A characterization of the Extended Serial Correspondence Eun Jeong Heo∗ October 27, 2011

Abstract We study the problem of assigning objects to a group of agents, when each agent has ordinal preferences over the objects. We allow for indifference among objects. We focus on probabilistic methods, in particular, the extended serial correspondence, introduced by Katta and Sethuraman (2006). Our main result is that (i) each correspondence satisfying stochastic dominance efficiency, stochastic dominance no-envy and “bounded invariance,” an axiom we adapt from Bogomolnaia and Heo (2011), is a subcorrespondence of the extended serial correspondence, and (ii) for each extended serial subcorrespondence, there is a “welfare-equivalent” correspondence satisfying the three properties. The key to these results is to introduce a profile of tie-breaking rules among indifferent objects, one for each agent, and then exploit “preference-decreasing consumption schedule”. JEL classification: C70, D61, D63. Keywords: the serial rule; sd efficiency; sd no-envy; the extended serial correspondence; bounded invariance.

∗

Department of Economics, University of Rochester, Rochester NY 14627, USA, E-mail: [email protected] I am indebted to William Thomson for his guidance and support. I am also grateful to Anna Bogomolnaia for helpful comments and discussions. I benefited from comments from the seminar participants at Columbia University, in particular, Yeon-koo Che, Fuhito Kojima, and Jay sethuraman. All errors are my own responsibility.

1. Introduction We study the problem of assigning objects to a group of agents, when each agent has ordinal preferences over objects. We focus on probabilistic methods: each agent receives a probability distribution over objects, whose coordinates add up to 1. An assignment matrix specifies, for each agent and each object, the probability that this agent will receive this object. A solution selects a set of assignment matrices for each preference profile that agents report. If a solution is single-valued, we call it a rule. If a solution is not necessarily single-valued, we call it a correspondence. On the domain of strict preferences, Bogomolnaia and Moulin (2001) propose to compare lotteries on the basis of first-order stochastic dominance. Given two lotteries, say p and p0 , we say that “p weakly first-order stochastically dominates p0 at an agent’s preference,” if (1) the probability of his receiving his most preferred object at p is at least as large as the corresponding probability at p0 , (2) the sum of the probabilities of his receiving his most preferred object and his second most preferred object at p is at least as large as the corresponding sum at p0 , (3) and so on. If p weakly first-order stochastically dominates p0 at an agent’s preference and p 6= p0 , then we say that “p stochastically dominates p0 at his preference”. Throughout the paper, the notation “sd” stands for “stochastic dominance”.1 We say that an agent is “sd better off” at an assignment than at another assignment if the former first-order stochastically dominates the latter at his preference. Similarly, we say that an agent is “sd worse off” at an assignment than at another assignment if the latter first-order stochastically dominates the former at his preference. Let us now discuss assignment matrices. Two central notions are based on stochastic dominance comparisons by agents. First is an efficiency notion. “Sd-efficiency” requires that there should be no other assignment matrix at which some agent is sd better off and no agent is sd worse off. Next is a fairness notion. “Sd no-envy” requires that each agent should be sd better off at his assignment than at that of each other agent at his preference, or that his assignments be the same. We say that a rule satisfies sd efficiency and sd no-envy, respectively, if for each economy, it selects an assignment matrix satisfying sd efficiency and sd no-envy. The “serial rule”, introduced by Bogomolnaia and Moulin (2001), satisfies these two properties, but it is not the only rule to do so. However, it is the only one to satisfy the 1

We adopt the terminology and notation of Thomson (2010a, 2010b).

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two properties together with one of several invariance properties (Bogomolnaia and Heo (2011), Hashimoto and Hirata (2011), Heo (2011a), and Kesten et al. (2011)).2 The serial rule was originally defined for strict preference profiles, and it is not obvious how to adapt it to the weak preference domain. Consider, for example, a three-agent economy in which agents’ preferences are given as follows. If each agent finds several objects indifferent, we place the objects between parenthesis. (i)

the first agent prefers (a, b) to c,

(ii)

the second agent prefers b to (a, c), and

(iii) the third agent prefers b to a and prefers a to c. Note that the first agent is indifferent between a and b, but the other agents prefer b to each other object. To achieve sd-efficiency, the first agent has to yield b to the others, in exchange for receiving a instead. Then, the other agents should share b in a way that sd no-envy is not violated. Next, the second agent is indifferent between the remaining objects a and c, but the other agents prefer a to each other remaining object. To achieve sd-efficiency, again, the second agent has to yield a to the others, in exchange for receiving c. Lastly, the first and the third agents should share a in a way that sd no-envy is not violated. If there are more than three agents and the structure of a preference profile is not as simple as in the above example, it is often not clear who should yield which object to whom. This problem was solved by Katta and Sethuraman (2006), who presented a way of determining assignment matrices by means of a sequence of graphs, thereby defining the “serial correspondence.” Due to the difficulties generated by indifference, their proposal is not single-valued. However, each assignment matrix selected by this correspondence is sd-equivalent in welfare to each other. Thus, this solution is what is sometimes called “essentially single-valued’. They proved that it satisfies sd-efficiency and sd no-envy. However, so far, there has been no characterization of it nor of any of its subcorrespondences.3 Our main contribution is to present a characterization of a family of the extended subcorrespondences up to Pareto indifference. 2

Heo (2011a) imposes one additional requirement called “consistency”. There are also two other characterizations of the rule that do not involve any invariance-type axiom (Hashimoto and Hirata (2011) and Kesten et al. (2011)). 3 On a restricted domain of preference profiles, however, Heo (2011b) shows that for each economy, the set of assignment matrices selected by the extended serial correspondence coincides with the set of matrices satisfying sd-efficiency and sd no-envy. This result is interesting, in particular, because both axioms are “punctual” (they apply to each economy separately).

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We first extend this solution, which is designed for the case in which there is only one copy of each object, to be applicable to situations in which there may be multiple copies of each object. Let us call this extension the “extended serial correspondence”. We refer to its subcorrespondences as the extended serial subcorrespondences. We then characterize a family of the extended serial subcorrespondences (up to Pareto indifference) by means of sd-efficiency, sd no-envy, and “bounded invariance,” a natural adaptation of the axiom of the same name in Bogomolnaia and Heo (2011) to the domain of weak preferences. On the strict preference domain, bounded invariance is defined as follows. Keeping each other agent’s preference fixed, suppose that the preference of an agent, say agent i, changes but his weak upper contour set at an object, say a, remains the same and so does his ranking over this set. Then, it requires that the probability of each agent receiving a should remain the same. On the weak preference domain, we instead consider the set of objects that agent i finds indifferent to a. Under the same hypothesis as above, the total probability of each agent receiving these objects should remain the same. The key to these results is to introduce a profile of tie-breaking rules among indifferent objects. We then work with the strict preference profile induced from each weak profile by the profile of tie-breaking rules. Our proof is obtained by using an alternative representation of assignment matrices, called “preference-decreasing consumption schedule”. After obtaining the result, we came across an independent characterization of the extended serial correspondence by Yilmaz (2011). He was the first to obtain this result. He also adapted a result due to Kesten et al. (2011) to the weak preference domain. We independently obtained the the same result. We provide a direct proof by exploiting a technique developed in Heo (2011a), which is also used in Bogomolnaia and Heo (2011). This paper is organized as follows. Section 2 introduces model and properties. Section 3 presents our main result. Section 4 contains two concluding remarks.

2. Model Let A ≡ {o1 , · · · , o|A|−1 , ∅} be a set of objects and N ≡ {1, 2, · · · , n} a set of agents. There may be multiple copies of each object; for each a ∈ A, let qa ∈ N+ be the number of copies of a. In this list, o1 , · · · , o|A|−1 are “real” objects and ∅ is the “null object”, that is, ∅ represents not receiving any real object. Let q∅ = |N |. 3

For each i ∈ N , Ri represents agent i’s weak preferences over A. For each i ∈ N and each pair a, b ∈ A, we write that a Ii b if and only if a Ri b and b Ri a. Define a strict preference over A such that for each a, b ∈ A, a Pi b if and only if a Ri b and ¬(a Ii b). Let R be the set of weak preferences over A and P be the set of strict preferences over A. For each i ∈ N , each Ri ∈ R and each S ⊆ A, let M (Ri , S) be the set of objects in S that agent i most prefers. For each R ∈ RN , each i ∈ N , and each a ∈ A, let U (Ri , a) ≡ {o ∈ A : o Ri a} be the weak upper contour set at a of Ri . For each a ∈ A, let I(Ri , a) ≡ {o ∈ A : a Ii o} be the indifference class of a at Ri . An economy is defined as a list (A, q, N, R). We fix A, q and N . An economy is then represented by R. Let R be the set of all economies. An assignment matrix is an |N |×|A| matrix π ≡ (πia )i∈N,a∈A , where πia is the probability of agent i receiving a, such P that (i) for each i ∈ N and each a ∈ A, πia ∈ [0, 1], (ii) for each a ∈ A, i∈N πia ≤ qa , P and (iii) for each i ∈ N , a∈A πia = 1. We say that π ∈ Π is a deterministic assignment matrix if for each i ∈ N and each a ∈ A, πia ∈ {0, 1}. By the Birkhoff-von Neumann theorem (Birkhoff (1946), von Neumann (1953)), each assignment matrix can be represented as a convex combination of deterministic assignment matrices. Let Π be the set of all assignment matrices at e. Let 2Π be the collection of subsets of assignment matrices. Let φ be the empty set. A rule is a mapping ϕ : R → Π. A correspondence is a mapping Φ : R → 2Π \ {φ}. A subcorrespondence Ψ of Φ is a correspondence such that for each R ∈ RN , Ψ(R) ⊆ Φ(R). Let R ∈ RN . Let π, π 0 ∈ Π. For each i ∈ N , we say that πi weakly stochastically P P 0 dominates πi0 at Ri , if for each a ∈ A, b∈U (Ri ,a) πib . We write b∈U (Ri ,a) πib ≥ that πi Riwsd πi0 . If there is at least one strict inequality, we write that πi Risd πi0 . Throughout the paper, the notation “sd” stands for “stochastic dominance.” For each pair of correspondences Φ and Ψ, we say that Φ and Ψ are equivalent in welfare if for each R ∈ RN , each π ∈ Φ(R), each π 0 ∈ Ψ(R), and each i ∈ N , πi Riwsd πi0 and w

πi0 Riwsd πi . We write this as Φ = Ψ. The following are properties of assignment matrices based on this relation. First is an efficiency property: there should be no other assignment matrix at which some agent is “sd better off” and no agent is sd worse off. Formally, π is sd-efficient at e if there is no π 0 ∈ Π such that for each i ∈ N , πi0 Riwsd πi and for some i ∈ N , πi0 Risd πi . The corresponding property of a correspondence Φ is as follows: Sd-efficiency: for each R ∈ RN and each π ∈ Φ(R), π is sd-efficient at e. 4

The following is a fairness property. Each agent should find his assignment at least as “sd-desirable” as that of each other agent. Formally, π is sd envy-free at e if for each pair i, j ∈ N , πi Riwsd πj . The corresponding property of a correspondence Φ is as follows: Sd-no-envy: for each R ∈ RN and each π ∈ Φ(R), π is sd envy-free at e. The last requirement is a natural adaptation of “bounded invariance” in Bogomolnaia and Heo (2011) to the domain of weak preferences. Let a ∈ A. Suppose that, keeping each other agent’s preference fixed, the preference of an agent, say agent i, changes but his upper contour set at an object, say a, remains the same and so does his ranking over this set. Consider an assignment matrix selected for the resulting profile. Consider the set of objects that agent i finds indifferent to a. We require that the total probability of each agent receiving these objects should remain the same.4 For each pair R0 , R00 ∈ R and each a ∈ A, we write Ri (a) = Ri0 (a) if U (Ri , a) = U (Ri0 , a) and Ri |U (Ri ,a) = Ri0 |U (Ri ,a) . Bounded invariance: For each R ∈ RN , each i ∈ N , each a ∈ A, and each Ri0 ∈ R, if Ri (a) = Ri0 (a), then for each π ∈ Φ(R), there is π 0 ∈ Φ(Ri0 , R−i ) such that for each 0 j ∈ N and each o ∈ I(Ri , a), πjo = πjo .

2.1. Consumption Schedule and the serial rule In this section, we describe an alternative representation of assignments, which is the key to our proofs. Let P ∈ P N and π ∈ Π. For each i ∈ N , we represent πi as a process of consuming objects at unit speed over the time interval [0, 1]. The process is in decreasing order of Pi . First, consider agent i’s most preferred object, say a. Imagine that he consumes a from time 0 to time πia . Second, consider his second most preferred object, say b. Imagine that he consumes b from time πia to time (πia +πib ), and so on. The list of probabilities that agent i consumes over [0, 1] is exactly (πia )a∈A . Thus, we obtain an alternative representation of πi at Pi as a list of breakpoints, (0, πia , πia + πib , · · · , 1). Note that the time at which agent i stops consuming an object, say o, is the sum of the probabilities of his receiving objects that he finds at least as desirable as o. Once an 4

This requirement is an implication of an axiom formulated for the probabilistic voting model (Gibbard, 1977). In our context, this can be rephrased as follows: keeping each other agent’ preference fixed, whenever the preference of an agent, say agent i, changes but his weak upper contour set at some object, say a, remains the same, the total probability of each agent receiving the objects that agent i finds at least as desirable as a should remain the same. It is easy to check that this axiom implies bounded invariance.

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agent switches from one object to another, he never returns to the former object. Formally, for each i ∈ N and each k ∈ {1, · · · , |A|}, denote by oki the k-th most preferred object of agent i with preference Pi .5 For each i ∈ N , we define the consumpP P tion schedule representing πi at Pi as t(πi , Pi ) ≡ (πio1i , 2t=1 πioti , 3t=1 πioti , · · · , P|A|−1 P|A| P|A| t=1 πioti , t=1 πioti ). Note that t=1 πioti = 1. Let t(π, P ) ≡ (t(πi , Pi ))i∈N be the profile of consumption schedules representing P at R . Let S ⊆ A and τ ∈]0, 1]. Given t(π, P ), we say that S reaches exhaustion at τ if τ is the earliest time at which the supply of all objects in S reaches exhaustion.6 We now define the serial rule, S. Let P ∈ P N . The probability supply of each a ∈ A is qa . The serial assignment matrix is selected by the following procedure: The serial rule, S: Given a preference profile, at time 0, each agent starts consuming his most preferred object in A. Consumption rates are equal across agents (we normalize this common rate to be 1). When the supply of his most preferred object reaches exhaustion, each agent switches to his most preferred object among those that are still available and then starts consuming it. When the supply of this object reaches exhaustion, he switches to his most preferred object among those that are still available and the starts consuming it, and so on, until time 1. Each agent’s assignment is defined as the list of probabilities that he has consumed. Let P ∈ P N and π ∈ Π. From the definition of the serial rule, we obtain that π = S(P ) if and only if t(π, P ) is such that each agent switches from one object, say o, to another object to which he prefers o, only when the supply of o reaches exhaustion.

2.2. The extended serial correspondence Katta and Sethuraman (2006) propose an adaptation of the serial rule to profiles of (possibly) weak preferences. As readers familiar with the difficulties generated by indifference might expect, their proposal is not single-valued. However, we should note that for each economy, each agent is indifferent between any two of the assignment matrices selected by the correspondence. Thus, this solution is “essentially single-valued”. The original formulation of Katta and Sethuraman (2011) was for the case in which there is only one copy of each object. We extend this correspondence to be applicable to situ5 6

Formally, it should be ok (Pi ), but for simplicity, we omit Pi otherwise specified. Equivalently, the supply of the last object(s) in S reaches exhaustion at τ .

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ations in which there may be multiple copies of each object. Let us call this extension the extended serial correspondence (ES). Let R ∈ RN . Consider a graph with nodes consisting of a source node α, a sink node β, agents in N , and objects in A. Each pair of nodes may be connected by a directed arc, and each arc has a capacity (for each pair of nodes, say k and l, let c(kl) be the capacity of the arc connecting them). The maximum flow is the largest amount of the resource that can be sent from α to β, respecting the capacity constraints. The set of assignment matrices selected by the extended serial correspondence is given by an iterative computation of maximum flows in a graph constructed in a particular way, as follows. Step 1. Let λ1 ∈ [0, 1]. Let A1 ≡ A. Construct (i) the arc from α to each agent i and set its capacity to be c1 (αi) ≡ λ1 , (ii) the arc from each agent i to each object o ∈ M (Ri , A1 ) and set its capacity to be c1 (io) ≡ ∞, and (iii) the arc from each object o to β and set its capacity to be c1 (oβ) ≡ qo . Increase λ1 gradually from 0 to 1 and calculate the maximum flow from α to β associated with λ1 . Let λ∗1 be the smallest number such that the maximum flow at the graph associated with λ∗1 is the largest. At the graph associated with λ∗1 , let X1 ⊆ {o ∈ A1 : the flow from o to β is qo } and S N (X1 ) ⊆ {i ∈ N : M (Ri , A1 ) ⊆ X1 } be such that (i) i∈N (X1 ) M (Ri , A1 ) = X1 and (ii) for each i ∈ N (X1 ), there are a ∈ M (Ri , A1 ) and j ∈ N (X1 ) \ {i} with a ∈ M (Rj , A1 ).7 Each i ∈ N (X1 ) is assigned a total probability c1 (αi) of receiving objects in M (Ri , A1 ). For each i ∈ N , let c1 (αi) ≡ λ∗1 . In general, Step t. Let λt ∈ [0, 1 −

Pt−1

s=1

λ∗s ]. Let At ≡ At−1 \ Xt−1 . Construct (i) the arc from α

to each agent i ∈ N (Xt−1 ) and set its capacity to be ct (αi) ≡ λt , (ii) the arc from α to each agent i ∈ / N (Xt−1 ) and set its capacity to be ct (αi) ≡ ct−1 (αi) + λt , (iii) the arc from each agent i to each object o ∈ M (Ri , At ) and set its capacity to be ct (io) ≡ ∞, and (iv) the arc from each object o ∈ At to β and set its capacity to be ct (oβ) = qo . P ∗ Increase λt gradually from 0 to (1 − t−1 s=1 λs ) and calculate the maximum flow from α to β associated with λt . Let λ∗t be the smallest number such that the maximum flow at the graph associated with λ∗t is the largest. At the graph associated with λ∗t , let Xt ⊆ {o ∈ At : the flow from o to β is qo } and N (Xt ) ⊆ {i ∈ N : M (Ri , At ) ⊆ Xt } If there are more than one such subset of objects, say X1 and X10 , with no common object, then let X1 be handled in Step 1 and X10 be handled in the next step. 7

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be such that (i)

S

i∈N (Xt )

M (Ri , At ) = Xt and (ii) for each i ∈ N (Xt ), there are a ∈

M (Ri , At ) and j ∈ N (Xt ) \ {i} with a ∈ M (Rj , At ).8 Each i ∈ N (Xt ) is assigned a total probability ct (αi), of receiving objects in M (Ri , At ). For each i ∈ N (Xt−1 ), let / N (Xt−1 ), let ct (αi) ≡ ct−1 (αi) + λ∗t . ct (αi) = λ∗t and for each i ∈ We say that at each Step t, the supply of the objects in Xt reaches exhaustion, or simply, Xt reaches exhaustion. This algorithm terminates in at most |A| steps. Remark 1. Let R ∈ RN . For each π, π 0 ∈ ES(R), each i ∈ N , each t ∈ {1, · · · , |A|}, P P and each a ∈ Xt , o∈A:o Ii a πi = o∈A:o Ii a πi0 .9 Let us call a subcorrespondence of the extended serial correspondence an extended serial subcorrespondence (ES subcorrespondence).

3. Main result We first present an important structural property of assignment matrices selected by the extended serial correspondence. Let R ∈ RN and π ∈ Π. For each i ∈ N , we introduce a way to break ties among indifferent objects. Let σi be a bijection from A to {1, · · · , |A|}. Suppose that agent i finds a, b ∈ A indifferent. Then, he break ties by ranking a higher than b if σi (a) < σi (b). Let Σ be the set of all bijections. Given σi ∈ Σ and Ri ∈ R, we induce a strict preference P σi (Ri ) ∈ P as follows: for each pair a, b ∈ A, a P σi (Ri ) b if and only if either (i) a Pi b or (ii) a Ii b and σi (a) < σi (b). Let P σ (R) ≡ (P σi (Ri ))i∈N . Given π ∈ Π and P σ (R), we define the profile of consumption schedules t(π, P σ (R)) as in Section 2.1. Given a profile of consumption schedules, we say that for each S ⊆ A, an agent stops consuming from S at τ ∈ [0, 1] if τ = maxa∈S {τ 0 ∈ [0, 1] : the agent switches from a to another object at τ 0 }. If there are a ∈ A and τ ∈ [0, 1[ such that agent i stops consuming from I(Ri , a) at τ , by which I(Ri , a) does not reach exhaustion, we say that he does not do his best at τ .10 Similarly, we say that an agent does his best during [0, τ [ if there is no t ∈ [0, τ [ at which he does not do his best. Let R ∈ RN . The following lemma says that an assignment matrix is selected by ES if and only if for each profile of tie-breaking rules, σ ∈ ΣN , the profile of consumption 8

The only exception is the last step. This implies that π and π 0 are sd-equivalent in welfare. 10 We adopt this expression from Bogomolnaia and Heo (2011). However, we switch from an object to an indifference class in the definition. 9

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schedules representing π at P σ (R) is such that each agent switches from one object, say a, to another only if the supply of objects that he finds indifferent to a is exhausted. Lemma 1. Let R ∈ RN and π ∈ Π. For each σ ∈ ΣN , each agent does his best during [0, 1[ at t(π, P σ (R)) if and only if π ∈ ES(R). Proof. (if): it directly comes from the definition of ES. (only if): let R ∈ RN and π ∈ Π be such that for each σ ∈ ΣN , each agent does his best during [0, 1[ at t(π, P σ (R)). Suppose, by contradiction, that π ∈ / ES(R). Let π ¯ ∈ ES(R). Then, there are s ∈ {1, · · · , |A|}, i ∈ N , and a ∈ A so that i ∈ N (Xs ), P P a ∈ M (Ri , As ), and o∈U (Ri ,a) πio < o∈U (Ri ,a) π ¯io . Choose (i∗ , a∗ , s∗ ) among them so P that o∈U (Ri ,a) πio is the smallest at (i∗ , a∗ ). If there are more than one such triple with the same smallest sum of probabilities, then choose a triple such that s∗ is the largest, namely, a∗ reaches exhaustion at the last step among them in the ES algorithm for R P (denote this statement by (†)).11 Let τ ∗ ≡ o∈U (Ri∗ ,a∗ ) πi∗ o . Let s¯ be the largest integer P in {1, · · · , s∗ } such that for each n ∈ N (Xs¯) and each x ∈ M (Rn , As¯), o∈U (Rn ,x) πno = P ¯no .12 We claim that at least one of the following statements holds: o∈U (Rn ,x) π (i) there are a pair i, j ∈ N (Xs∗ ) and a ∈ M (Ri , As∗ ) ∩ M (Rj , As∗ ) such that P P o∈U (Rj ,a) πjo and πja > 0. o∈U (Ri ,a) πio < (ii) there are k ∈ / N (Xs∗ ), i ∈ N (Xs∗ ), and a ∈ M (Ri , As∗ ) such that πka > 0 and P P o∈aU (Ri ,a) πio . o∈U (Rk ,a) πko > P (iii) there are i ∈ N (Xs∗ ) and a ∈ M (Ri , As∗ ) such that n∈N πna < qa . If at least one statement holds, then let σ ∈ ΣN be such that for each n ∈ N , σn (a) = |A|. Then, agent i stops consuming from U (Ri , a) before a reaches exhaustion at t(π, P σ (R)), completing the proof of Lemma 1. Suppose that none of statements (i) to (iii) holds. There are two possibilities. 11

If there are still more than one such pair, choose any pair among them. Thus, for s + 1,P s¯ + 2, · · · , s∗ }, each nP∈ N (Xs ), and each x ∈ M (Rn , As ), P each s ∈ {¯ (i) if o∈U (Rn ,x) πno < o∈U (Rn ,x) π ¯no , then o∈U (Rn ,x) πno = τ ∗ and, P P P (ii) if o∈U (Rn ,x) πno > o∈U (Rn ,x) π ¯no , then o∈U (Rn ,x) π ¯no ≥ τ ∗ . ∗ ∗ ∗ (i) directly comes from the way we chose (i , a , s ). Suppose that (ii) does not hold. Then, it is easy s +P1, s¯ + 2, · · · , s∗ }, n ∈ N (Xs ), and x ∈ (Rn , As ), such that P to check thatPthere are s ∈ {¯ ¯no and o∈U (Rn ,x) πno < τ ∗ . This contradicts that τ ∗ is chosen to be o∈U (Rn ,x) πno < o∈U (Rn ,x) π the smallest such number. 12

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P Case 1. For each pair i, j ∈ N (Xs∗ ) and each b ∈ M (Ri , As∗ )∩M (Rj , As∗ ), o∈U (Ri ,b) πio = P ∗ o∈U (Rj ,b) πjo (= τ ). There are two subcases. First, for each k ∈ N \ N (Xs∗ ) and each P P c ∈ Xs∗ , πkc = 0. That is, some object in Xs∗ is not exhausted: n∈N o∈Xs∗ πno < P o∈Xs∗ qo , which corresponds to (iii). Second, there is k ∈ N \ N (Xs∗ ) and c ∈ Xs∗ such that πkc > 0. Let γ be the smallest positive integer such that k ∈ N (Xs∗ +γ ). P Since k ∈ N \ N (Xs∗ ), for each d ∈ M (Rk , As∗ +γ ), d Rk c and thus, o∈U (Rk ,d) πko ≤ P P P ∗ ¯i∗ o ≤ o∈U (Rk ,d) π ¯ko . o∈U (Rk ,c) πko . Since a ∈ Xs∗ and d ∈ Xs∗ +γ , we have o∈U (Ri∗ ,a∗ ) π P We also have τ ∗ ≥ o∈U (Rk ,c) πko : otherwise, it corresponds to the statement (ii) which does not hold. Altogether, P P P P ∗ ¯ i∗ o . o∈U (Rk ,d) πko ≤ o∈U (Rk ,c) πko ≤ τ = o∈U (Ri∗ ,a∗ ) πi∗ o < o∈U (Ri∗ ,a∗ ) π P P P P ¯i∗ o , a contra¯ko < ¯ko ≤ If o∈U (Ri∗ ,a∗ ) π o∈U (Rk ,d) π o∈U (Rk ,d) πko , then o∈U (Rk ,d) π P P∗ P Ps∗ +γ ¯i∗ o = sν=1 λν . If diction to the fact that o∈U (Rk ,d) π ¯ko = ν=1 λν ≥ o∈U (Ri∗ ,a∗ ) π P P P ¯ko , then since τ ∗ is the smallest, o∈U (Rk ,d) πko = τ ∗ . Howo∈U (Rk ,d) π o∈U (Rk ,d) πko < ever, this contradicts (†): (k, d, s∗ + γ) should have been chosen instead of (i∗ , a∗ , s∗ ). Case 2. There are a pair i, j ∈ N (Xs∗ ) and b ∈ M (Ri , As∗ ) ∩ M (Rj , As∗ ) such that P P π < io o∈U (Rj ,b) πjo . Since (i) does not hold, for each such b, πjb = 0 (denote o∈U (Ri ,b) this statement by (§)). Note that i∗ is chosen such that for each j ∈ N (Xs∗ ) and each P P a ∈ M (Ri∗ , As∗ ) ∩ M (Rj , As∗ ), o∈U (Ri∗ ,a) πi∗ o (= τ ∗ ) ≤ o∈U (Rj ,a) πjo . Let ∗ ¯ ≡ {j ∈ N (Xs∗ ) : for each a ∈ M (Rj , As∗ ), P N o∈U (Rj ,a) πjo = τ }, ∗ ¯ ≡ {j ∈ N (Xs∗ ) : for each a ∈ M (Rj , As∗ ), P N (Xs∗ ) \ N o∈U (Rj ,a) πjo < τ }, ¯ and each x ∈ X, ¯ ¯ ≡ S ¯ M (Ri , As∗ ). By (§), for each k ∈ N (Xs∗ ) \ N and X i∈N

πkx = 0. Recall that s¯ and s∗ are chosen as follows: (1) for each t ∈ {1, · · · , s¯}, each P P n ∈ N (Xt ), and each x ∈ M (Rn , At ), ¯no , (2) for each o∈U (Rn ,x) πno = o∈U (Rn ,x) π P ∗ ∗ t ∈ {¯ s + 1, · · · , s }, each n ∈ N (Xt ), and each x ∈ M (Rn , At ), τ ≤ o∈U (Rn ,x) π ¯no . ¯ ¯ However, at π, we have N ( N (Xs∗ ) and X ( Xs∗ such that ¯ , M (Ri , As∗ ) ⊆ X ¯ and S ¯ M (Ri , As∗ ) = X, ¯ (a) for each i ∈ N i∈N ¯ and each a ∈ M (Ri , As∗ ) ∩ M (Rj , As∗ ), (b) for each pair i, j ∈ N P

o∈U (Ri ,a)

πio =

P

o∈U (Rj ,a)

πjo < τ ∗ , and

¯ and each b ∈ X, ¯ πkb = 0. (c) For each k ∈ N \ N ¯ ( Xs∗ should reach This contradicts the way Step s∗ of the ES algorithm is defined: X exhaustion at a step earlier than Step s∗ of the ES algorithm. 10

Remark 2. Let π, π 0 ∈ Π and R ∈ RN . Suppose that for each σ ∈ ΣN , each agent does his best during [0, 1[ at t(π, P σ (R)) and t(π 0 , P σ (R)). By Lemma 1, π and π 0 are welfareP P 0 equivalent (that is, for each i ∈ N and each a ∈ A, o∈I(Ri ,a) πio = o∈I(Ri ,a) πio ). Remark 3. A “local” property of the profile of consumption schedules representing each ES assignment matrix follows from Lemma 1. Let R ∈ RN , a ∈ A, and τ ∈ [0, 1]. Let π ∈ Π be such that for each σ ∈ ΣN , each agent does his best during [0, τ ] at t(π, P σ (R)). Then, the following statements are equivalent: (1) for each σ ∈ ΣN , a reaches exhaustion no later than τ at t(π, P σ (R)). (2) there is π ¯ ∈ ES(R) such that for each σ ∈ ΣN , a reaches exhaustion no later than τ at t(¯ π , P σ (R)).13 We next present a characterization of a family of the ES subcorrespondences (up to Pareto indifference) by means of sd efficiency, sd no-envy, and bounded invariance. We emphasize that ES is not the only correspondence that satisfies these properties, as shown in the following example. Example 1. An example of a single-valued correspondence satisfying sd efficiency, sd no-envy, and bounded invariance Consider the set of pairs consisting of an agent and an object, N × A ≡ {(i, o) : i ∈ N and o ∈ A}. Let χ be a strict ordering defined over N ×A. Let Φχ be a subcorrespondence of the ES correspondence that selects, for each economy, an assignment matrix that lexicographically maximizes the probability assigned to each agent receiving each object in order of χ. That is, for each R ∈ RN , if (i, a) is the first pair at χ, we choose Π0 ≡ 0 }. If |Π0 | > 1, then we consider the second {π ∈ ES(R) : for each π 0 ∈ ES(R), πia ≥ πia 0 pair at χ, say (j, b). Then, we choose Π1 ≡ {π ∈ Π0 : for each π 0 ∈ Π0 , πjb ≥ πjb }. We

continue until we obtain a single assignment matrix. It is easy to check that this rule satisfies the three properties. Next is our main result. Theorem 1. Let Φ be a correspondence. w

(i) If Φ ⊆ ES, then there is a correspondence Ψ = Φ that satisfies sd efficiency, sd no-envy, and bounded invariance. (ii) If Φ satisfies sd efficiency, sd no-envy, and bounded invariance, then Φ ⊆ ES. 13

Equivalently, a reaches exhaustion at Step s of the ES algorithm for R such that

11

Ps

t=1

λt ≤ τ .

Proof. (i) First, ES satisfies sd efficiency and sd no-envy (Katta and Sethuraman, 2006). For each R ∈ RN and each π ∈ ES(R), π is sd efficient and sd envy-free at R, each ES subcorrespondence Φ satisfies the two properties. Next, we let Ψ ≡ ES and show that ES satisfies bounded invariance. Let R ∈ RN , i ∈ N , and a ∈ A. Let Ri0 ∈ R be such that Ri (a) = Ri0 (a) and let R0 ≡ (Ri0 , R−i ). Let s ∈ {1, · · · , |A|} be the step of the ES algorithm, at which a reaches exhaustion. It is easy to check that from Step 1 to Step s, the ES algorithm works in the same way for R and R0 . Remark 1 completes the proof. (ii) Let Φ be a correspondence satisfying sd efficiency and bounded invariance. Suppose that Φ * ES. Then, there are R ∈ RN and π ∈ Φ(R) such that π ∈ / ES(R). By Lemma 1, there is σ ≡ (σi )i∈N ∈ ΣN such that t(π, P σ (R)) is as follows: there are a ∈ A, τ ∈ [0, 1[, i ∈ N , and x ∈ I(Ri , a) such that agent i stops consuming from I(Ri , a) at τ , by which x does not reach exhaustion. Let R ∈ RN and σ ∗ ∈ ΣN be such that the associated τ is the smallest. Denote this smallest τ by τ ∗ . By sd efficiency, there is ¯ ≡ {k ∈ N : πkx > 0} j ∈ N \ {i} who is assigned a positive probability of x.14 Let N S ¯ πis = 0. Now, let and S¯ ≡ k∈N {o ∈ A : x Pi o Rk x}. By sd efficiency, for each s ∈ S, Ri0 ∈ R be such that Ri (x) = Ri0 (x), and all objects in S¯ are indifferent for agent i and are placed just below x. Let R0 ≡ (Ri0 , R−i ). By bounded invariance, there is π 0 ∈ Φ(R0 ) P 0 0 = τ ∗ ). such that for each k ∈ N and each b ∈ U (Ri , x), πko = πko (thus, o∈U (R0 ,c)∪S¯ πio i 0 ¯ πio By sd efficiency, for each o ∈ S, = 0. Next, we claim that there is σ ¯ ∈ ΣN such that I(Ri0 , x) does not reach exhaustion by τ ∗ at t(π 0 , P σ¯ (R0 )). Suppose otherwise. For each σ ∈ ΣN , t(π 0 , P σ (R0 )) is such that I(Ri0 , x) reaches exhaustion no later than τ ∗ . Recall that τ ∗ is chosen so that for each R00 ∈ RN , each σ 00 ∈ ΣN and each π 00 ∈ Φ(R), each agent does his best during 00

[0, τ ∗ [ at t(π 00 , P σ (R00 )) (denote this statement by (‡)). Thus, for each σ ∈ ΣN , each agent does his best during [0, τ ∗ [ at t(π 0 , P σ (R0 )). By Remark 3, there is π ¯ 0 ∈ ES(R0 ) such that for each σ ∈ ΣN , I(Ri0 , x) reaches exhaustion no later than τ ∗ at t(¯ π 0 , P σ (R0 )). Let Step s be the step of the ES algorithm for R0 , at which I(Ri0 , x) reaches exhaustion. As shown in (i), Step 1 to Step s of the ES algorithm are the same for R and R0 . Thus, I(Ri , x)(= I(Ri0 , x)) also reaches exhaustion at Step s and for each π ¯ ∈ ES(R) and each σ ∈ ΣN , I(Ri , x) reaches exhaustion no later than τ ∗ at t(¯ π , P σ (R)). By (‡) and ∗

That is, I(Ri , a) reaches exhaustion after τ ∗ at t(π, P σ (R)). If there is no such agent j, then there is δ > 0 such that by increasing by δ the probability of agent i receiving x, decreasing by δ the total probability of his receiving the objects to which he prefers x, and keeping each other agent’s assignment the same, we make agent i sd-better off without sd-hurting any other agent. 14

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Remark 3, we obtain that for each σ ∈ ΣN , I(Ri , x) reaches exhaustion no later than τ ∗ at t(π, P σ (R)), a contradiction. By sd efficiency, at t(π 0 , P σ¯ (R0 )), there is an agent l ∈ N who consumes a positive P P 0 0 . < o∈U (Rl ,c) πlo probability of some object c ∈ I(Ri0 , x) after τ ∗ . Thus, o∈U (R0 ,c)∪S¯ πio i Since U (R0 , x) ∪ S¯ ⊇ U (R0 , c), we obtain a violation of sd no-envy. i

l

4. Concluding Remarks We establish that the axioms listed in Theorem 1 are independent. In each case, we indicate which axiom is violated. • Sd efficiency : consider the rule that for each economy, assigns probability 1 of receiving ∅ to each agent. • Sd no-envy : select a strict ordering over the agents. Consider the rule that maximizes agents’ welfare lexicographically in that order. • Bounded invariance: let N = {1, 2, · · · , |N |}, A ≡ {a, · · · , z}, q = (1, 1, · · · , 1), and R be such that (i) a R1 b R1 c R1 ∅, a R2 c R2 b R2 ∅, b R3 c R3 a R3 ∅, and (ii) each other agent prefers ∅ to each other object. Let π be such that π1 = (1/2, 1/6, 1/3, 0, · · · , 0), π2 = (1/2, 0, 1/2, 0, · · · , 0), π3 = (0, 5/6, 1/6, 0, · · · , 0), and each other agent is assigned ∅ with probability 1. Let Φ be a correspondence such that Φ(R) = {π} and for each R0 6= R, Φ(R0 ) = ES(R0 ). The characterization of the serial rule in Bogomolnaia and Heo (2011) is implied by Theorem 1. This is because (i) strict preference is included in our preference domain, (ii) for each strict preference profile, the extended serial correspondence is single-valued and coincides with the serial rule, and (iii) indifference among objects is not invoked in the proof of Theorem 1.

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[3] Bogomolnaia, A., and Moulin, H., “New Solution to the Random Assignment Problem,” Journal of Economic Theory, 100 (2001), 295-328. [4] Gibbard. A., “Straightforwardness of Game Forms with Lotteries as Outcomes,” Econometrica 46 (1978), 595-614. [5] Hashimoto, T., and Hirata, D., “Characterizations of the Probabilistic Serial Mechanism,” (2011), mimeo [6] Heo, E.-J., “Random Assignment Problem with Multiple Demands: A Generalization and a Characterization of the Serial Rule,” (2011a), mimeo [7] Heo, E.-J., “Consumption Schedules and the Serial Rule: on the Full Preference Domain,” (2011b), mimeo [8] Katta, A.-K., and Sethuraman, J., “A Solution to the Random Assignment Problem on the Full Preference Domain,” Journal of Economic Theory, 131 (2006), 231-250. ¨ [9] Kesten, O., Kurino, M., and Unver, U., “Fair and Efficient Assignment via the Probabilistic Serial Mechanism,” (2011), mimeo [10] Thomson, W., “Consistency for the Probabilistic Assignment of Objects,” (2010a), mimeo [11] Thomson, W., “Strategy-proof Allocation Rules,” (2010b), mimeo. [12] Von Neumann, J., “A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem,” Contributions to the Theory of Games Vol.2 (1953) Princeton University Press, Princeton, New Jersey. [13] Yilmaz, O., “Characterizations of the Extended Probabilistic Serial Rule,” (2011), mimeo.

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