties is a strong argument for recommending its use whenever agents havedichotomous preferences, instead of the competitive equilibrium with equalincomes.The dichotomous preference domain is admittedly simple, and is not suit-able for modeling some multi-unit assignment problems in which agents mayconsider objects as complements, such as the allocation of courses to stu-dents. However, this set-up is helpful to represent scheduling problems (seefor example the tennis allocation problem in Maher, 2016), in which agentsare either compatible or incompatible with each object and want to maxi-mize the number of objects they obtain, or for the aforementioned problemsof assigning job interviews to candidates or seats for performances to thepublic, among others. 3oreover, focusing on this particular domain of preferences will be helpfulto show the properties of the egalitarian solution, while, at the same time, itwill make the problem complicated enough to identify why the competitiveequilibrium with equal incomes fails to be unique and group strategy-proof.The reason behind its non-uniqueness is that for some objects, the number ofidentical copies available of them (their supply) equals their total demand.I call these objects perfect. Although there is no doubt on how perfectobjects should be allocated, the question of how to price them becomes tricky.Because their demand is always equal to their supply, they can have a zerocompetitive price. However, they could also have a positive price, hencereducing the available budget of those agents who buy them.Perfect objects are also the reason why the competitive solution is notgroup strategy-proof. A coalition of agents can agree to misrepresent theirpreferences in order to make a set of objects perfect. This allows those objectsto have a lower price (even a zero price), thus allowing agents to increasetheir budget, and, consequently, their share of other over-demanded objects.Manipulating agents benefit unambiguously, meaning that every competitiveequilibrium of the allocation problem with misrepresented preferences yieldsa weakly better allocation than the unique competitive equilibrium of theoriginal problem.More generally, perfect objects also raise the issue of how they should af-fect the final allocation. Some allocation procedures can be decomposed intothe allocation of perfect and over-demanded objects, meaning that the shareof over-demanded objects that agents obtain is independent of their demandfor perfect objects. I call this property independence of perfect objects. Thisis a desirable property in scenarios where agents can claim that perfect ob-jects belong unambiguously to them and the number of perfect objects theyobtain should not diminish their shares of over-demanded objects. Althoughthe egalitarian solution does not satisfy this requirement, we can construct arefined egalitarian solution that does and is also Lorenz dominant for the as-signment problem with over-demanded objects only. However, independenceof perfect objects comes at a price: the refined egalitarian solution is notgroup strategy-proof. The theoretical model I study is closely related to two existing problemsin the literature: 4.
Single-unit random assignment with dichotomous preferences by Bogomol-naia and Moulin (2004), henceforth BM04. Our model generalizes theirs inthat agents can receive more than one object. They study the egalitarianand the equal income competitive solution. They show that the egalitar-ian solution is Lorenz dominant and can always be supported by competitiveprices. Because the competitive solution is Lorenz dominant, the competitivesolution can easily be computed as the maximization of the Nash productof agents’ utilities. They also prove that the egalitarian solution is groupstrategy-proof.Roth et al. (2005) show that the egalitarian solution is also Lorenz dom-inant in assignment problems on arbitrary graphs that are not necessarilybipartite. They use dichotomous preferences to model whether a person iscompatible with a particular organ for transplantation. In their words, “ theexperience of American surgeons suggests that preferences over kidneys canbe well approximated as 0 - 1, i.e. that patients and surgeons should be moreor less indifferent among kidneys from healthy donors that are blood type andimmunologically compatible with the patient” .Assignment on the dichotomous domain of preferences has been further stud-ied by Bogomolnaia et al. (2005), Katta and Sethuraman (2006), and Bou-veret and Lang (2008).Kurokawa et al. (2015) also study a single-unit assignment problem in whichagents can derive a utility equal to one or zero if their demand is met. This is,if an agent demands 10 objects, he obtains the same zero utility if he receives9, 2 or 0 objects, whereas in this paper agents’ utility is linear on the goodsthey find acceptable: each extra unit has the same marginal utility. Theyconsider a broader preference domain which satisfies four conditions, namelyconvexity, equality, shifting allocations, and optimal utilization. They showthat the egalitarian solution is group strategy-proof, envy-free and uniquein utilities for every multi-unit assignment problem in this domain of pref-erences. They show that the egalitarian solution is not Lorenz dominant inthis larger domain.They allow for non individually rational allocations. In their set-up, an agentwho wants an apple but who dislikes a pear may in fact get the pear. Fur-thermore, they impose that the agent derives the same utility from obtainingthe apple and the pear, or the apple alone. In contrast, in BM04 and themodel in this paper, agents cannot receive objects they do not find accept-5ble (and thus they do not have preferences over them). More technically, theproperty of shifting allocations does not apply in our set-up because agentsmay receive non individually rational allocations.2.
The course allocation problem (CAP) described by Brams and Kilgour(2001); Budish (2011); Budish and Cantillon (2012); Kominers et al. (2010);Krishna and ¨Unver (2008); and S¨onmez and ¨Unver (2010), with some impor-tant differences. First, in CAP, students may have arbitrary preferences overthe set of objects, which are considerably more complex than those I studyin this paper. However, reporting combinatorial preferences is infeasible foreven few alternatives, and, in practice, combinatorial mechanisms never al-low agents to fully report such preferences, not only because such revelationwould be complicated, but also because agents may not know their prefer-ences in such detail. Consequently, a new strand of theory has focused onallocation mechanisms with simpler preferences (including linear utilities anddichotomous preferences, e.g. Bouveret and Lemaˆıtre, 2016 and Bogomolnaiaet al., 2017). Although the dichotomous preference domain is smaller thanthose considered in CAP, it is not contained in any of those because CAPrules out indifferences.Furthermore, Budish (2011) only considers deterministic assignments. I in-stead study randomized assignments: in practice, many allocation mecha-nisms use some degree of randomization to achieve a higher degree of fair-ness. I define the egalitarian and the constrained competitive solution. Theegalitarian one is Lorenz dominant in the set of efficient utility profiles (The-orem 1), while the competitive one exists (Theorem 2) but is multi-valued(Example 1). The egalitarian solution is group strategy-proof, but the com-petitive one is not (Theorem 3). Both solutions are disjoint (Example 2),which is a stark difference between this model and BM04. Randomization is used to assign both permanent visas and housing subsidies in theUS, or school places in the UK. Sources: “A one in a million chance at a better life”,
The Guardian , 2/5/2017, “Why does random chance decide who gets housing subsidies?”,
NPR , 3/5/2016, and “School admissions: is a lottery a fairer system?”,
The Guardian ,14/3/2017.
6s a consequence, the classic result stating that the competitive solutioncan be computed as the maximizer of the Nash product of utilities no longerholds: a result known as the Eisenberg-Gale program. Its failure is importantnot only because it leaves us with no known algorithm for computing com-petitive equilibria, but also because the Eisenberg-Gale program is a ratherrobust result that applies to a large class of utility functions beyond the linearcase (Vazirani, 2007) and to the division of goods and bads (Bogomolnaiaet al., 2017). The fact that the competitive solution is not unique is alsointeresting, as a unique utility profile is always obtained in Fisher markets.I show that the egalitarian solution is not independent of perfect objects,and propose a refined egalitarian solution that achieves this property, while atthe same time being Lorenz dominant for the assignment of over-demandedobjects (Lemma 5). This refined solution, while appealing, violates groupstrategy-proofness, unlike the classical egalitarian solution (Example 3).This paper is structured as follows. Sections 2 and 3 formalize the modeland the solutions I consider, respectively. Section 4 analyses the solutions’manipulation by groups, whereas Section 5 introduces the property of inde-pendence of perfect objects. Section 6 concludes the paper. All proofs areprovided in the Appendix.
2. Model
I consider the allocation of m objects (each with possibly several copiesof itself) to n agents. Up to q k copies of object k ∈ M can be assigned tothe set of agents N . I refer to the integer vector q = ( q , . . . , q m ) as objects’capacities .Agents’ preferences over objects are given by a m × n binary matrix R .Each entry r ik = 1 if agent i finds object k acceptable and 0 otherwise. Slightly abusing the notation, R iM (resp. R Nk ) denotes both the i -th row(resp. k -th column) of R and the set of objects (resp. agents) for which r ik = 1. I assume | R Nk | ≥ q k for each object k . Depending on the application we have in mind, R can also be understood to representeither allocation or physical constraints. If an object is under-demanded, the additional copies can always be thrown away.If we eliminate this assumption, then we need to consider assignments which are notindividually rational, in which agents receive objects that they do not want. random assignment matrix (RAM) for an MAP ( R, q ) is an m × n matrix satisfying the following conditions ∀ i ∈ N, k ∈ M Feasibility ( ≤ z ik ≤ P k ∈ M z ik ≤ q k (1)Individual Rationality n z ik > r ik = 1 (2)An RAM’s entries indicate what probability each agent has of obtainingone unit of each object. The feasibility conditions ensure that no agentobtains more than one unit of each object, and that the total number ofunits assigned of each object is less than its capacity. Similarly, individualrationality guarantees that each agent only obtains shares from acceptableobjects. Throughout the paper, I only consider assignments satisfying thesetwo properties. As before, the notation Z iM (resp. Z Nk ) denotes both the i -th row (resp. k -th column) of Z and the set of objects (resp. players) forwhich z ik = 1. F ( R, q ) denotes the set of all RAMs for the MAP (
R, q ).The matching size ν ( R, q ) = P k ∈ M q k of an MAP represents the maximumnumber of object units that can be assigned.Several random assignments can have the same corresponding RAM. The-orem 1 in Budish et al. (2013) implies that Lemma 1.
Any RAM can be decomposed into a convex combination of binaryRAMs, and can thus be implemented.
I assume that agents are indifferent between objects that they find ac-ceptable, and that they want to maximize the number of acceptable objectsthey obtain. The canonical utility function representing those preferences is u i ( Z ) = X k ∈ M z ik (3)for an arbitrary agent i . This function is clearly not unique but it isconvenient to work with. The preference relation represented by this functionis a complete order over all RAMs, and implies that an RAM Z is Paretooptimal for an MAP ( R, q ) if and only if P i ∈ N P k ∈ M z ik = ν ( R, q ). The set The implication follows because the set of feasibility constraints is a hierarchy. Lemma1 is an extension of the well-known Birkhoff-von Neumann decomposition theorem.
8f efficient utility profiles U ( R, q ) can be described as U ( R, q ) = { U ∈ R n | ∃ Z ∈ F ( R, q ) : U i = X k ∈ M z ik , ∀ i ∈ N } (4)I do not distinguish between ex-ante and ex-post efficiency because inthe dichotomous preference domain they coincide. This equivalence occursbecause the sum of utilities is constant in all efficient assignments. In ourset-up, efficiency simply requires that no object is wasted.A welfarist solution is a mapping Φ from (
R, q ) to a set of efficientutility profiles in U ( R, q ), and hence, it only focuses on the expected numberof objects received by an agent and not on the exact probability distribu-tion over deterministic assignments. Whenever a solution is single-valued Iinstead use the notation φ . We can partition the corresponding set of objects M into two subsets P ( R, q ) and O ( R, q ), which are called perfect and over-demanded , re-spectively. The set of perfect objects is defined as P ( R, q ) = { k ∈ M : | R Nk | = q k } (5)The vectors q P ( R,q ) and q O ( R,q ) denote the capacities of perfect and over-demanded goods, respectively.Given a MAP ( R, q ), a perfect extension for agent i represents addingan arbitrarily perfect object k ′ that agent i finds acceptable. Formally, aperfect extension for agent i in a MAP ( R, q ) is a pair ([
R R Nk ′ ] , q ) where[ R R Nk ′ ] denotes the n × ( m + 1) juxtaposition of the two matrices and q = ( q , . . . , q m , | R Nk ′ | ).
3. Three Efficient Solutions
An intuitive solution equalizes agents’ utilities as much as possible re-specting efficiency and individual rationality: this is the well-known leximin Ex-ante and ex-post efficiency are equivalent in assignment problems with dichotomouspreferences (BM04, Roth et al., 2005).
Egalitarian Solution (ES) , proposed the-oretically by BM04, and applied to the exchange of live donor kidneys fortransplant by Roth et al. (2005) and Yılmaz (2011).To define it formally, let ≻ l be the well-known lexicographic order. Foreach U ∈ R n , let γ ( U ) ∈ R n be the vector containing the same elements as U but sorted in ascending order, i.e. γ ( U ) ≤ . . . ≤ γ n ( U ). The leximin order ≻ LM is defined by U ≻ LM U ′ if and only if γ ( U ) ≻ l γ ( U ′ ). The ES is definedby φ ES ( R, q ) = arg max ≻ LM U ( R, q ) (6)The ES satisfies a strong fairness notion called
Lorenz dominance ,defined as follows. Define the order ≻ ld on R n so that for any two vectors U and U ′ , U ≻ ld U ′ only if P ti =1 U i ≥ P ti =1 U ′ i ∀ t ≤ n , with strict inequalityfor some t . We say that U Lorenz dominates U ′ , written U ≻ LD U ′ , if γ ( U ) ≻ ld γ ( U ′ ). A vector U ∈ U ( R, q ) is Lorenz dominant for an MAP(
R, q ) if it Lorenz dominates any other vector in U ( R, q ).Lorenz dominance is a partial order in U ( R, q ) and therefore a Lorenzdominant utility profile need not exist. Nevertheless, the ES solution isLorenz dominant.
Theorem 1.
The ES solution is Lorenz dominant in the set of efficient utilityprofiles.
I prove Theorem 1 using Theorem 3 in Dutta and Ray (1989), whichstates that the core of every supermodular cooperative game has a Lorenzdominant element. The construction of the corresponding cooperative gamecan be found in the Appendix.
A second solution, which is substantially more complicated, requires tobalance the supply and demand for goods when agents are endowed withequal budgets. These equal budgets are often normalized to one currencyunit, a normalization that I also use. This solution is known as the
Compet-itive Equilibrium with Equal Incomes (CEEI) (Varian, 1974; Hyllandand Zeckhauser, 1979). In MAPs, each agent can consume at most one unit So that for any two vectors
U, U ′ ∈ R n , U ≻ l U ′ only if U t > U ′ t for some integer t ≤ n , and U p = U ′ p for any positive integer p < t .
10f each object, hence having particular constraints on their consumption set.I use the term
Constrained Competitive Equilibrium ( CCE , still withequal incomes) from now on to make this distinction obvious. The CCE solu-tion is different from the CEEI as defined in Hylland and Zeckhauser (1979)in that in our case agents never partially consume objects that have differentprices (see Table 1 in their paper). This distinction justifies the differentterminology of CCE.
Definition 1.
A CCE for an MAP (
R, q ) is a pair of an RAM Z ∗ and a non-negative price vector p ∗ such that, ∀ i ∈ N , agents maximize their utilities Z ∗ iM ∈ arg max Z iM ∈ β i ( p ∗ ) u i ( Z iM ) (7)where β i ( p ) is the budget set defined as β i ( p ) = { Z iM | P k ∈ M z ik ≤| R iM | : p · Z iM ≤ } , and the market clears, so that Z ∗ ∈ F ( R, q ) (8)As we shall see in Theorem 2, the set of CCE is never empty but may belarge. The optimality conditions of CCE imply k / ∈ P ( R, q ) = ⇒ p ∗ k > z ∗ ik , z ∗ ik ′ ∈ (0 ,
1) = ⇒ p ∗ k = p ∗ k ′ (10)[ p ∗ k < p ∗ k ′ ] ∧ [0 < z ∗ ik ′ ] = ⇒ z ∗ ik = 1 (11) X k z ∗ ik < | R iM | = ⇒ X k p ∗ k · z ∗ ik = 1 (12)These are the equivalent of the Fisher equations in our model, see Brainardand Scarf (2005). Condition (9) allows a zero price only for perfect objects,while expression (10) forces the same marginal benefit for every object thatagents obtain partially but not fully.The CCE is in general multivalued. Given an MAP, I denote the set of That is, if an apple pie has a higher price than a pear pie, and the agent values themequally, the agent either fully consumes the pear pie and eats some or all of the apple pie,or the agent only consumes some of the pear pie and none of the apple pie. What cannever occur in a CCE is that an agent consumes some, but not all, of the pear pie andsome, but not all, of the apple pie. Z ∗ , p ∗ ) as C ( R, q ). The CCE solution is defined byΦ
CCE ( R, q ) = { u ( Z ′ ) | ∃ p ′ : ( Z ′ , p ′ ) ∈ C ( R, q ) } (13) Finally, a naive and highly intuitive solution (that I use as a benchmarkonly) breaks up the allocation problem into m sub-problems of assigning q k units of object k into R Nk , distributing an equal share of object k among allagents who find it acceptable. I call this solution Egalitarian Per Object(EPO) . Given an MAP (
R, q ), the EPO solution assigns to each agent φ EP Oi ( R, q ) = X k ∈ M r ik · q k | R Nk | (14)In the dichotomous preference domain, EPO is equivalent to the well-known random priority mechanism, also known as random serial dictator-ship. I do not consider EPO to be an appropriate solution for MAPsbecause it ignores the interaction between the m assignment problems corre-sponding to each object. EPO also fails the following basic fairness property:if n − n -th agent also receives at leastone object; see Example 1 for an illustration.One could also consider other solutions discussed in the literature, inparticular the probabilistic serial rule, defined by Bogomolnaia and Moulin(2001). I do not consider this solution for two reasons. First, the probabilisticserial rule is appealing in scenarios where different notions of efficiency do notcoincide. This is not the case for MAPs. Second, it was originally defined forassignment problems with strict preferences. Even though Katta and Sethu-raman (2006) extend the probabilistic serial rule to allow for indifferences,their extension is only defined for single-unit assignment problems. Example 1 (Multivalued CCE differs from EPO) . Table 1 shows the dif-ferent outcomes that our three solutions produce for a problem with n = 6, m = 3, and ( R, q ) given in subtable 1a. The CCE utilities are written inbrackets in subtable 1b because there are CCE that support utility profiles EPO would not be efficient in a more general domain of preferences. The equivalencewith random priority would also disappear. . , . ,
1) and (2 . , ,
1) with 0 ≤ p γ ≤ . This multiplicity isinteresting: the competitive solution is always unique in the correspondingutility profile in Fisher markets (Jain and Vazirani, 2010, p.87). It is alsoproblematic, as there is no obvious selection from the CCE. Table 1: CCE is multi-valued. N \ M α β γ
Total a : d e f q (a) Corresponding R matrix. N ES CCE EPO a : d e f (b) Utility profiles for each solution. Any CCE in Example 1 gives one unit of object α to agent f . Thisimplies that there are no CCE prices that support the EPO outcome andthus is a strong argument against this solution, as competitive equilibriaare considered “ essentially the description of perfect justice ” (Arnsperger,1994), and the base of Dworkin’s “ equality of resources ” (Dworkin, 1981).In consequence, the EPO solution is not ideal. But interestingly, the ESsolution can also produce outcomes that cannot be supported as a CCE. Example 2 (ES differs from CCE) . I show this using a MAP with n = 9, m = 6, and ( R, q ) given in subtable 2a. Note that in the single-unit case(Theorem 1 in BM04), the ES is always supported by competitive prices.If the ES solution (2, 2.5, 2.5, 3.25) could be supported as a CCE, then p α = p γ = p δ = p ǫ = p ζ because agents f : i obtain those objects with positiveprobability but do not exhaust them. Furthermore, agents d : i must spendtheir whole budget, implying prices p α = and p β = . However, at suchprices, the ES utility for agents a : c is unaffordable ( > able 2: ES and CCE are disjoint. N \ M α β γ, δ ǫ, ζ
Total a : c d e f : i q (a) Corresponding R matrix. α β γ, δ ǫ, ζ Total1 0.97 0 0 1.970 0.54 1 0 2.540 0.54 0 1 2.540.25 0 0.75 0.75 3.25 (b) Corresponding Z ∗ (CCE). algorithm for computing the competitive equilibrium. The Eisenberg-Galeprogram is otherwise a rather robust result since it extends to a large familyof utility functions beyond the linear case (Jain and Vazirani, 2010), as wellas to the mixed division of objects and bads (Bogomolnaia et al., 2017).The multiplicity of the competitive solution and its non-equivalence withthe egalitarian outcome justify the new terminology of CCE. For any MAP,the set of CCE is non-empty, a result that I prove in the Appendix using astandard fixed point argument. I summarize these findings in Theorem 2. Theorem 2.
The ES solution is well-defined and single-valued, and the CCEsolution exists. Their intersection can be empty.3.5. Envy
It is easy to see that both the ES and CCE solutions are envy-free. Asolution φ is envy-free if, for any MAP ( R, q ) with agents i and j such that R iM ⊆ R jM , φ i ( R, q ) ≤ φ j ( R, q ). For the multi-valued CCE, envy-freenessholds for any selection from it.
Lemma 2.
ES and CCE are envy-free.
I postpone an easy proof. Note that there is no efficient solution that isstrongly envy-free, i.e. that for any MAP (
R, q ) with agents i and j suchthat | R iM | < | R jM | , φ i ( R, q ) ≤ φ j ( R, q ), see Ortega (2016).
4. Manipulation by Groups
I consider agents’ manipulation in the direct revelation mechanism asso-ciated with each solution. For this purpose, we need to know exactly how14bjects are assigned and not just the total utility that each agent receives.A detailed solution ψ maps every MAP ( R, q ) into an RAM Z ∈ F ( R, q ),specifying which share of each object is allocated to each agent, whereas awelfarist solution φ maps every MAP into a utility profile U ∈ U ( R, q ) andonly tells us the expected number of objects received by each agent. Everydetailed solution ψ projects onto the welfarist solution φ ( R, q ) = u ( ψ ( R, q )).The direct revelation mechanism associated with a detailed solution ψ issuch that all agents reveal their preferences R iM , and then ψ is applied tothe corresponding MAP ( R, q ), implementing the RAM ψ ( R, q ) = Z .I assume that agent i with the true preferences R iM can only misrepresenther preferences by understating the number of objects that she finds accept-able, i.e. by declaring a preference profile R ′ iM ⊂ R iM (we then say that R ′ iM is IR for R iM ). I use this assumption for two reasons. The first is theoretical:I have not specified the dis-utility that the consumption of an undesirableobject brings to an agent, as I have only focused on individually rationalassignments. I would need to specify such dis-utility to analyse the manipu-lation of a solution by exaggerating the set of acceptable objects. The secondreason is that such an assumption has already been imposed in the study ofscheduling problems (e.g. Koutsoupias, 2014). In many scheduling problemsmotivating MAPs, cancelling consumption could be strongly punished by thecentral clearinghouse, particularly when other agents’ consumption dependson other agents fully exhausting their bundles (no double tennis match canbe made with only 3 out of 4 players). , A detailed solution ψ is group strategy-proof if for every MAP ( R, q )and every coalition S ⊂ N , ∄ R ′ satisfying i) R ′ jM = R jM ∀ j / ∈ S , and ii) R ′ SM is IR for R SM , such that ∀ i ∈ S, u i ( ψ ( R ′ , q )) ≥ u i ( ψ ( R, q )) (15)with strict inequality for at least one agent in S . A welfarist solution φ is group strategy-proof if every detailed solution ψ projecting onto φ isgroup strategy-proof. If agents are allowed to report undesirable objects as desirable, the egalitarian andeven priority solutions are manipulable by groups (an agent with a higher priority requestsan object she does not want and gives it to a member of the manipulating coalition). BM04 impose an equivalent assumption: they require that the RAM of every MAPmust be individually rational according to the agents’ true preferences.
Lemma 3.
Any deterministic priority solution is group strategy-proof.
The previous Lemma shows that group strategy-proofness is relativelyeasy to achieve for MAPs in the dichotomous domain. In fact, I show belowthat the ES solution is also group strategy-proof. Is CCE also group strategy-proof? There are two extensions of our group strategy-proofness definitionto set valued solutions.The first requires that for every MAP (
R, q ), there is no competitiveequilibrium of the manipulated MAP ( R ′ , q ) that is weakly better than everycompetitive equilibria of the original problem ( R, q ), for every member of themanipulating coalition S . A stronger extension is that there is at least onecompetitive equilibrium of ( R, q ) which yields a weakly higher utility thansome competitive equilibrium of ( R ′ , q ), with strict inequality for at leastone member of the deviating coalition S . It turns out that CCE violatesboth conditions. The reason for this is that a group can coordinate to makeseveral objects perfect, thus allowing those objects to have a zero price. Theorem 3.
ES is group strategy-proof but CCE is not.
The proof of ES being group strategy-proof can be found in the Appendix,but I show that CCE is unambiguously manipulable by groups below.
Example 3 (CCE not group strategy-proof) . Let n = 7, m = 4, and ( R, q )given by Table 3. 16 able 3: Example 3. N \ M α β γ δ Φ CCE aaa bbb ccc d e f g q (a) True preferences R . α β γ δ Φ CCE [2.5 - 2.57] [2.5 - 2.57] q (b) Misreport R ′ for S = { a, b, c } . Consider the coalition S = { a, b, c } . When agents submit their real pref-erences, there exists a unique CCE that supports the ES solution: agents a, b, and c obtain 2.5 expected objects. By changing their report each fora different object, as in subtable 3b, they make objects β , γ and δ perfect,consequently enlarging the set of CCE solutions, which includes utilities thatare always weakly above 2.5 and up to 2.57. By misrepresenting and creatingartificially perfect objects, they allow those to be priced at 0, weakly increas-ing the number of expected objects received in any competitive equilibria of( R ′ , q ), at the expense of agents with limited acceptable objects, in this case g . I do not discuss strategy-proofness (manipulation by individuals on theirown) since it is immediate that ES and CCE (and EPO) are strategy-proof.For CCE, we can construct a selection of it that is strategy-proof, sincereducing the total demand for an object either reduces its price, relativelyincreasing the price of other objects, or leaves its price unchanged.Efficiency, fairness, and non-manipulability are standard goals in the de-sign of resource allocation mechanisms. Before concluding, I discuss a newgoal that arises naturally for MAPs.
5. Independence of Perfect Objects
Some solutions do not depend on the number of perfect objects desiredby an agent. If an agent finds a new perfect object to be acceptable, we could17xpect that she would always receive one extra expected unit. This is whatour following property captures.A solution φ is independent of perfect objects (IPO) if, for everyMAP, every i ∈ N and for any of its perfect extensions ([ R R Nk ′ ] , q ), φ i ( R, q ) + 1 = φ i ([ R R Nk ′ ] , q ) (16)IPO is a desirable property for two reasons. First, perfect objects belongunambiguously to agents who find them acceptable, so they can argue thatthey should obtain them fully, irrespective of the share they obtain fromover-demanded objects. Second, if the clearinghouse uses a solution that wasnot IPO, the set of agents who find perfect objects acceptable could avoidreporting their demand for perfect objects and obtain them fully outside thecentralized mechanism, a real concern for scheduling applications in whichagents may organize teamwork activities on their own.CCE (partially) satisfies this requirement. Lemma 4.
Although ES is not IPO, there exists a selection of CCE thatsatisfies IPO.
Lemma 4 highlights that CCE can always assign a zero price to all perfectobjects: this is how we construct the selection of CCE that satisfies IPO. Butit may also assign a zero price to some perfect objects only, or to no perfectobject at all. The designer has a high degree of flexibility in choosing theequilibrium prices.The selection problem extends to Budish’s (2011) competitive mechanismfor CAP in which students reveal their preferences to a centralized clear-inghouse which announces a corresponding equilibrium allocation. Budishargues that this mechanism is transparent, meaning that students can ver-ify that the allocation is an equilibrium. But the mechanism can be “ma-nipulated from the inside”, selectively assigning zero prices to hand-pickedcourses, while at the same time rightly arguing that it produces a competitiveallocation.If IPO must be achieved (a decision depending on the context and thedesigner’s objectives), we would like to have a solution that, at the sametime, avoids the multiplicity problem of the CCE, while being envy-free and EPO also satisfies IPO. Once more, EPO performs very poorly with respect to fairnessconsiderations so I do not analyse it further.
18s fair as possible. Such a solution exists: we call it the refined egalitariansolution (ES*) . To define it, we use the partition of M into P ( R, q ) and O ( R, q ), and split the original MAP (
R, q ) into two independent problems( R N P ( R,q ) , q P ( R,q ) ) and ( R N O ( R,q ) , q O ( R,q ) ), which correspond to the indepen-dent MAPs with perfect and over-demanded objects, respectively. ES* isgiven by φ ES ∗ i ( R, q ) = φ ES ( R N O ( R,q ) , q O ( R,q ) ) + (cid:12)(cid:12) R i P ( R,q ) (cid:12)(cid:12) (17)ES* takes the egalitarian solution for the MAP with over-demanded ob-jects only, and adds the number of perfect objects in which a player is avail-able. ES* is close to a suggestion in Budish (2011). Noting that some coursesmay be in excess supply, he proposes to run the allocation mechanism onlyon the set of over-demanded courses: “ if some courses are known to be insubstantial excess supply, we can reformulate the problem as one of allocat-ing only the potential scarce courses ”. ES* formalizes this suggestion. It alsosatisfies several desiderata. Lemma 5.
The ES* solution is well-defined and single-valued, efficient, IPO,envy-free, and Lorenz dominant for the problem ( R N O ( R,q ) , q ) . It is immediate that ES* is single-valued, efficient and IPO. The remain-ing properties are straightforward modifications of the proofs of Lemmas 1and 2 and Theorem 1. Unfortunately, the properties in Lemma 5 come at acost: ES* is not group strategy-proof. ES* can be manipulated by groupsreducing their availability in order to make some objects perfect. There-fore, the members of the manipulating coalition obtain those objects fully,while also obtaining an egalitarian fraction of the remaining over-demandedproblem.Group strategy-proofness and IPO are compatible. EPO satisfy themboth (plus envy-freeness and Pareto efficiency). However, its poor perfor-mance with respect to fairness makes it inappropriate for the problems Ihave considered in this paper, as argued in subsection 3.3.
6. Conclusion
For multi-unit assignment problems in the dichotomous preference do-main, the egalitarian solution is Lorenz dominant, single-valued and group For an example, use the MAP and manipulation R ′ illustrated in Example 3. ppendix: ProofsTheorem 1 The ES solution is Lorenz dominant in the set of efficient utilityprofiles.Proof.
Fix a MAP (
R, q ). Consider the concave cooperative game (
N, µ )where µ : 2 N → R is a function that assigns, to each subset of agents, themaximum number of objects that they can obtain together. To formalizethis intuitive function, given a coalition S ⊂ N , let us partition the set ofobjects M into M + ( S ) and M − ( S ), defined as M + ( S ) = { k ∈ M : | R Sk | < q k } (18)The function µ is given by µ ( S ) = X k ∈ M + ( S ) X i ∈ S r ik + X k ∈ M − ( S ) q k (19)This function is clearly submodular, i.e. for any two subsets T, S ⊂ Nµ ( S ) + µ ( T ) ≥ µ ( S ∪ T ) + µ ( S ∩ T ) (20)The “core from above” is defined as the following set of profiles C ( R, q ) = { x ∈ R n | X i ∈ N x i = ν ( R, q ) and ∄ S ⊂ N : X i ∈ S x i > µ ( S ) } (21)It follows from Theorem 3 in Dutta and Ray (1989) that the set C ( R, q )has a Lorenz dominant element and is the egalitarian solution. But by con-struction of the “core from above”, U ( R, q ) ⊂ C ( R, q ), the ES solution isalso Lorenz dominant in the set of efficient utility profiles U ( R, q ). Theorem 2
For generalized tennis problems, the ES solution is well-definedand single-valued, and the CCE solution exists. Their intersection can beempty.Proof.
Fix a MAP (
R, q ). Let p ∈ R m + be an arbitrary price vector such that p · c = n , and use the notation y i = R iM to denote the optimal consumptionbundle for agent i ∈ N , and y N = ( | R N | , . . . , | R Nm | ). Note that p · y N ≥ p · q (22)21efine the vector ~λ as ~λ ( p ) = ( λ , . . . , λ n ) = UNIF { p · y i ; n } (23)where UNIF denotes the uniform rationing rule: a mapping that gives toevery agent the money needed to buy her preferred bundle of objects as longas it is less than λ , chosen so that p · ~λ = n . Define the sets of satiated andnon-satiated agents N ( p ) = { i ∈ N | λ i = p · y i } (24) N + ( p ) = { i ∈ N | λ i < p · y i } (25)So that λ i = λ ∀ i ∈ N + . Define the demand correspondence d i ( p ) as d i ( p ) = arg max Z iM ∈I ( R iM ) { p · Z iM ≤ λ i } (26)where I ( R iM ) denotes the set of individually rational assignments for R iM . Note that d i ( p ) = { y i } for every i ∈ N ( p ), while for agents in N + ( p ),any vector z i ∈ d i ( p ) satisfies p · z i = λ . By Berge’s maximum theorem,the demand correspondence is upper hemi-continuous and convex valued.The excess demand correspondence for the whole society, which inherits theproperties of d i , is given by e ( p ) = d N ( p ) − q (27)where d N ( p ) denotes the aggregate demand correspondence for each ob-ject. Using the Gale-Nikaido-Debreu theorem (Theorem 7 in pp. 716-718 ofDebreu (1982)), we know that there exists both a price vector p ∗ ∈ R + andan excess demand vector x ∗ ∈ e ( p ∗ ) for which the following two conditionsare satisfied x ∗ = ~ p ∗ · x ∗ = 0 (29)Where Walras’ law in equation (29) holds by construction of ~λ and d .Finally, ∀ i ∈ N Z ∗ iM = d i ( p ∗ ) (30)so that the corresponding Z ∗ ∈ F ( R, q ) by equation (28), concluding the22roof of existence of CCE. That ES is single-valued follows from Theorem 1.I have shown in Example 3 that for some MAP there do not exist prices thatsupport the ES as a CCE.
Lemma 2
ES and CCE are envy-free.Proof.
For an arbitrary MAP, let φ ES ( R, q ) = ( U , . . . , U i , U j , . . . , U n ), andassume agent i is envious of j , which means that R jM ⊆ R iM and that thereexists a Pigou-Dalton transfer ǫ so that the utility profile U ′ = ( U , . . . , U i + ǫ, U j − ǫ, . . . , U n ) ∈ U ( R, q ). But U ′ Lorenz dominates φ ES ( R, q ), so φ ES ( R, q )was not the ES solution, a contradiction.Any selection of the CCE solution is envy-free because of the standardargument: if there is any agent who is envious, she could afford the scheduleof the agent she envies.
Theorem 3
ES is group strategy-proof but CCE is not.
I have shown that CCE is not group strategy-proof in the main text. Toshow that ES is group strategy-proof, I start with a few preliminaries. Let Z denote the set of all feasible RAMs supporting the egalitarian solution, i.e. Z = { Z ∈ F ( R, q ) | ∀ i ∈ N : X k ∈ M z ik = φ ES i ( R, q ) } (31)A rule is non-bossy if no agent can affect someone else’s allocation withoutchanging her own utility. That is, a solution φ is non-bossy if, for everyMAP ( R, q ), ∀ i ∈ N , and any manipulation R ′ such that 1) ∀ j = i, R jM = R ′ jM , and 2) R ′ iM ( R iM , we have φ i ( R, q ) = φ i ( R ′ , q ) only if φ ( R, q ) = φ ( R ′ , q ) (32)We prove a useful auxiliary Lemma below. Lemma 6.
ES is non-bossy.Proof.
We proceed by way of contradiction. Let R ′ be as specified in the pre-vious definition. The manipulation may come from a reduction of availabilityin three types of objects:1. k ∈ O ( R, q ) and { k ∈ M | ∃ Z ∈ Z : z ik = 0 } , and hence there is a wayto implement the ES solution even when agent i misreported, so her changein availability is inconsequential and all utilities remain the same, so agent i cannot be bossy. 23. k ∈ O ( R, q ) and { k ∈ M | ∀ Z ∈ Z : z ik > } , so clearly agent i ’sutility changes, so she cannot be bossy.31. k ∈ P ( R, q ), but if agent i reduces the number of perfect goods, shealways reduces the utility she obtains (as I prove below), so her utility is notconstant and she cannot be bossy.Now I prove that reducing the number of perfect objects in which agent i is available always strictly reduces her utility. The certain loss of the perfectobject(s) must be exactly compensated by an increase of the shares she getsfrom all over-demanded objects, which is constant in any Z ∈ Z . Agent i was not getting full shares of those objects (as otherwise we obtain a con-tradiction) so another agent(s) j must be obtaining shares for those objects,implying φ ES j ( R, q ) ≤ φ ES i ( R, q ) (because otherwise the ES would give thoseshares to agent i ). Some of the shares obtained by agent j in φ ( R, q ) mustbe transferred to agent i in φ ( R ′ , q ): this is a Pigou-Dalton transfer becauseif agent i did not obtain a lower utility in the misrepresented problem thenhe would not obtain the shares of j . Moreover, φ ES i ( R, q ) − < φ ES j ( R, q ) ≤ φ ES i ( R, q ) (33)as otherwise j does not transfer any shares to i when i reduces the numberof perfect objects. Let γ be the Pigou-Dalton transfer from j to i requiredso that the utility of i is kept constant. We have φ ES i ( R ′ , q ) = φ ES i ( R, q ) − γ = φ ES j ( R, q ) − γ < φ ES i ( R, j ) (34)showing that indeed reducing the number of perfect objects always yieldslower utility, and thus concluding the proof that ES is non-bossy.We are now ready to prove that ES is group strategy-proof. We will dothis by showing that nobody can join a manipulating coalition.
Proof.
By way of contradiction, assume there exists a MAP (
R, q ), a coalition S ( N , and a manipulation R ′ such that, for all i ∈ S φ ES i ( R ′ , q ) ≥ φ ES i ( R, q ),and for some j ∈ S φ ES j ( R ′ , q ) > φ ES j ( R, q ).Let φ ES ( R, q ) = U ES and order the agents such that U ES1 ≤ . . . ≤ U ES n .We will show by induction on the order of agents the following property i / ∈ S (35)24here are two cases in which an agent i can be in S . Case 1) either shegets more utility, φ ES i ( R ′ , q ) > φ ES i ( R, q ), or case 2) she gets the same utilitybut she changes her reported preferences to help another member of S . Thisis ruled out by the non-bossiness of ES so we focus on case 1) only.We prove it for i = 1 first, i.e. the agent with lowest utility. Agent 1 getsa strictly higher number of objects with the new profile R ′ , which must comefrom a set of objects K ⊆ O ( R, q ) from which he was not getting full shares( K = { k ∈ M | ∃ Z ∈ Z : 0 < z ik < } ), for which agents 2 , . . . , t are alsoavailable and U ES1 = U ES2 = . . . = U ES t . Those agents exhaust q k entirely; i.e. ∀ k ∈ K, ∀ Z ∈ Z , P t z ik = q k .Let T = { , . . . , t }∩ S . For any preference matrix R ′ T M that is individuallyrational for R T M , the objects { k ∈ K | R Nk = R ′ Nk } become less over-demanded for agents { , . . . , t } \ T , and therefore the agents in T get lessobjects as a whole. Therefore there must be at least one agent in T who isworst off, and the coalition S is not viable. Therefore 1 / ∈ S .Now we assume that i / ∈ S for agent i = h − h . We must have that U ES h < | R hM | . We assume φ ES1 ( R, q ) ES1 <φ ES h ( R, q ) as otherwise our argument for agent 1 works exactly the same.If agent h ∈ S , it must be that there exists a manipulation R ′ so that φ h ( R ′ , q ) > φ h ( R, q ). The increase in her utility must come from more objectshares on over-demanded objects which she was not obtaining fully, i.e. K h = { k ∈ M | ∃ Z ∈ Z : 0 < z hk < } . Some of these objects are exhausted byagents 1 , . . . , h −
1. There is no way agent h could get more shares from anyof those objects because { , . . . , h − } ∩ S = ∅ by our induction step.Therefore, the increase must come from objects that are not exhausted by { , . . . , h − } . Those objects become less over-demanded for { h, . . . , n } \ S ,and therefore agents in S get less object shares as a whole. It follows thatthere must be a agent in S who gets less utility, so coalition S is not viable.Therefore h / ∈ S , and this concludes the proof. Lemma 4
Although ES is not IPO, there exists a selection of CCE thatsatisfies IPO.Proof.
It is straightforward to show that ES is not IPO. Let n = 5 , M = { α } , q = 4, and R ⊤ = [1 1 1 1 1]. φ ES i ( R, q ) = 0 . k ′ with capacity 4 for any agent i changes φ ES i ([ R R Nk ′ ] , (4 , . = 2.To show that there is a selection of Φ CCE that is IPO, let ( Z ∗ , p ∗ ) be aCCE of ( R, q ) and ([
R R Nk ′ ] , q ) be a perfect extension of ( R, q ) . Then fix25 ∗ k ′ = 0 and, for every i ∈ N let z ∗ ik ′ = 1 if r ik ′ = 1, and 0 otherwise. Thepair ([ Z ∗ Z ∗ Nk ′ ] , ( p ∗ , . . . , p ∗ n , R R Nk ′ ] , q ),because everybody interested in the perfect object is able to afford it, andthe demand for k ′ equals its supply, because the new object k ′ is perfect. ReferencesReferences
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