Accepted Manuscript

Probabilistic assignment problem with multi-unit demands: Ageneralization of the serial rule and its characterization

Eun Jeong Heo

PII: S0304-4068(14)00105-0DOI: http://dx.doi.org/10.1016/j.jmateco.2014.08.003Reference: MATECO 1913

To appear in: Journal of Mathematical Economics

Received date: 17 April 2013Revised date: 6 May 2014Accepted date: 15 August 2014

Please cite this article as: Heo, E.J., Probabilistic assignment problem with multi-unitdemands: A generalization of the serial rule and its characterization. Journal of MathematicalEconomics (2014), http://dx.doi.org/10.1016/j.jmateco.2014.08.003

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Probabilistic Assignment Problem with Multi-unit Demands:

A Generalization of the Serial Rule and its Characterization

Eun Jeong Heo∗†

First draft: November 6, 2009This version: August 21, 2014

Abstract

We study a probabilistic assignment problem when agents have multi-unit demands for ob-jects. We first introduce two fairness requirements to accommodate different demands acrossagents. We show that each of these requirements is incompatible with stochastic dominanceefficiency (henceforth, we use the prefix “sd” for stochastic dominance) and weak sd-strategy-proofness, unless all agents have unitary demands. We next introduce a new incentive re-quirement which we call limited invariance. We explore implications of these requirements incombination of consistency or converse consistency.

Our main result is that the generalized serial rule, which we propose as an adaptationof the serial rule to our setting, is the only rule satisfying sd-efficiency, the sd proportional-division lower-bound, limited invariance, and consistency. Uniqueness persists if we replacethe sd proportional-division lower-bound by sd normalized-no-envy, or consistency by converseconsistency, or both. The serial rule in Bogomolnaia and Moulin (2001) is characterized as aspecial case of our generalized serial rule.

JEL classification: C70, D61, D63.

Keywords: the generalized serial rule; sd-efficiency; sd proportional-division lower-bound; sdnormalized-no-envy; limited invariance; consistency; converse consistency; weak sd-strategy-proofness.

∗Department of Economics, University of Rochester, Rochester NY 14627, USA and Department of Economics,University of Bonn, Lennestr. 37, Bonn 53113, Germany, E-mail: heoeunjeong@gmail.com†I am grateful to William Thomson for his support and guidance. I have also benefited from useful comments

by Daisuke Hirata, Yoichi Kasajima, Bettina Klaus, Fuhito Kojima, Vikram Manjunath, Benny Moldovanu, JohnWeymark, and the participants of the WCU/BK Summer Economics Program at Yonsei University in 2009, theparticipants of the seminar at the University of Rochester in 2010, and the participants of the 10th Meeting of theSociety for Social Choice and Welfare in 2010. All errors are my own responsibility.

1. Introduction

We study the problem of assigning to agents a set of indivisible resources or objects. Each agentis assumed to receive a fixed number of objects. This number can be given several interpretations:a demand that he may have for the objects, a claim on these resources, or an obligation that hehas to fulfill when the objects are tasks. We simply refer to it as this agent’s demand and assumethat it is given exogenously and observed publicly. Real-life examples include assigning courses tostudents, assigning projects to workers, and assigning lectures to professors. We assume that agentshave strict rankings over objects and their preferences over sets of objects are additively separable.We focus on probabilistic rules that assign lotteries to agents for each reported preference profile.1

We propose a generalization of the serial rule (Bogomolnaia and Moulin, 2001) and show that it isthe unique one satisfying a certain set of requirements.

Several standard requirements of rules have been proposed and studied when all agents demandonly one object (unitary demands) (Bogomolnaia and Moulin, 2001). These requirements areformulated on the basis of the first-order stochastic dominance relation, associated with the ordinalpreferences for objects that agents report. First, stochastic dominance efficiency (sd-efficiency)requires that there should be no Pareto improvement possible at each selection made by a rule.2

Second, sd no-envy requires that each agent should find his own assignment at least as desirableto any other agent’s. Lastly, weak sd-strategy-proofness requires that each agent should not benefitby misreporting his preferences. When agents have unitary demands, the serial rule satisfies theserequirements (Bogomolnaia and Moulin, 2001).3

When agents receive more than one object, however, these requirements are no longer compat-ible. If multi-unit demands are constrained to be equal across agents, as a partial generalization ofunitary demands, there is an impossibility result involving the same set of requirements (Kasajima(2009) and Kojima (2009)).4

In this paper, we fully generalize the unitary demands by allowing (multi-unit) demands todiffer across agents.5 The same definitions of efficiency and of the incentive requirement in thecase of unitary demands apply to this generalization. However, the fairness requirements haveto be adapted, since one cannot directly evaluate whether certain assignments are “fair” whenagents demand different numbers of objects. We propose one plausible way of doing so: it is tofirst “normalize” each agent’s assignment by dividing it by his demand, and then to compare hisnormalized assignment with a reference assignment.6 This formulation is based on the assumption

1Given the indivisibility of objects, probabilistic rules are often used in practice to achieve (ex ante) fairness.The first formal paper on probabilistic assignment (Hylland and Zeckhauser, 1979) considers rules that take agents’cardinal preferences as input. Unfortunately, no rule satisfies the incentive requirement, ex ante efficiency, and“symmetry,” a fairness notion that is significantly weaker than no-envy (Zhou, 1990).

2For short, we use the prefix “sd” for stochastic dominance in other expressions below. We adopt the terminologyand notation of Thomson (2010a, 2010b).

3For the formal definitions of this rule and the axioms, see Sections 2 and 3.4When agents may regard several objects as being indifferent, these requirements are also incompatible even if

agents have unitary demands (Katta and Sethuraman, 2006).5See Papai (2001), Hatfield (2009), and Kojima (2012) for the rules assigning “sure” objects to agents in multi-unit

demand setting.6Liu and Pycia (2012) propose another way of evaluating fairness in this setting. Their fairness requirement

is based on the assumption that each agent compares his assignment with some part of a reference assignment

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that an agent with, say k-times more demand than another, has k-times more claim on the resources.We introduce two fairness requirements. The first one is sd normalized-no-envy, an adaptation

of “no-envy” (Foley, 1967): each agent should find his normalized assignment at least as desirableas that of any other agent. The second requirement is the sd proportional-division lower-bound, anadaptation of the “equal-division lower-bound” in classical fair division problems (Thomson, 2011b):each agent should find his normalized assignment at least as desirable as the equal probability sharesof all objects.7 In contrast to what is the case for classical fair division problems, these requirementsare logically related: the former implies the latter (Proposition 1).

As an extension of the previous impossibility results (Kasajima (2009) and Kojima (2009)),we show that, for each demand vector except for the unitary demand vector, (i) sd-efficiency,sd normalized-no-envy, and weak sd-strategy-proofness are incompatible and (ii) for two-agenteconomies, sd-efficiency, the sd proportional-division lower-bound, and weak sd-strategy-proofnessare incompatible (Proposition 2).

In what follows, we drop weak sd-strategy-proofness and replace it with a new incentive require-ment and one of two robustness requirements. The new incentive requirement is limited invariance.It rules out profitable misrepresentations for certain types of preferences, but not for all. With allother agents’ preferences fixed, suppose that the preference of an agent changes but the rankingsfrom his most preferred object down to a certain object, say a, do not change. Then, the probabilityof his receiving a should remain the same. This invariance requirement is also used in other recentpapers on the serial rule (Hashimoto et al. (2014) and Heo and Yılmaz (2012)). We discuss it indetail in Section 2.

The two robustness requirements are consistency and converse consistency.8 As probabilities ofobjects are infinitely divisible, we can easily adapt to our model two important variable-populationrequirements that have been studied in other models with divisible goods. Intuitively, both “con-firm” the desirability of choices made by a rule by comparing them across economies. Consis-tency relates the choice made by a rule for an economy to those it makes for associated “reducedeconomies” with smaller populations. Converse consistency, on the other hand, relates the collectionof choices made by a rule for two-agent economies to that it makes for the whole economy.9

To summarize, the requirements that we impose are (i) sd-efficiency, as an efficiency requirement,(ii) sd normalized-no-envy and the sd proportional-division lower-bound, as fairness requirements,(iii) limited invariance, as an incentive requirement, (iv) and consistency and converse consistency,as robustness requirements pertaining to variations in population. We show that these requirementsare satisfied by the “generalized serial rule” we propose. This rule is defined by means of analgorithm in which agents consume probabilities of objects over time: each agent consumes oneobject at a time in decreasing order of his preferences, switching from one object to another whenthe supply of the former reaches exhaustion. Unlike Bogomolnaia and Moulin (2001), however,

corresponding to his demand.7Equivalently, he should find his assignment (that is not normalized) at least as desirable as the “proportional-

division” of probabilities of all objects, which we define in Section 2.8Consistency can also be interpreted as a “fairness” requirement or a “reinforcement” requirement. For various

interpretations of consistency, see Thomson (2012).9For the formal definitions, see Section 2. For a survey on consistency and converse consistency, see Thomson

(2005). Thomson (2010a) also studies these two requirements in probabilistic assignment problems.

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the consumption rates differ across agents: an agent with, say k-times more demand than another,consumes probabilities k-times faster than the other does.

Our main result is that the generalized serial rule is the only rule satisfying sd-efficiency, the sdproportional-division lower-bound, limited invariance, and consistency (Theorem 1). Uniquenesspersists if we replace the sd proportional-division lower-bound by sd normalized-no-envy, or con-sistency by converse consistency, or both (Theorem 2 and Remark 4). These results can be appliedto the case of unitary demands (Bogomolnaia and Moulin, 2001): the serial rule is axiomaticallycharacterized as a special case of our generalized serial rule.

The current paper is one of the first papers characterizing the serial rule (Kesten et al. (2011)and Hashimoto and Hirata (2011)). Bogomolnaia and Heo (2012) is developed along the lines ofthe current manuscript, strengthening limited invariance. Hashimoto et al. (2014), evolved fromKesten et al. (2011) and Hashimoto and Hirata (2011), sharpens the result of Bogomolnaia andHeo (2012) in one of their characterizations.10 Our results are different from the others, however,in that (i) we allow multi-unit demands and work with a generalization of the serial rule and (ii) westudy variable-population axioms and a new fairness axiom that is weaker than sd no-envy. On theweak preference domain, on the other hand, Heo and Yılmaz (2012) extend the result of Hashimotoet al. (2014).

This paper is organized as follows. Section 2 presents the model, a set of axioms, and therelations between these axioms. Section 3 introduces the generalized serial rule and other rules.Section 4 presents our main results.

2. Model

Let A be a set of potential objects. Let A be the collection of all finite sets of potential objectsand A ≡ {a1, . . . , am} ∈ A. For each a ∈ A, let ωa ∈ [0, 1] be the probability that a is available.Let ω ≡ (ωa)a∈A be the endowment vector. Let ω be the unitary endowment vector composed of1’s.

Let N be a set of potential agents. Let N be the collection of all finite sets of potential agentsand N ≡ {i1, i2, . . . , in} ∈ N . Agent i ∈ N should receive a certain (finite) number of objects,denoted by di ∈ N+. We refer to di as his demand. Let d ≡ (di)i∈N be the demand vector. Let dbe the unitary demand vector composed of 1’s. We assume that

∑a∈A ωa =

∑i∈N di.

A lottery for each i ∈ N is defined as a list of numbers, πi ≡ (πia)a∈A, where πia is theprobability that agent i receives a, such that for each a ∈ A, πia ∈ [0, 1] and

∑a∈A πia = di.

We assume that (i) each i ∈ N has a strict preference Pi over A, (ii) there exists a von Neumann-Morgenstern utility function consistent with Pi, namely, ui : A→ R such that for each pair a, b ∈ A,a Pi b if and only if ui(a) > ui(b), and (iii) his utility from receiving a lottery πi ≡ (πia)a∈A is∑a∈A πiaui(a).11 For each a ∈ A, denote by U(Pi, a) ≡ {b ∈ A : b Pi a} ∪ {a} the weak upper

10Their first characterization is by means of sd-efficiency, sd no-envy, and limited invariance. The second is bymeans of “ordinal fairness” and a mild efficiency requirement.

11Kasajima (2009) and Kojima (2009) also adopt the same assumption of additively separable preferences. Liuand Pycia (2012) introduce a class of “responsive” preferences, which subsumes the class of additively separablepreferences. A preference (over lotteries) is responsive if (i) there is a strict ranking over objects, say Pi, and

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contour set of agent i with preference Pi at a. Let PA be the set of all strict preferences over A.Let P ≡ (Pi)i∈N ∈ PNA be the preference profile. For each i ∈ N , let P−i ≡ (Pj)j∈N\{i}.

An economy is a list e ≡ (A,ω,N, d, P ). Let E be the set of all economies. An assignment

matrix, (πia)i∈N,a∈A, is a collection of agents’ lotteries satisfying the feasibility constraint: foreach i ∈ N and each a ∈ A, (i) πia ∈ [0, 1], (ii)

∑i∈N πia = ωa, and (iii)

∑b∈A πib = di.12 For

each e ∈ E , let Π(e) be the set of all assignment matrices at e. For each π ∈ Π(e), each i ∈ N ,and each B ⊆ A, let πi|B ≡ (πia)a∈B be the list of probabilities for agent i of receiving the objectsin B. We say that an |N | × |A| matrix π ≡ (πia)i∈N,a∈A is deterministic if for each i ∈ N andeach a ∈ A, πia ∈ {0, 1},

∑i∈N πia ≤ 1 and

∑a∈A πia = di. Each π ∈ Π(e) can be represented

as a convex combination of deterministic matrices (Budish et al. 2013).13 A rule is a mappingϕ : E → ⋃

e∈E Π(e) such that for each e ∈ E , ϕ(e) ∈ Π(e). Note that agents are asked to reporttheir ordinal preferences even when they have cardinal preferences.

2.1. Axioms

Let e ≡ (A,ω,N, d, P ) and i ∈ N . Let πi ≡ (πia)a∈A, and π′i ≡ (π′ia)a∈A be two lotteries for agent i.We say that πi weakly stochastically dominates π′

i at Pi if for each a ∈ A,∑b∈U(Pi,a) πib ≥∑

b∈U(Pi,a) π′ib. We say that πi stochastically dominates π′

i at Pi if at least one of theseinequalities is strict. Let π, π′ ∈ Π(e) be two assignment matrices. We say that π stochastically

dominates π′ at P if for each i ∈ N , πi weakly stochastically dominates π′i at Pi and π 6= π′.Henceforth, we use the prefix “sd” for stochastic dominance for short.

The first axiom pertains to efficiency: no Pareto improvements should be possible. That is, itshould not be possible for everyone to be at least as well off and someone to be better off in thesd-sense. Formally, an assignment matrix π ∈ Π(e) is sd-efficient14 at e, if it is not stochasticallydominated by any other π′ ∈ Π(e).15 The corresponding requirement for a rule ϕ is the following:

Sd-efficiency: For each e = (A,ω,N, d, P ) ∈ E , ϕ(e) is sd-efficient at e.

Next are two fairness requirements. First, each agent should find his normalized assignment atleast as desirable as that of any other. For each π ∈ Π(e), agent i’s normalized assignment is 1

diπi.

We say that π ∈ Π(e) is sd normalized-envy-free at e, if for each pair i, j ∈ N , 1diπi weakly

stochastically dominates 1djπj at Pi. The corresponding requirement for a rule ϕ is the following:

(ii) given a pair of lotteries whose components differ only for two objects, say a and b, if a Pi b, then the lottery witha greater probability of a is preferred to the other. Our result extends to this domain immediately. This is becauseagents are asked to report their underlying ordinal preferences in either case and all the axioms and the rules aredefined based on the ordinal preferences.

12We can easily adapt our model for the cases with∑a∈A ωa ≤

∑i∈N di: conditions (ii) and (iii) change to

(ii′)∑i∈N πia ≤ ωa and (iii′)

∑b∈A πib ≤ di. Our results readily extend to such cases. See footnote 16.

13Each assignment matrix is represented by a convex combination of deterministic matrices in the following way:we first introduce a set of virtual agents (from outside the economy) whose total assignment is (ω − ω). Then, bythe generalization of the Birkhoff-von Neumann theorem (Budish et al, 2013), the assignment matrix with virtualagents can be represented by a convex combination of deterministic matrices. We then remove these virtual agents’components from each deterministic matrix in the support.

14Bogomolnaia and Moulin (2001) call this requirement ordinal efficiency. In this paper, we adopt the terminologyof Thomson (2010b).

15See Bogomolnaia and Moulin (2001) and Abdulkadiroglu and Sonmez (2003) for characterizations of sd-efficientassignment matrices.

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Sd normalized-no-envy: For each e = (A,ω,N, d, P ) ∈ E , ϕ(e) is sd normalized-envy-free at e.

The other fairness axiom requires that each agent should find his normalized assignment atleast as desirable as the equal-division of probabilities of all objects. Equivalently, each agentshould find his assignment at least as desirable as the “proportional-division” of probabilities of allobjects, where the proportional-division at e, π(e), is defined by setting, for each i ∈ N and eacha ∈ A, πia(e) = di∑

djωa. An assignment π ∈ Π(e) satisfies the sd proportional-division lower-

bound at e, if for each i ∈ N , πi weakly stochastically dominates πi(e) at Pi. The correspondingrequirement for a rule ϕ is the following:

Sd proportional-division lower-bound: For each e = (A,ω,N, d, P ) ∈ E , ϕ(e) satisfies the sdproportional-division lower-bound at e.

When there are more than two agents, for classical problems of fair division or for fair divisionproblems with single-peaked preferences, the corresponding concepts called “no-envy” and the“equal-division lower-bound” are not logically related to each other (Kolm (1972) and Thomson(2010b)). In our model, however, these requirements, adapted to our setting in the manner thatwe propose, are logically related. Here is how.

Proposition 1. For each e = (A,ω,N, d, P ) ∈ E ,16

(i) sd normalized-no-envy implies the sd proportional-division lower-bound.(ii) if |N | = 2, the sd proportional-division lower-bound implies sd normalized-no-envy.

Proof. (i) Let e = (A,ω,N, d, P ) ∈ E and π ∈ Π(e) be an sd normalized-envy-free assignmentmatrix at e. Then, for each a ∈ A,

1di

∑b∈U(Pi,a) πib ≥ 1

dj

∑b∈U(Pi,a) πjb, equivalently, dj

∑b∈U(Pi,a) πib ≥ di

∑b∈U(Pi,a) πjb.

Summing over j ∈ N and dividing both sides by∑j∈N dj ,

∑b∈U(Pi,a) πib ≥ di∑

j∈N dj

∑j∈N (

∑b∈U(Pi,a) πjb) = di∑

j∈N dj

∑b∈U(Pi,a) ωb =

∑b∈U(Pi,a) πib(e).

(ii) Let e = (A,ω,N, d, P ) with |N | = 2. Let N ≡ {i, j}. Let π ∈ Π(e) be an assignment matrixsatisfying the sd proportional-division lower-bound at e. Then, for each a ∈ A,

∑b∈U(Pi,a) πib ≥ di

di+dj

∑b∈U(Pi,a) ωb = di

di+dj

∑b∈U(Pi,a)(πib + πjb),

from which we obtain 1di

∑b∈U(Pi,a) πib ≥ 1

dj

∑b∈U(Pi,a) πjb.

Next are the incentive requirements discussed in the literature. The first one is that each agent’sassignment resulting from reporting his true preferences should weakly stochastically dominate anyassignment resulting from misreporting his preferences, no matter what the other agents’ prefer-ences are.17

Sd-strategy-proofness: For each e = (A,ω,N, d, P ) ∈ E , each i ∈ N , and each P ′i ∈ PA, ϕi(e)weakly stochastically dominates ϕi(A,ω,N, d, (P ′i , P−i)) at Pi.

16Proposition 1 extends to the cases in which∑a∈A ωa ≤

∑i∈N di by imposing “non-wastefulness”: π ∈ Π is

non-wasteful if for each pair a, b ∈ A and each i ∈ N , if a Pi b and πib > 0, then∑j∈N πja = ωa.

17Sd-strategy-proofness is referred to as straightforwardness in Gibbard (1978).

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Next is a weaker axiom than sd-strategy-proofness: no agent’s assignment resulting from misre-porting his preferences should stochastically dominate his assignment resulting from reporting histrue preferences, no matter what the other agents’ preferences are.

Weak sd-strategy-proofness: For each e = (A,ω,N, d, P ) ∈ E , each i ∈ N , and each P ′i ∈ PA,ϕi(A,ω,N, d, (P ′i , P−i)) does not stochastically dominate ϕi(e) at Pi.

In the case of unitary demands, there is a rule satisfying sd-efficiency, sd normalized-no-envy,and weak sd-strategy-proofness (Bogomolnaia and Moulin, 2001). However, when the agents receivethe same number of multiple objects, no rule satisfies these requirements (Kojima, 2009). We extendthis impossibility result to any demand vectors except for the unitary demand vector. In the proof,we employ the same proof strategy as Kojima (2009) and adapt his preference profiles to our setting(Appendix 1).

Proposition 2. Whenever there is an agent who demands more than one object, no rule satisfiessd-efficiency, sd normalized-no-envy, and weak sd-strategy-proofness.

By Propositions 1(ii) and 2, we obtain another incompatibility for two-agent economies.

Corollary 1. In two-agent economies, whenever there is an agent who demands more than oneobject, no rule satisfies sd-efficiency, the sd proportional-division lower-bound, and weak sd-strategy-proofness.

In view of these impossibilities involving weak sd-strategy-proofness, we introduce a new incentivecondition. Let i ∈ N and Pi ∈ PA. For each a ∈ A, let Pi(a) be the preference relation over theobjects in U(Pi, a) such that for each pair b, c ∈ U(Pi, a), b Pi c if and only if b Pi(a) c.

Limited invariance: For each e = (A,ω,N, d, P ) ∈ E , each P ′i ∈ PA, and each a ∈ A, ifU(Pi, a) = U(P ′i , a) and Pi(a) = P ′i (a), then ϕia(e) = ϕia(A,ω,N, d, (P ′i , P−i)).

Recall that each agent is asked to report his ordinal preferences, but the preference intensitiesover the objects are kept as private information. Suppose, for example, that an agent misreports hispreferences but his most preferred object remains the same. There is a possibility that this agentfinds his most preferred object much more desirable than the remaining objects. If he is assigned agreater probability of his most preferred object, then he is obviously better off misrepresenting hispreferences; if he is assigned a smaller probability for the same object, then a symmetric argumentapplies by switching his true preferences and his misrepresented preferences. Thus, to prevent sucha profitable misrepresentation, the probability assigned to his most preferred object should remainthe same, as limited invariance requires.

Now, suppose that an agent misreports his preferences but the rankings over his most preferredtwo objects remain the same. As above, there is a possibility that this agent finds his most preferredobject much more desirable than the remaining objects. By the same argument, the probability ofreceiving this object should remain the same. However, there is also the possibility that this agentfinds his most preferred two objects much more desirable than the remaining objects.18 Given that

18From the social planner’s perspective, all possibilities have to be considered, since this cardinal information

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the probability of receiving his most preferred object remains the same, the probability of receivinghis second most preferred object should remain the same as well, to prevent him from being betteroff misrepresenting his preferences. This is exactly what limited invariance requires.

These arguments can be generalized to any misrepresentations keeping the rankings over hismost preferred k objects invariant, where k ∈ {1, . . . , |A|}. For this reason, we regard this axiom asan incentive requirement that rules out profitable misrepresentations for certain types of preferences,but not for all. This is obviously a weakening of sd-strategy-proofness. Other interpretations oflimited invariance are also available in Hashimoto et al. (2014).

Remark 1. Sd-strategy-proofness implies limited invariance, but there is no logical relation betweenweak strategy-proofness and limited invariance.

Next, we introduce two central requirements pertaining to variable populations. Both requirerobustness of choices that a rule makes for an economy and a set of “sub-economies”. In definingthese requirements, (i) we treat the probability of each object’s availability as an infinitely divis-ible good and thus, (ii) we allow for fractional endowment of resources, instead of restricting ourattention to economies with integer endowments (Thomson, 2010a).19

First, consider an economy and the assignment matrix chosen for it by a rule. Consider anysubset of agents and identify the resources that they have collectively received. Then, consider theproblem of assigning these resources to these agents. The requirement is that the rule should recom-mend the same assignments for them as it did in the initial economy. Formally, let e = (A,ω,N, d, P )and π ∈ Π(e). For each T ⊆ N , the reduced economy of e relative to T and π, rπT (e), is thelist (A,

∑i∈T πi, T, (di)i∈T , (Pi)i∈T ).

Consistency: For each e = (A,ω,N, d, P ) ∈ E and each T ⊆ N , (ϕi(e))i∈T = ϕ(rϕ(e)T (e)).20

Another requirement is “converse consistency”. Consider an economy and a candidate assign-ment matrix for it. Choose any two agents, and identify the resources that they have collectivelyreceived. Then, consider the problem of assigning these resources to these two agents. Supposethat the rule recommends the very same assignments for these agents as it did at the candidateassignment matrix, and that this holds for all two-agent pairs. The requirement is that the ruleshould recommend the candidate assignment matrix for the initial economy.

Converse consistency: For each e = (A,ω,N, d, P ) ∈ E and π ∈ Π(e), if for each pair i, j ∈ N ,(πi, πj) = ϕ(rπ{i,j}(e)), then π = ϕ(e).

cannot be elicited publicly. For each l ∈ N+, let al be this agent’s l-th most preferred object and ε ∈]0, 1[ besufficiently small. Such cardinal preferences, for example, are represented by functions u, u′ : A → R+ such thatu(a1) = 10, u(a2) = ε, u(a3) = ε2, u(a4) = ε3, and so on, and u′(a1) = 10, u′(a2) = 9, u′(a3) = ε, u′(a4) = ε2, andso on.

19It is often the case that probability shares of objects have already been distributed to some agents outside theeconomy, and that the resources available are the remaining shares. In the case of public housing, for example,applicants are divided into several groups with different priorities depending on the number of family members,income level, and so on. The group with the highest priority comes first to receive lotteries, households with thesecond highest priority come next, and so on. The resources available to a group, say the group with the secondhighest priority, will be the probabilities that are not yet assigned to the group with the highest priority. Theresources available to this group can be fractional. Athanassoglou and Sethuraman (2011) and Yılmaz (2010) alsoallow fractional endowments, but these are owned by individuals.

20A weaker axiom, bilateral consistency, requires that this equality hold for each T ⊆ N with |T | = 2.

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These axioms are the natural adaptations of the consistency notions that have been studied forother classes of problems, such as standard fair division problems or claims problems.21

3. Rules

To present our main result, we first introduce a special representation of assignments by means of“consumption schedules”. Using this representation, we define our generalization of the serial rule.It is also a key component of the intuitive and direct proofs of our results.22

3.1. Consumption Schedule

Let e = (A,ω,N, d, P ) and π ∈ Π(e). We represent π as a process in which (i) each agent consumesprobability shares of objects in decreasing order of his preference over time period [0, 1] and (ii) hisconsumption rate is equal to his demand. Consider i ∈ N and his assignment πi. This agentstarts consuming his most preferred object, say a, at time 0 at the consumption rate di, and endsconsuming it at time 1

diπia (then the probability that he has consumed his most preferred object is

exactly πia); he then switches to his second most preferred object, say b, and starts consuming it attime 1

diπia with the same rate, and ends consuming it at time 1

di(πia + πib) (again, the probability

that he has consumed his second most preferred object is exactly πib); and so on. The list of timesat which agent i starts/ends consuming objects determines the probabilities that he has consumed,which coincide with (πia)a∈A. We call such a process the preference-decreasing consumption

schedule at (Pi, di) representing πi. For brevity, we call it a “consumption schedule”.

Remark 2. Once an agent switches from one object to another along his consumption schedule, henever returns to the former object. Also, if an agent stops consuming object a at time τ along theconsumption schedule at (Pi, di) representing πi, the sum of probabilities of his receiving objectsin U(Pi, a) is diτ .

Let the collection of all agents’ consumption schedules be the profile of consumption sched-

ules at (P,d) representing π. Note that for each π ∈ Π(e), the profile of consumption schedulesat (P, d) representing π is uniquely determined.

3.2. The Generalized Serial Rule

We now define a generalization of the serial rule, which we call the generalized serial rule, S.Let e = (A,ω,N, d, P ) ∈ E and i ∈ N . Agent i consumes probability shares of objects, one objectat a time, in decreasing order of his preference Pi. His consumption rate is di. At time t = 0,each agent starts consuming his most preferred object in A. When the supply of his most preferredobject is exhausted, he switches to his most preferred object among the ones still available at that

21There is another way to define consistency in probabilistic assignment problems. Chambers (2004) uses Bayesianupdating to determine the assignment for each reduced economy. His axiom turns out to be very demanding: in thecase of unitary demands, if we require consistency in this sense and equal treatment of equals, we obtain the equaldivision rule.

22This representation is also used in other characterizations of the serial rule: see Bogomolnaia and Heo (2012),Heo and Yılmaz (2012), and Heo (2013).

8

sequential random generalized proportionalpriority rule priority rule serial rule rule

sd-efficiency + − +∗,† −sd normalized-no-envy − − + +

sd proportional-division LB − − +∗,† +sd-strategy-proofness + + − +

weak sd-strategy-proofness + + − +limited invariance + + +∗,† +

consistency + − +∗ +converse consistency + − +† +

Table 1. The properties of the four rules (∗: Theorem 1, †: Theorem 2)

point, and starts consuming it. Again, when the supply of the object that he has been consumingis exhausted, he switches to his most preferred object among the ones still available at that point,and starts consuming it, and so on. The assignment selected for each agent by the generalized serialrule is the list of probabilities that he has consumed over [0, 1].

Remark 3. For each e = (A,ω,N, d, P ) ∈ E , the profile of consumption schedules at (P, d)represents S(e) if and only if each agent switches from one object to another only when the formerreaches exhaustion. This is equivalent to saying that there is no pair of agents i, j ∈ N and an objecta ∈ A such that agent j consumes a positive probability of a even after agent i ends consuming a.

3.3. Other Rules

We present three other rules. Let � be an order over all potential agents. For each N ∈ N , let �Nbe the order restricted to N . The sequential priority rule associated with � is defined asfollows: for each e = (A,ω,N, d, P ) ∈ E , let the first agent in �N choose his most preferred lotteryamong all available probability supplies in e; let the second agent in �N choose his most preferredlottery among the remaining probability supplies; and so on; the assignment matrix is chosen byrepeating this process up to the agent ordered last in �N . The random priority rule, RP isdefined for each N ∈ N , by averaging the sequential priority rules over all possible �N ’s. Theproportional division rule, Prop, selects for each economy e ∈ E , the proportional division ofprobabilities of all objects, i.e., Prop(e) = π(e).

Table 1 summarizes the properties of all the rules that we have discussed. Appendix 3 pro-vides examples to show that the random priority rule violates converse consistency and the sdproportional-division lower-bound.

4. Characterization of the Generalized Serial Rule

In this section, we present our main results.

Theorem 1. The generalized serial rule is the only rule satisfying sd-efficiency, the sd proportional-division lower-bound, limited invariance, and consistency.

9

Proof. We first show that the generalized serial rule satisfies these four axioms. Bogomolnaia andMoulin (2001) show that the serial rule is sd-efficient and sd-envy-free, and the same argumentscan be applied to show that the generalized serial rule is sd-efficient and sd normalized-envy-free.By Proposition 1(i), it satisfies the sd proportional-division lower-bound. The generalized serialrule satisfies limited invariance because the object each agent consumes at each t ∈ [0, 1] does notdepend on the objects to which he switches later (that is, his less preferred objects). Next, weprove that the rule is consistent.23 Let e ≡ (A,ω,N, d, P ) ∈ E . By Remark 3, along the profile ofconsumption schedules at (P, d) representing S(e), no agent ends consuming an object that is notexhausted. Let T ⊆ N . Consider the profile of consumption schedules at (Pi, di)i∈T representing(Si(e))i∈T . Again, no agent ends consuming an object that is not exhausted. Again, by Remark 3,(Si(e))i∈T = S(rS(e)

T (e)). Conversely, let ϕ be a rule satisfying the four axioms. We prove thatϕ = S.

Claim 1. In two-agent economies, if ϕ satisfies sd-efficiency, the sd proportional-division lower-bound, and limited invariance, then ϕ = S.

Let e ≡ (A,ω,N, d, P ) with |N | = 2. By Proposition 1(ii), ϕ is sd normalized-envy-free. Sup-pose, by contradiction, that ϕ(e) 6= S(e). Consider the profile of consumption schedule at (P, d)representing ϕ(e). By Remark 3, there are a pair i, j ∈ N and a ∈ A such that agent i endsconsuming a at a time τ ∈ [0, 1[, after which agent j consumes a positive probability of a. Letτ ′ ∈ [0, 1] be the time at which agent j ends consuming a. Obviously, τ < τ ′. By sd-efficiency,for each b ∈ {c ∈ A : a Pi c Pj a}, ϕib(e) = 0; as ϕib(e) + ϕjb(e) = ωb, we have ϕjb(e) = ωb. LetU ≡ U(Pi, a) ∪ U(Pj , a). Let P ′i ∈ PA be such that (i) U(Pi, a) = U(P ′i , a) and Pi(a) = P ′i (a), and(ii) for each b ∈ U \U(Pi, a) and each c ∈ A\U , b P ′i c. Let e′ ≡ (A,ω, {i, j}, d, (P ′i , Pj)). By limitedinvariance, for each b ∈ U(Pi, a), ϕib(e) = ϕib(e′) and thus, by feasibility, ϕjb(e′) = ϕjb(e). Sinceϕja(e′) = ϕja(e) > 0, by sd-efficiency, for each b ∈ U \ U(Pi, a), ϕib(e′) = 0, and by feasibility,ϕjb(e′) = ωb. Altogether,

1di

∑b∈U ϕib(e

′) = 1di

∑b∈U(Pi,a) ϕib(e) = τ < τ ′ = 1

dj

∑b∈U(Pj ,a) ϕjb(e) = 1

dj

∑b∈U(Pj ,a) ϕjb(e

′),

Since U(Pj , a) ⊆ U , we have 1dj

∑b∈U(Pj ,a) ϕjb(e

′) ≤ 1dj

∑b∈U ϕjb(e

′). Thus, 1diϕi(e′) does not

weakly stochastically dominate 1djϕj(e′) at P ′i , in violation of sd normalized-no-envy.

Now, we consider general economies with |N | ≥ 2. We show that S is conversely consistent.

Claim 2. The generalized serial rule is conversely consistent.

Let e ≡ (A,ω,N, d, P ) ∈ E . Let π ∈ Π(e) be such that for each pair i, j ∈ N , (πi, πj) =S(rπ{i,j}(e)). Suppose, by contradiction, that π 6= S(e). By Remark 3, along the profile of con-sumption schedules at (P, d) representing π, there are a pair k, l ∈ N and a ∈ A such thatagent l consumes a positive probability of a even after agent k ends consuming a. The samestatement holds at the profile of consumption schedules at (Pi, di)i∈{k,l} representing (πi)i∈{k,l}.Thus, (πk, πl) 6= S(rπ{k,l}(e)), a contradiction.

23In the case of unitary demands, Thomson (2010a) proves that the serial rule is consistent and converselyconsistent. Here, we prove it by using the consumption schedules.

10

Now, let ϕ be a rule satisfying sd-efficiency, the sd proportional-division lower-bound, limitedinvariance, and consistency. By Claim 1, for two-agent problems, ϕ = S. By Claim 2, S isconversely consistent. We now invoke the following lemma:

Elevator Lemma (Thomson, 2011a) Let ψ be a consistent rule and ψ be a converselyconsistent rule. If ψ = ψ for each two-agent problem, then ψ = ψ for any problem.24

By setting ψ ≡ ϕ and ψ ≡ S and applying the Elevator Lemma, we obtain the desired result.

If we replace consistency by converse consistency in Theorem 1, then, again, the generalizedserial rule is the only possible rule satisfying these axioms.

Theorem 2. If a rule satisfies sd-efficiency, the sd proportional-division lower-bound, limited in-variance, and converse consistency, then it is the generalized serial rule.

Proof. Let ϕ be a rule satisfying the four axioms listed in the theorem. In the proof of Theorem 1,we show that S is consistent and for two-agent problems, ϕ = S. By setting ψ ≡ S and ψ ≡ ϕ andthen applying the Elevator lemma, we obtain the desired result.

Remark 4. By Proposition 1(i), the characterization results in Theorems 1 and 2 still hold if wereplace the sd proportional-division lower-bound by sd normalized-no-envy.

Appendix 1: Proof of Proposition 2

Let ϕ be a rule satisfying the three requirements. Consider an economy e ≡ (A, ω,N, d, P ) inwhich there is at least one agent who demands more than one object. Let i, j ∈ N be such that(di, dj) 6= (1, 1). Without loss of generality, let di ≤ dj and α ≡ di

di+dj. Suppose that

Pi : a2 a3 a1 a4 a5 a6 · · · am−1 am and Pj : a1 a2 a3 a4 a5 a6 · · · am−1 am.

Let A be agent i’s most preferred (di+dj) objects at Pi. Suppose that all agents other than i and jhave the same preferences and they prefer each object in A \ A to each object in A. Now, consider

e′ ≡ (A, ω,N, d, (P ′i , Pj , (Pk)k 6=i,j))e′′ ≡ (A, ω,N, d, (Pi, P ′j , (Pk)k 6=i,j))

where P ′i = P ′j : a2 a1 a3 a4 a5 a6 · · · am−1 am.

Let π ≡ ϕ(e), π′ ≡ ϕ(e′), and π′′ ≡ ϕ(e′′). By sd-efficiency,

for each k ∈ N \ {i, j} and each b ∈ A, πkb = π′kb = π′′kb = 0for each k ∈ {i, j} and each b ∈ A \ A, πkb = π′kb = π′′kb = 0.

By sd-efficiency and sd normalized-no-envy,

π′i|A = (0, 2α, α, α, . . . , α) and π′j |A = (1, 1− 2α, 1− α, 1− α, . . . , 1− α)π′′i |A = (0, α, 2α, α, . . . , α) and π′′j |A = (1, 1− α, 1− 2α, 1− α, . . . , 1− α)

24The Elevator Lemma applies in general to the models pertaining to variable populations. The proof in our setting

is as follows. Let e ∈ E. Since ψ is consistent, for each pair of agents, say i and j, (ψi(e), ψj(e)) = ψ(rψ(e){i,j}(e)).

Since ψ and ψ coincide in each two-agent problem, ψ(rψ(e){i,j}(e)) = ψ(r

ψ(e){i,j}(e)). Thus, ψ(r

ψ(e){i,j}(e)) = (ψi(e), ψj(e)).

Since ψ is conversely consistent, ψ(e) = ψ(e), as desired. Note that we only invoke bilateral consistency of ψ in theproof. For other applications of the same logic, see Peleg (1986).

11

Lastly, we calculate π. By sd-efficiency, πia1 = 0. By weak sd-strategy-proofness, given agent j’spreference Pj , agent i with true preference P ′i should not be better off reporting Pi: πia2 ≥ 2α. Byfeasibility, πja1 + πja2 = 2− πia2 . By sd normalized-no-envy, 2−πia2

dj≥ πia2

di. Therefore, πia2 ≤ 2α.

Altogether, πia2 = 2α. Thus,

πi|A = (0, 2α, α, α, . . . , α) and πj |A = (1, 1− 2α, 1− α, 1− α, . . . , 1− α)

Given agent i’s preference Pi, agent j with true preference Pj is better off reporting P ′j . This is aviolation of weak sd-strategy-proofness. �

Appendix 2: Independence of Axioms

We establish the independence of the axioms in our results.

Independence of Axioms in Proposition 2

The proportional division rule satisfies all but sd-efficiency. Each sequential priority rule satisfiesall but sd normalized-no-envy (or the sd proportional-division lower-bound). Lastly, the generalizedserial rule satisfies all but weak sd-strategy-proofness (Kojima, 2009).

Independence of Axioms in Theorem 1.

The proportional division rule satisfies all but sd-efficiency. Each sequential priority rule satisfiesall but the sd proportional-division lower-bound. The following rule ψ satisfies all but limitedinvariance. Recall that d is the unitary demand vector.

Example 1. A rule satisfying all but limited invariance.

Consider

e ≡ ({a1, a2, a3}, ω, {i, j, k}, d, (Pi, Pj , Pk))

e′ ≡ ({a1, a2, a3}, ( 45 ,

710 ,

12 ), {i, j}, d, (Pi, Pj))

e′′ ≡ ({a1, a2, a3}, ( 25 ,

35 , 1), {j, k}, d, (Pj , Pk))

wherePi : a1 a2 a3

Pj : a3 a1 a2

Pk : a3 a1 a2.

Let ψ be the anonymous and neutral rule25 such that

ψ(e) =

35

25 0

15

310

12

15

310

12

ψ(e′) =

(35

25 0

15

310

12

)ψ(e′′) =

(15

310

12

15

310

12

)

and for each e ∈ E \ {e, e′, e′′, e′′′}, ψ(e) = S(e). If the preference of agent i changes from Pi

to P ′i : a1 a3 a2, then ψia1({a1, a2, a3}, ω, {i, j, k}, d, (P ′i , Pj , Pk)) = 23 . This is a violation of limited

invariance. �

The following rule ϕ satisfies all but consistency.25A rule is anonymous if the labels of agents do not matter; a rule is neutral if the labels of objects do not

matter. Thanks to these properties, we do not need to specify all the assignment matrices to define ψ: we canderive choices that the rule makes for some preference profiles from these two properties, if applicable. For example,let e′ ≡ ({a1, a2, a3}, ( 4

5, 710, 12

), {i, k}, d, (Pi, Pk)); by anonymity, we derive ψ(e′) from ψ(e′): ψi(e′) = ψi(e′) and

ψk(e′) = ψj(e′).

12

Example 2. A rule satisfying all but consistency.

Considere ≡ ({a1, a2, a3}, ω, {i, j, k}, d, (Pi, Pj , Pk))e′ ≡ ({a1, a2, a3}, ω, {i, j, k}, d, (Pi, Pj , P ′k))

wherePi : a1 a2 a3 Pk : a2 a1 a3

Pj : a1 a3 a2 P ′k : a2 a3 a1.

Let ϕ be the anonymous and neutral rule such that ϕ(e) = ϕ(e′) =

12

15

310

12 0 1

2

0 45

15

and for each

e′′ ∈ E \ {e, e′}, ϕ(e′′) = S(e′′). Then, (ϕi(e), ϕk(e)) 6= ϕ(rϕ(e){i,k}(e)). �

Independence of Axioms in Theorem 2.

The proportional division rule satisfies all but sd-efficiency. Each sequential priority rule satisfiesall but the sd proportional-division lower-bound. The following rule ψ satisfies all but converseconsistency.

Example 3. A rule satisfying all but converse consistency.

Considere ≡ ({a1, a2, a3}, ω, {i, j, k}, d, (Pi, Pj , Pk))e′ ≡ ({a1, a2, a3}, ω, {i, j, k}, d, (Pi, Pj , P ′k))

wherePi : a1 a2 a3 Pk : a2 a1 a3

Pj : a1 a3 a2 P ′k : a2 a3 a1.

Let ψ be the anonymous and neutral rule such that ψ(e) = ψ(e′) =

12

16

13

12 0 1

2

0 56

16

, and for each

e′′ ∈ E \{e, e′}, ψ(e′′) = S(e′′). Let π ≡ S(e). For each pair l,m ∈ {i, j, k}, (πl, πm) = ψ(rπ{l,m}(e)).However, π 6= ψ(e). �

The following rule ϕ satisfies all but limited invariance.

Example 4. A rule satisfying all but limited invariance.

Consider e ≡ ({a1, a2, a3}, ( 34 ,

12 ,

34 ), {i, j}, d, (Pi, Pj)) where Pi : a1 a2 a3, Pj : a2 a3 a1, and

P ′j : a2 a1 a3. Let ϕ be such that ϕ(e) =(

34

14 0

0 14

34

), and for each e′ ∈ E \{e}, ϕ(e′) = S(e′). If

the preference of agent j changes from Pj to P ′j , then ϕja2({a1, a2, a3}, ( 34 ,

12 ,

34 ), {i, j}, d, (Pi, P ′j)) = 1

2 .This is a violation of limited invariance. �

Appendix 3. The Random Priority Rule

We present two examples to show that the random priority rule violates converse consistency andthe sd proportional-division lower-bound.

Example 5. RP violating converse consistency

Consider e ≡ ({a, b, c}, ω, {1, 2, 3}, d, P ) where P1 = P2 : a b c and P3 : b a c. Let π ≡

49

29

13

49

29

13

19

59

13

. It is easy to check that for each pair i, j ∈ N , (πi, πj) = RP (rπ{i,j}(e)), but

π 6= RP (A,ω,N, d, P ). �

13

Example 6. RP violating the sd proportional-division lower-bound

Consider e ≡ ({a, b, c}, ω, N = {1, 2}, d = (2, 1), P ) where P1 = P2 : a b c. We have RP (e) =(12 1 1

212 0 1

2

): RP1(e′) does not weakly stochastically dominate π1 = ( 2

3 ,23 ,

23 ) at P1. �

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