The Extended Serial Correspondence on the Rich Preference Domain Eun Jeong Heo

∗ †

First draft: October 24, 2011 This version: April 8, 2013

Abstract We study the problem of assigning objects to a set of agents. We focus on probabilistic solutions that only take agents’ preferences over objects as input. Importantly, agents may be indifferent among several objects. The “extended serial correspondence” is proposed by Katta and Sethuraman (2006) to solve this problem. As a follow-up of Liu and Pycia (2012) that introduce the notion of profiles with “full support”, we work with two interesting classes of preference profiles: profiles that (i) have rich support on a partition or (ii) are single-peaked with rich support on a partition. For each profile in these classes, an assignment matrix is selected by the extended serial correspondence if and only if it is sd-efficient and sd envy-free. We also provide an asymptotic result. JEL classification: C70, D61, D63. Keywords: sd-efficiency; sd no-envy; rich support on a partition; single-peaked preference profiles with rich support on a partition; the extended serial correspondence.

∗

Department of Economics, University of Rochester, Rochester NY 14627, USA ([email protected]) I am grateful to William Thomson for his support and encouragement. I also benefited from useful comments from Anna Bogomolnaia, Fuhito Kojima, Qingmin Liu, Vikram Manjunath, Marek Pycia, and seminar audience at Columbia University in October 2011. †

1. Introduction We study the problem of assigning indivisible goods — we call them “objects” — to a set of agents. Importantly, agents may be indifferent among several objects, that is, agents may have “weak preferences” over the objects. Each agent is to receive a “lottery,” which specifies for each object the probability that he will receive this object. An assignment matrix is a collection of all agents’ lotteries. The problem is how to select an assignment matrix (or a set of assignment matrices). A “solution” is a systematic way of doing so. We focus on probabilistic solutions that only take agents’ preferences over objects as input. We work with two interesting classes of preference profiles for this problem: profiles that (i) have “rich support on a partition” or (ii) are “single-peaked with rich support on a partition”. In both, roughly speaking, there is a tier-structure over the objects and within each tier, the preferences are diversified across agents. Our main result is that, for each profile in these classes, the assignment matrices selected by the “extended serial” correspondence, proposed by Katta and Sethuraman (2006) are the only ones that satisfy the requirements of efficiency and fairness defined below. Bogomolnaia and Moulin (2001) assume that each agent has strict preferences over objects and compares lotteries on the basis of first-order stochastic dominance as follows; suppose that this agent is given two lotteries, π0 and π00 ; we say that “π0 weakly first-order stochastically dominates π00 at his preference,” if (1) the probability of his receiving his most preferred object at π0 is at least as large as the corresponding probability at π00 , (2) the sum of the probabilities of his receiving either his most preferred object or his second most preferred object at π0 is at least as large as the corresponding sum at π00 , (3) and so on. Two requirements based on such comparisons have been central in the literature: “stochastic dominance efficiency (sd-efficiency)”1 and “sd no-envy”. Sd-efficiency requires that there should be no other assignment matrix that makes some agent “sd better off” without making someone else “sd worse off”. Sd no-envy requires that each agent should find his assignment at least as “sd-desirable” as that of each other agent. If each matrix selected by a solution satisfies sd-efficiency (or sd no-envy), we say that the solution satisfies sd-efficiency (or sd no-envy). The “serial rule” has been extensively studied in the literature as a solution satisfying these two requirements (Bogomolnaia and Moulin, 2001). This rule is designed for preference profiles that exhibit no indifferences: for each problem, each object is interpreted as a divisible good with (probability) supply of 1; each agent simultaneously “consumes” the probability 1

Throughout the paper, the prefix “sd” stands for stochastic dominance. In Bogomolnaia and Moulin (2001), this requirement is referred to as ordinal efficiency. We adopt the terminology and notation of Thomson (2010a, 2010b).

1

supplies of objects over time, one object at each moment, in the decreasing-order of his preferences; the assignment matrix selected by the serial rule is the collection of what agents have consumed. As an adaptation of the serial rule for the preferences exhibiting indifference, Katta and Sethuraman (2006) propose the “extended serial (ES) correspondence” by using max-flow graph. We call an assignment matrix an “ES matrix” if it is selected by the extended serial correspondence. The notions of sd-efficiency and sd no-envy in Bogomolnaia and Moulin (2001) are also adapted to this domain. Although the extended serial correspondence satisfies sd-efficiency and sd no-envy, it is not the only one to do so. When an additional requirement of “limited invariance” is imposed, however, the extended serial correspondence remains as the unique admissible one (Heo and Yılmaz, 2012). This result is an extension of a characterization of the serial rule that Hashimoto et al. (2013) provides for strict preferences. Other characterizations of the serial rule by means of different sets of requirements are also available (Bogomolnaia and Heo (2012), Hashimoto et al. (2013), Hashimoto and Hirata (2011), Heo (2011), and Kesten et al. (2011)).2 Instead of working on the unrestricted domain of preference profiles as in the previous literature, we introduce two classes of preference profiles. Both classes exhibit a certain correlation of preferences among agents that Kandori et al. (2010) and Kesten (2010) addressed in the context of school choice: the set of objects are partitioned into ordered “tiers”, each object in a tier being preferred to each object in any lower ranked tier. Let A be a set of objects, k ∈ {1, · · · , |A|}, and A ≡ (A1 , · · · , Ak ) be a partition of A. Each agent prefers each object in A1 to each object in A2 , each object in A2 to each object in A3 , and so on. Profiles of this type are common in practice. Consider preferences over schools or colleges: schools are often partitioned in this way; everyone agrees on the first-tier schools and all students prefer them to the others; next come the second-tier schools and all students prefer them to the others except for the first-tier schools, and so on.3 Within each tier, however, students may have different preferences. We consider two specifications of preference profiles in each tier. First are “profiles with rich support on a partition”: for each partition A and each component of A, all strict prefer2 Hashimoto and Hirata (2011), Heo (2011), Kesten et al. (2011) are the first papers characterizing the serial rule. Hashimoto et al. (2013) is evolved from Hashimoto and Hirata (2011) and Kesten et al. (2011), sharpening the result of Bogomolnaia and Heo (2012). Heo (2011) generalizes the model to accommodate the possibility that agents receive more than one object. 3 If there is no such tier-structure within the set of schools, then the partition has to be composed of single component, A. If preferences are perfectly correlated, then the partition has to be such that the cardinality of each component is one. Such a tier structure among colleges is often publicly published as well. For example, see U.S. News and World Report at http://www.usnewsuniversitydirectory.com/undergraduate-colleges/nationaluniversities.aspx.

2

ences over the objects in the component are represented in the profile. That is, within each tier, there are “fully diversified” strict preferences. This condition is demanding for economies with a small number of agents, but it is rather lenient when the number of agents is large. To present our second class of profiles, we need a preliminary concept: a profile is “singlepeaked over a set of objects”, say B ⊆ A, if there is a linear order over B such that for each agent, (i) there is an object that he most prefers in B — we call this object his “peak” in B — and (ii) on each side of his peak according to the linear order, the further an object is from his peak, the lower he ranks that object. Preferences of this type have been widely studied in social choice and resource allocation problem (Black (1958), Moulin (1980), Sprumont (1991), and Thomson (2010b)). For the problems considered in this paper, Kasajima (2012) is the first to explore the implication of preferences being single-peaked. Our second specification is profiles that are“single-peaked with rich support on a partition”: for each partition A and each component of A, that is, each tier, (i) the profile is single-peaked over the objects in the component, and (ii) “most” of the single-peaked preferences over the objects in the component are represented in the profile. That is, within each tier, preferences are single-peaked and “well diversified”.4 For a profile as just defined, we study the implication of sd-efficiency and sd no-envy. Our main result is that for each preference profile in either of these classes, an assignment matrix is an ES matrix if and only if it is sd-efficient and sd envy-free. This result is of particular interest because both of these requirements are “punctual”, namely, they apply to each preference profile separately. By contrast, the existing characterizations of the serial rule, or of the extended serial correspondence, all involve at least one “relational” requirement. A key ingredient to our result is a structural property of the ES assignment matrices (Heo and Yılmaz, 2012). For each agent, we introduce a tie-breaker for equally desirable objects and invoke it to induce a strict preference relation from each weak preference. Based on the resulting strict preferences, we represent an assignment matrix by means of a profile of “preference-decreasing consumption schedules”.5 This representation allows us to obtain a straightforward proof. This paper is inspired by Liu and Pycia (2012) which work with strict preferences. Their main result is that all sequences of rules asymptotically coincide6 if they are sd-efficient, symmetric, and “sd strategy-proof,” all properties being defined in an asymptotic sense. This coincidence is based on an important result for finite economies: for each profile with “full 4

For the formal definition of these conditions, see Section 3. This representation is introduced by Heo (2011) and it is also used in Bogomolnaia and Heo (2012). 6 Formally, given any two sequences of rules, say (φν )ν∈N+ and (ϕν )ν∈N+ , the assignment matrix selected by φν converges to the one selected by ϕν as ν → ∞. Liu and Pycia (2012) impose a mild assumption of “regularity” on these sequences. 5

3

support” where all preferences are represented at the profile, the serial assignment matrix is the unique one that is sd-efficient and sd envy-free. We also consider finite economies, but we allow for indifferences among objects, working with the extended serial correspondence. One of the classes of preference profiles we define includes those in Liu and Pycia (2012). The other includes profiles that generalize the singlepeaked preference profiles. The case of strict preference is a special case of ours. We also provide asymptotic results along the lines of Liu and Pycia (2012). This paper is organized as follows. Section 2 introduces the model and the extended serial correspondence. Section 3 presents two domains of preference profiles. Section 4 is devoted to our main result. Section 5 presents our asymptotic results.

2. Model Let A ≡ {o1 , · · · , om } be a set of objects. There may be several copies of each object: for each a ∈ A, let qa ∈ N+ be the number of copies of a ∈ A. We interpret this number as the “probability supply” of the object.7 Let q ≡ (qo )o∈A . Let N ≡ {1, 2, · · · , n} be a set of agents. Each agent i ∈ N has a weak preference Ri over A. For each B ⊆ A, we denote the restriction of Ri to B by Ri |B . Let RB be the set of all weak preferences over B. Let R ≡ (Ri )i∈N ∈ RN A . For each pair a, b ∈ A, we write a Pi b if and only if a Ri b and ¬(b Ri a). For each i ∈ N and each a ∈ A, let U (Ri , a) ≡ {o ∈ A : o Ri a} be the weak upper contour set of a at Ri , U 0 (Ri , a) ≡ {o ∈ A : o Pi a} be the strict upper contour set of a at Ri , and I(Ri , a) ≡ {o ∈ A : o Ri a and a Ri o} be the set of objects that are as equally desirable as a at Ri . For each i ∈ N and each B ⊆ A, let M (Ri , B) be the set S of objects that agent i most prefers among B. For each T ⊆ N , M (RT , B) ≡ i∈T M (Ri , B). For each B ⊆ A, we denote the restriction of Pi to B by Pi |B . Let PB be the set of all strict preferences over B. For each P0 ∈ PA and each pair B, B 0 ⊆ A, we write B P0 B 0 if and only if for each b ∈ B and each b0 ∈ B 0 , b P0 b0 . An economy is a list (A, q, N, R). Unless otherwise specified, we fix A, q, and N and simply write an economy as a preference profile R. 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 that (i) for P each i ∈ N and each a ∈ A, πia ∈ [0, 1], (ii) for each a ∈ A, i∈N πia ≤ qa , and (iii) for each P 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 7

The set of objects may include the “null object,” ∅, which represents not receiving any object in A \ {∅}. The number of copies of the null object is equal to the number of agents: q∅ = |N |.

4

of deterministic assignment matrices. Let Π be the set of all assignment matrices. A rule Π is a mapping ϕ : RN A → Π. Let 2 be the collection of subsets of assignment matrices. A Π correspondence is a mapping Φ : RN A → 2 \ ∅.

Let R ∈ RA and 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 ≥ b∈U (Ri ,a) πib . We write this as πi Riwsd πi0 . If there is at least one strict inequality, we say that πi stochastically dominates πi0 at Ri . We write this as πi Risd πi0 . Throughout the paper, the prefix “sd” stands for “stochastic dominance.” If πi Risd πi0 , we say that agent i with preference Ri is sd better off at πi than at πi0 . If ¬ (πi Riwsd πi0 ), we say that agent i with preference Ri is sd worse off at πi than at πi0 . Next are two key requirements based on sd comparisons. Let R ∈ RN A and π ∈ Π. First is an efficiency requirement: there should be no other assignment matrix that makes some agent sd better off without making someone else sd worse off. Sd-efficiency: π is sd-efficient at R 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 following is a fairness requirement, due to Foley (1967). Each agent should find his assignment at least as “sd-desirable” as that of each other agent. Sd-no-envy: π is sd envy-free at R if for each pair i, j ∈ N , πi Riwsd πj .

2.1. Consumption schedule and the serial rule In this section, we describe an alternative representation of assignments, introduced by Heo (2011). Consider a strict preference profile P ∈ PAN and an assignment matrix 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. Thus, we obtain 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 agent switches from one object to another, he never returns to the former object. The formal definition is as follows. For each strict preference P ∈ PAN , each i ∈ N , each a ∈ A, and each k ∈ {1, · · · , |A|}, let oki be agent i’s k-th most preferred object at Pi .8 For each i ∈ N , the consumption schedule representing πi at Pi is defined as the list 8

Formally, it should be ok (Pi ), but for simplicity, we omit Pi , unless otherwise specified.

5

t(πi , Pi ) ≡ (πio1 , i

P2

t=1 πioti ,

P3

t=1 πioti , · · ·

,

P|A|−1 t=1

πioti ,

P|A|

t=1 πioti ).

Note that

P|A|

t=1 πioti

= 1.

Let t(π, P ) ≡ (t(πi , Pi ))i∈N be the profile of consumption schedules representing the components of π at P . For each P ∈ PAN and each π ∈ Π, t(π, P ) is uniquely defined. Given a consumption schedule t(π, P ), we say that for each B ⊆ A, (i) an agent stops consuming from B at τ ∈ [0, 1] if τ = maxa∈B {τ 0 ∈ [0, 1] : the agent switches from a to another object at τ 0 }9 and (ii) B reaches exhaustion at τ ∈ [0, 1] if τ = min{τ 0 ∈ [0, 1] : the remaining probability of each object in B is zero at τ 0 }. We now define the serial rule, S. Let P ∈ PAN . The serial assignment matrix at P is determined by the following procedure. The serial rule, S: The probability supply of each object is the number of copies of the object. 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 is exhausted, he switches to his most preferred object among those that are still available and starts consuming it. When the supply of this object is exhausted, he switches to his most preferred object among those that are still available and starts consuming it, and so on, until time 1. His assignment is defined as the list of probabilities with which he has consumed various objects by time 1. Note that an agent switches from one object to another only when the supply of the former is exhausted. Let P ∈ PAN 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 is exhausted.

2.2. The extended serial correspondence We next introduce the extended serial (ES) correspondence. 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, say k and l, may be connected by a directed arc denoted by kl. The arc has a capacity denoted by c(kl). The maximal 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 maximal flows in the graph constructed as follows. Let A0 = A, X0 = ∅, λ∗0 = 0, and for each i ∈ N , s0 (αi) = 0. For each m ∈ N+ , 9

Equivalently, the supply of the last object(s) in B reaches exhaustion at τ .

6

Step m. Let λm ∈ [0, 1 −

Pm−1 s=1

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

to each agent i ∈ N and set its capacity equal to cm (αi) ≡ sm−1 (αi) + λm , (ii) the arc from each agent i to each object o ∈ M (Ri , Am ) and set its capacity equal to cm (io) = ∞, and the arc from each object o to β and set its capacity equal to cm (oβ) ≡ qo . Let P P q − o o∈M (R ,A ) i∈T sm−1 (αi) m T T ∗ ∈ argmin |T | T ⊆N P

and call such

T∗

a “bottleneck set” at Step m. Let

λ∗m

≡

P

o∈M (RT ,Am ) qo − i∈T ∗ |T ∗ |

sm−1 (αi)

.

Let Nm ⊆ N be a bottleneck set at Step m with the smallest cardinality and let Xm ≡ P i∈Nm M (Ri , Am ). For each i ∈ Nm , let sm (αi) = 0 and assign the probability (sm−1 (αi) + λ∗m ) to objects in M (Ri , Am ). For each i ∈ / Nm , let sm (αi) ≡ sm−1 (αi)+λ∗m , without assigning any probability at Step m. We say that at Step m, the supply of each object in Xm reaches exhaustion, or simply, that Xm reaches exhaustion. The algorithm terminates at Step m ¯ if either Am ¯ = ∅ or Pm ¯ N ∗ ¯ ≤ |A|. For each R ∈ RA , we say that an assignment matrix is an s=1 λs ≥ 1. Note that m ES matrix for R if it is selected by the ES correspondence for R. Example 1. The extended serial correspondence Let A ≡ {a, b, c, d}, N ≡ {1, 2, 3, 4}, and q = (1, 1, 1, 1). Let R ∈ RN A be such that R1

R2

a, b

a

R3

R4

a, b a, b, c

c, d b, c

d

d

d

c

We start with A0 = A, X0 = ∅, λ∗0 = 0, and for each i ∈ N , s0 (αi) = 0. Step 1. Let λ1 ∈ [0, 1]. Construct a set of arcs, ∪i∈N {αi} ∪ {1a, 1b, 2a, 3a, 3b, 4a, 4b, 4c} ∪ ∪o∈A {oβ} such that for each i ∈ N , c1 (αi) = λ1 , for each o ∈ A, c1 (oβ) = 1, and the capacities of the remaining arcs equal to ∞. It is easy to check that λ∗1 = 23 , N1 = {1, 2, 3}, and X1 = {a, b}. Step 2. Let λ2 ∈ [0, 31 ]. Construct a set of arcs, ∪i∈N {αi} ∪ {1c, 1d, 2c, 3d, 4c} ∪ {cβ, dβ} such that c2 (α1) = c2 (α2) = c2 (α3) = λ2 , c2 (α4) =

2 3

+ λ2 , c2 (cβ) = c2 (dβ) = 1, and the

capacities of the remaining arcs equal to ∞. It is easy to check that λ∗2 = 16 , N2 = {2, 4}, and X2 = {c}. Step 3. Let λ3 ∈ [0, 16 ]. Construct a set of arcs, ∪i∈N {αi} ∪ {1d, 2d, 3d, 4d} ∪ {dβ} such that c3 (α1) = c3 (α3) =

1 6

+ λ3 , c3 (α2) = c3 (α4) = λ3 , c2 (dβ) = 1, and and the capacities 7

of the remaining arcs equal to ∞. It is easy to check that λ∗3 =

1 6,

N3 = {1, 2, 3, 4}, and

X3 = {d}. 

2 3 1 3

   Then, we have   0  0

0

0

1 3 2 3

1 6

0

0

5 6

1 3 1 6 1 3 1 6

     ,  

     

1 3 1 3 1 3

1 3 1 3 1 3

0

0

0

5 6

1 6

0

1 3 1 6 1 3 1 6

     ∈ ES(R).  



Proposition KS. (Katta and Sethuraman, 2006) For each R ∈ RN A , each π ∈ ES(R) is sd-efficient and sd envy-free at R. Note that for each R ∈ RN A , the ES matrices are welfare-equivalent to each other, namely, P P 0 for each pair π, π ∈ ES(R), each i ∈ N , and each a ∈ A, o∈I(Ri ,a) πi = o∈I(Ri ,a) πi0 . We now induce a strict preference from a weak preference by using a “tie-breaker”. Let Σ be the set of all bijections from A to {1, · · · , |A|}. Let R ∈ RN A , i ∈ N , and σi ∈ Σ. Let P σi (Ri ) ∈ PA be the strict preference induced from Ri such that 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 . Let π ∈ Π. Given P σ (R), we define the consumption schedule t(π, P σ (R)) as in Section 2.1. Next is an important structural property of ES matrices (Heo and Yılmaz, 2012). Proposition HY. (Heo and Yılmaz, 2012) Let R ∈ RN A , π ∈ Π, τ ∈ [0, 1], and a ∈ A. The following statements are equivalent: (i) π ∈ ES(R). (ii) For each σ ∈ ΣN , t(π, P σ (R)) is such that each agent stops consuming I(Ri , a) at τ only when τ = 1 or I(Ri , a) reaches exhaustion at τ .

3. Two Classes of Preference Profiles Let k ∈ {1, · · · , |A|}. We say that a list A ≡ (A1 , A2 , · · · , Ak ) is a partition of A if each S component of the list is non-empty, kl=1 Al = A, and for each pair l, m ∈ {1, · · · , k} with l 6= m, if any, Al ∩ Am = ∅. Let A¯ be the set of all partitions of A.

3.1. Profiles with rich support on a partition Let R ∈ RN A . We say that R has rich support on a partition, if for some k ∈ {1, · · · , |A|}, there is A ≡ (A1 , · · · , Ak ) ∈ A¯ such that (i) for each i ∈ N , A1 Pi A2 Pi · · · Pi Ak and (ii) for each t ∈ {1, · · · , k} and each P0 ∈ PAt , there is i ∈ N such that Ri |At = P0 . 8

Note that on the weak preference domain, there are many more preferences than on strict preference domain. However, (ii) states that in each component of the partition, all strict preferences over the objects in the component are represented. Note also that if an agent is indifferent between several objects, then these objects belong to the same component of the partition. This class includes the profiles in Liu and Pycia (2012), which they call “profiles with full support”. Those are special case of ours in which k = 1 (namely, A ≡ (A)) for the cases where preferences exhibit no indifference. For a profile to have full support in that case, there has ¯ to be at least |A|! agents. For a profile to have rich support on some A ≡ (A1 , · · · , Ak ) ∈ A, there has to be at least (argmaxt∈{1,··· ,k} |At |)! agents.

3.2. Single-peaked profiles with rich support on a partition Let B ⊆ A. We say that a preference profile is “single-peaked” over B if (i) each agent has a most preferred object in B — which we call his “peak” in B — and (ii) there is a linear order over B such that on each side of his peak according to the order, the further an object is from his peak, the lower he ranks that object. The formal definition is as follows. Consider a linear order ≺B over B. A preference profile R is single-peaked over B with respect to ≺B if (i) for each i ∈ N , |M (Ri , B)| = 1 and (ii) for each i ∈ N and each pair a, b ∈ B, if either a ≺B b ≺B M (Ri , B) or M (Ri , B) ≺B b ≺B a, then b Pi a. Let ≺ be a linear order over A. For each B ⊆ A, let ≺B be the order ≺ restricted to B.10 Now, we define our second class of preference profiles. Let R ∈ RN A . We say that R is single-peaked with rich support on a partition if for some k ∈ {1, · · · , |A|}, there are A ≡ (A1 , · · · , Ak ) ∈ A¯ and a linear order ≺ over A such that (i) for each i ∈ N , A1 Pi A2 Pi · · · Pi Ak , for each t ∈ {1, · · · , k}, (ii) R is single-peaked over At with respect to ≺At , (iii) for each a ∈ At , there is i ∈ N such that {a} = M (Ri , At ), and (iv) for each triple a, b, c ∈ At with either a ≺ b ≺ c or c ≺ b ≺ a, there is i ∈ N such that {b} = M (Ri , At ) and |U 0 (Ri , c)| − |U 0 (Ri , a)| = 1. 10 Katta and Sethuraman (2006) show that on the weak preference domain, no solution satisfies sd-efficiency, sd no-envy, “weak sd strategy-proofness”, which requires that, given any preference profile of the other agents, no agent should ever be sd better off by misreporting his preferences. Even if we restrict our attention to the single-peaked preference domain, this impossibility still holds. To see this, consider the following example. Let A ≡ {a, b, c, d}, N ≡ {1, 2, 3, 4}, q = (1, 1, 1, 1), and R ∈ RN A be such that R1 : b (a, c) d, R2 = R3 : b c a d, and R4 : b c d a, where the equally desirable objects are represented in the parenthesis. The profile is single-peaked with respect to a ≺ b ≺ c ≺ d. Let π ∈ Π be an sd-efficient and sd envy-free matrix. It is easy to check that 7 π1 = ( 59 , 14 , 0, 36 ). Suppose that the preference of agent 1 changes to R10 : b c a d. The resulting assignment 0 1 1 1 1 π1 has to be ( 3 , 4 , 4 , 6 ), which stochastically dominates π1 at R1 , in violation of weak sd strategy-proofness. I thank the referee for pointing out the strategic issue on this domain.

9

(iii) and (iv) are the conditions for “rich support” for single-peaked preferences. (iii) states that for each object in each component, there should be at least one agent whose peak is this object. (iv) states that for each triple of objects in each component of A that are linearly ordered according to ≺, say a ≺ b ≺ c, there should be an agent whose peak in this component is b and he ranks a just above c. When R satisfies (i) to (iv), we say that the profile is singlepeaked with respect to ≺ with rich support on A. Example 2. Single-peaked profiles with rich support on a partition |A|

Let A ≡ {a, b, c, d, e, f, g}, N ≡ {1, 2, · · · , 14}, and q ∈ N+ . Let A ≡ ({a, b, c, d}, {e, f, g}) and ≺

: a ≺ b ≺ c ≺ d ≺ e ≺ f ≺ g. The following (incomplete) profile R ∈ RN A is

single-peaked with respect to ≺ with rich support on A, no matter how preferences in the dotted area are completed:  R1 R2 R3 R4 R5 R6 R7 R8 R9 , · · · , R14  a b b b c c c d · ·   b a c c b b d c · ·   c c a d a d b b · · R≡  d d d a d a a a · ·   e f f g · · · · · ·   f e g f · · · · · · g g e e · · · · · ·

           

Let A0 ≡ ({a, b, c, d, e}, {f, g}). The following (incomplete) profile R0 ∈ RN A is single-peaked with respect to ≺ with rich support on A0 , no matter how preferences in the dotted area are completed:       0 R ≡     

0 0 0 0 0 R10 R20 R30 R40 R50 R60 R70 R80 R90 R10 R11 R12 R13 R14 a b b b b c c c c d d d d e b a c c c a b, d b d a c c e d c c a d d d a d b e b e c c d d d a e e e a, e e b e b b b e e e e a b a c a a a a f g · · · · · · · · · · · · g f · · · · · · · · · · · ·

            

4. Main Results Here is our main result. Theorem 1. Let R ∈ RN A be a profile that either (i) has rich support on a partition or (ii) is single-peaked with rich support on a partition. An assignment matrix is an ES matrix for R if and only if it is sd-efficient and sd envy-free at R. 10

Proof. By Proposition KS, it suffices to prove “if” part of the theorem. Let π be an assignment matrix satisfying the two requirements. Suppose, by contradiction, that π ∈ / ES(R). By Proposition HY, there are σ ≡ (σi )i∈N ∈ ΣN , τ¯ ∈ [0, 1[, a ∈ A, and i ∈ N such that under t(π, P σ (R)), agent i stops consuming from I(Ri , a) at τ¯, although some object in I(Ri , a) has not reached exhaustion at τ¯. Let a ¯ ∈ I(Ri , a) be an object that is not exhausted at τ¯. Then, there is j ∈ N \ {i} who consumes a positive probability of a ¯ after τ¯. Otherwise, it is easy to check that we can make agent i sd better off without making anyone else sd worse off, in P violation of sd-efficiency. Note that πj¯a > 0 and o∈U (¯a,Pj ) πjo > τ¯. By sd-efficiency, for each o0 ∈ {o ∈ A : a ¯ Pi o Rj a ¯}, πio0 = 0. There are two cases. ¯ Let m ∈ Case 1: R is a profile that has rich support on some A ≡ (A1 , · · · , Al ) ∈ A. {1, · · · , l} be such that a ¯ ∈ Am . By sd no-envy, for each pair n, n0 ∈ N and each t ∈ P P {1, · · · , l}, o∈At πno = o∈At πn0 o . Since R has rich support on A, there is k ∈ N such that U (Rk , a ¯) = U (Ri , a ¯) and at Rk , the objects in {o ∈ A : a ¯ Pi o Rj a ¯} are contiguous and are ranked just below a ¯. By sd-efficiency, for each o0 ∈ {o ∈ A : a ¯ P i o Rj a ¯}, πko0 = 0. P P P By sd no-envy, o∈U (Pk ,¯a) πko = o∈U (Pi ,¯a) πio (= τ¯). Altogether, o∈U (Pk ,¯a)∪U (Pj ,¯a) πko = τ¯. P However, o∈U (Pj ,¯a) πjo > τ¯, in violation of sd no-envy. ¯ Case 2: R is single-peaked profile with rich support on some A ≡ (A1 , · · · , Al ) ∈ A. Let m ∈ {1, · · · , l} be such that a ¯ ∈ Am . By sd no-envy, for each pair n, n0 ∈ N and each P P t ∈ {1, · · · , l}, ¯. Otherwise, o∈At πno = o∈At πn0 o . We first claim that M (Rj , Am ) 6= a P P a > o∈U (Ri ,¯ a) πio , in violation of sd no-envy. Let ≺ be a linear order o∈U (Ri ,¯ a) πjo ≥ πj¯ over A such that R is single-peaked over Am with respect to ≺Am . Let t ∈ {1, · · · , l} be the integer such that a ¯ is the t-th object in the linear order ≺Am . There are two subcases (Figure 1). Subcase 2.1: t = |Am | or t = 1. Suppose that t = |Am |. Since R is single-peaked with rich support on a partition, there is k ∈ N such that {¯ a} = M (Rk , Am ). By sd-efficiency, for each o ∈ U (Rj , a ¯) \ {¯ a}, πko = 0. Note that at Rk , the objects in U (Rj , a ¯) \ {¯ a} are contiguous and are ranked just below a ¯. Altogether, P P P πk¯a = ¯ = o∈U (Ri ,¯a) πio , o∈U (Rj ,¯ a) πko = o∈U (Rj ,¯ a) πjo > τ where the second equality comes from sd no-envy and the strict inequality from the way in which we chose agents i and j. Since {¯ a} ⊆ U (Ri , a ¯), this is in violation of sd no-envy. A symmetric argument applies to the case t = 1. Subcase 2.2: 1 < t < |Am |. Let b ∈ U (Rj , a ¯) be the object such that a ¯ and b are the two endpoints of U (Rj , a ¯) with respect to ≺Am .11 Next, let c ∈ Am \ U (Rj , Am ) be the object 11 Formally, b is the object such that (i) if M (Rj , Am ) ≺Am a ¯, then for each o ∈ U (Rj , a ¯) \ {b}, b ≺Am o, and (ii) if a ¯ ≺Am M (Rj , Am ), then for each o ∈ U (Rj , a ¯) \ {b}, o ≺Am b.

11

Subcase 2.1

Two possibilities of Subcase 2.2 • Rk

• • •

•

•

•

Rk •

Rk •

• Rj •

•

• Ri • • • • • p p •p p p p •p • • a ¯ • | {z } Am

•

•

•

• Rj •

•

• •

• Ri • • • • p p •p p p p •p • • b a ¯ c • | {z } Am •

•

• •

•

• Rj •

• • • • • Ri •

• p p •p p p • • b a ¯ • | {z Am

p •p c }

Figure 1. Case 2 in the proof of Theorem 1

that is next to a ¯ in the order ≺Am . Consider the triple, b, a ¯, and c. From the way in which we chose b and c, either b ≺ a ¯ ≺ c or c ≺ a ¯ ≺ b. Since R is single-peaked with rich support on a partition, there is k ∈ N such that {¯ a} = M (Rk , Am ) and |U 0 (Rk , c)| − |U 0 (Rk , b)| = 1. Note that at Rk , the objects in U (Rj , a ¯) \ {¯ a} are contiguous and are ranked just below a ¯. The same argument as in Subcase 2.1 applies and we obtain a violation of sd no-envy. Note that in defining the two classes, we do not invoke indifferences among objects. Thus, the definition of the two classes above directly carries over to the domain of strict preferences. A characterization of the serial assignment matrix for each economy follows from Theorem 1. Corollary 1. Let P ∈ PAN be a profile that either (i) has rich support on a partition or (ii) is single-peaked with rich support on a partition. The serial assignment matrix for P is the only one that is sd-efficient and sd envy-free at P .

5. Asymptotic Result Along the lines of Liu and Pycia (2012), we present asymptotic results concerning the ES correspondence. We maintain our notation of Section 2, except that (i) we return to the definition of an economy as a list e ≡ (A, q, N, R) and (ii) we let Π(e) be the set of assignment matrices at e. Let E be the set of economies. A correspondence is a mapping Φ : E → S Π(e) \ ∅) such that for each e ∈ E, Φ(e) ∈ 2Π(e) \ ∅. e∈E (2 ν ν ν ∞ ν Let (eν )∞ ν=1 ≡ (A, q , N , P )ν=1 be a sequence of economies such that |N | → ∞ as

ν → ∞. Let (π ν )∞ ν=1 be a sequence of assignment matrices such that for each ν ∈ N+ , π ν ∈ Π(eν ). 12

The following requirements are due to Liu and Pycia (2012). To define a notion of “asymptotic efficiency” for such a sequence of matrices, we first present an approximate notion of sd-efficiency. Consider an economy e ≡ (A, q, N, R) ∈ E and an assignment matrix π ∈ Π(e). Let  > 0. We say that π is -sd-efficient at e if there is no π 0 ∈ Π(e) such that (i) for each i ∈ N , πi0 Riwsd πi and (ii) there are i ∈ N and a ∈ A such that

0 o∈U (Pi ,a) πio

P

−

P

o∈U (Pi ,a) πio

> .

The number  can be interpreted as a “degree of inefficiency” (in the sd-sense) that is allowed: if  = ∞, the property is vacuous, and if  = 0, it coincides with sd-efficiency. Now, consider a sequence of economies and a sequence of matrices. We require that inefficiency should disappear in the limit: ν ν ν ∞ Asymptotic sd-efficiency: Let (eν )∞ ν=1 ≡ (A, q , N , P )ν=1 . A sequence of matrices ν ∞ ∞ (π ν )∞ ν=1 is asymptotically sd-efficient at (e )ν=1 if there is a sequence ((ν))ν=1 such that

for each ν ∈ N+ , (ν) > 0, π ν is (ν)-sd-efficient at eν , and limν→∞ (ν) = 0. Next is a fairness requirement in the limit. ν ν ν ∞ ν ∞ Asymptotic sd no-envy: Let (eν )∞ ν=1 ≡ (A, q , N , P )ν=1 . A sequence of matrices (π )ν=1

is asymptotically sd envy-free at (eν )∞ ν=1 if   P P ν ν ≥ 0. lim inf min o∈U (P ν ,a) πjo o∈U (P ν ,a) πio − ν ν∈N+

i

i

i,j∈N ,a∈A

ν ∞ Consider two sequences of matrices, (π ν )∞ π ν )∞ ν=1 and (¯ ν=1 . We say that (π )ν=1 con-

verges to (¯ π ν )∞ ν=1 if  lim

ν→∞

max

i∈N ν ,a∈A

|

P

ν o∈I(Riν ,a) (πio



−

ν )| π ¯io

= 0.

Now, we introduce two classes of sequences of economies corresponding to those of the ν ν ν ∞ previous section. Here is the first class. A sequence of economies (eν )∞ ν=1 ≡ (A, q , N , R )ν=1

has asymptotic rich support on a partition, if for some k ∈ {1, · · · , |A|}, there is A ≡ (A1 , · · · , Ak ) ∈ A¯ such that (i) for each ν ∈ N+ and each i ∈ N ν , A1 Piν A2 Piν · · · Piν Ak and (ii) for each t ∈ {1, · · · , k}, there is δ > 0 and ν¯ ∈ N+ such that for each ν > ν¯ and each Po ∈ PAt ,

|{i∈N ν :Riν |At =P0 }| ≥ δ. |N ν |

ν ν ν ∞ Next, a sequence of economies (eν )∞ ν=1 ≡ (A, q , N , R )ν=1 is single-peaked with

asymptotic rich support on a partition, if for some k ∈ {1, · · · , |A|}, there is A ≡ (A1 , · · · , Ak ) ∈ A¯ and a linear order ≺ over A such that 13

(i) for each ν ∈ N+ and each i ∈ N ν , A1 Piν A2 Piν · · · Piν Ak and for each t ∈ {1, · · · , k}, (ii) Rν is single-peaked over At with respect to ≺At , (iii) for each a ∈ At , there are δ > 0 and ν¯ ∈ N+ such that for each ν > ν¯, |{i∈N ν :M (Riν ,At )={a}}| ≥ δ, and |N ν |

(iv) for each triple a, b, c ∈ At with either a ≺ b ≺ c or c ≺ b ≺ a, there are δ 0 > 0 and ν¯0 ∈ N+ such that for each ν > ν¯0 , |{i∈N ν :M (Riν ,At )={b} and |U 0 (Riν ,c)|−|U 0 (Riν ,a)|=1}| ≥ δ0. |N ν | ν )∞ be a sequence of matrices such that for each ν ∈ N , π ν ∈ ES(eν ). Let (πes + es ν=1 ν ν ν ∞ Theorem 2. Let (eν )∞ ν=1 ≡ (A, q , N , R )ν=1 be a sequence of economies that either (i) has

asymptotic rich support on a partition or (ii) is single-peaked with asymptotic rich support on a partition. Then, each sequence of matrices (π ν )∞ ν=1 that is asymptotically sd-efficient ν ∞ and asymptotically sd envy-free at (eν )∞ ν=1 converges to (πes )ν=1 .

Proof. There are two possibilities. ν ∞ ¯ (i) (eν )∞ ν=1 has asymptotic rich support on some A ≡ (A1 , · · ·, Al ) ∈ A. Let (π )ν=1

be a sequence of matrices that is asymptotically sd-efficient and asymptotically sd envy-free ν ∞ ν ∞ at (eν )∞ ν=1 . We show that (π )ν=1 converges to (πes )ν=1 . By asymptotic sd-efficiency, there ν ν is a sequence ((ν))∞ ν=1 such that for each ν ∈ N+ , (ν) > 0, π is (ν)-sd-efficient at e ,

and limν→∞ (ν) = 0. Without loss of generality, suppose that (ν) is decreasing in ν. By asymptotic sd no-envy, there is an integer ν¯ ∈ N+ such that for each ν ≥ ν¯, each pair n, n0 ∈ N ν , and each b ∈ A, ν ν ,b) πno o∈U (Rn

P

−

P

ν ν ,b) πn0 o o∈U (Rn

> −(ν)

(∗)

ν

Now, let σ ∈ ΣN and consider t(π ν¯ , (P σ (Rν¯ )). Choose a pair of an agent and an object, say i and a, such that agent i stops consuming a at some time, say time τ¯, even though some object in I(Riν¯ , a) is not exhausted at τ¯ under t(π ν¯ , P σ (Rν¯ )).12 Let t ∈ {1, · · · , l} be an integer such that a ∈ At . Three cases can occur at t(π ν¯ , P σ (Rν¯ )). Case 1: 1 − τ¯ < (¯ ν ).13 Case 2: 1− τ¯ ≥ (¯ ν ) and no agent consumes a positive probability of I(Riν¯ , a) after time τ¯. We claim that what remains of the supply of I(Riν¯ , a) at τ¯ is less than (¯ ν ). Otherwise, by 12

If there is no such pair, (¯ ν ) = 0, and we are done. That is, we cannot make some agent sd better off without making someone else sd worse off, by increasing his consumption of I(Riν¯ , a) by (¯ ν ) and decreasing his consumption of the objects to which he prefers a by the same amount in total. 13

14

increasing agent i’s consumption of I(Riν¯ , a) by (¯ ν ) and decreasing his consumption of the other objects to which he prefers a by the same amount in total, we can make him sd better off without making anyone else sd worse off. This is a violation of (¯ ν )-sd-efficiency. Case 3: 1 − τ¯ ≥ (¯ ν ) and there is another agent, say agent j, who consumes a positive probability of I(Riν¯ , a) after time τ¯. Let a ¯ ∈ I(Riν¯ , a) be an object that agent j consumes after time τ¯. Let δ¯ be the probability that agent j consumed from a ¯ after τ¯. We claim that δ¯ < (¯ ν )(|A| + 1). Suppose, by contradiction, that δ¯ ≥ (¯ ν )(|A| + 1). Then, τ¯ + (¯ ν )(|A| + 1) ≤ P ν ¯ ν ¯ 0 ν ¯ 0 ν ¯ ν¯ < (¯ ν )-sd-efficiency of π , for each o ∈ {o ∈ A : a ¯ Pi o Rj a ¯}, πio ν ). o∈U (Rν¯ ,¯ a) πjo . By (¯ j

ν¯ ν¯ ¯) = U (Rν¯ , a Since (eν )∞ ν=1 has asymptotic rich support on A, there is k ∈ N such that U (Rk , a i ¯)

and at Rkν¯ , the objects in {o ∈ A : a ¯ Piν¯ o Rjν¯ a ¯} are contiguous and are ranked just below a ¯. St−1 ν ¯ ν ¯ ν ¯ ν Moreover, {o ∈ A : a ¯ Pi o Rj a a}. By (∗), ¯} ⊆ At and for each n ∈ N , λ=1 Aλ Pn {¯ P P ν¯ ν¯ ν )(= τ¯ + (¯ ν )). o∈U (Rν¯ ,¯ a) πio + (¯ o∈U (Rν¯ ,¯ a) πko < k

i

ν¯ < (¯ ν ). Since By (¯ ν )-sd-efficiency of π ν¯ , for each o ∈ {o0 ∈ A : a ¯ Piν¯ o0 Rjν¯ a ¯}, we have πko P ν ¯ ν ν ν )|A|. Since |{o ∈ A : a ¯ Pi o Rj a ¯}| ≤ |A| − 1, it follows that o∈U (Rν¯ ,¯a)∪U (Rν¯ ,¯a) πko < τ¯ + (¯ j k P ν¯ , we have τ¯ + (¯ ν )(|A| + 1) ≤ o∈U (Rν¯ ,¯a) πjo j P P P ν¯ ν¯ (≤ ν¯ ν ) < o∈U (Rν¯ ,¯a) πjo o∈U (Rν¯ ,¯ a)∪U (Rν¯ ,¯ a) πjo ), o∈U (Rν¯ ,¯ a)∪U (Rν¯ ,¯ a) πko + (¯ k

j

j

k

j

in violation of (∗). Thus, if there is any such pair i and a, then it corresponds to at least one of these three cases. Since (ν) is decreasing in ν and limν→∞ (ν) = 0, if agent i stops consuming object a that is not exhausted, either the time at which he stops consuming a converges to 1 (Case 1), or the remaining probability supply of objects in I(Ri , a) converges to 0 (Cases 2 and 3). By ν , P σ (Rν )) is such that for each o ∈ A, each agent j Proposition HY, for each ν ∈ N+ , t(πes

stops consuming I(Rjν , o) at τ only when τ = 1 or I(Rjν , o) reaches exhaustion at τ . Thus, ν ∞ (π ν )∞ ν=1 converges to (πes )ν=1 .

(ii) (eν )∞ ν=1 is single-peaked with asymptotic rich support on some A ≡ (A1 , · · ·, Al ) ¯ ∈ A. Let ≺ be the linear order over A such that for each ν ∈ N+ and each t ∈ {1, · · · , l}, Rν is single-peaked over At with respect to ≺At . Let (π ν )∞ ν=1 be a sequence of matrices that is asymptotically sd-efficient and asymptotically sd envy-free at (eν )∞ ν=1 . The same argument as in (i) applies except for Case 3: 1 − τ¯ ≥ (¯ ν ) and there is another agent, say agent j, who consumes a positive probability of I(Riν¯ , a) after time τ¯. Let a ¯ ∈ I(Riν¯ , a) be an object that agent j consumes after time τ¯. Let δ¯ be the probability that agent j consumed from a ¯ after τ¯. We claim that δ¯ < (¯ ν )(|A| + 1). Suppose, by contradiction, that δ¯ ≥ (¯ ν )(|A| + 1). Then, P ν¯ . Since (eν )∞ is single-peaked with asymptotic rich support τ¯ +(¯ ν )(|A|+1) ≤ o∈U (Rν¯ ,¯a) πjo ν=1 j

on A, there is k ∈ N ν¯ such that M (Rkν¯ , At ) = {¯ a} and at Rkν¯ , the objects in U (Rjν¯ , a ¯) \ {¯ a} are 15

P P ν¯ − (¯ ν¯ . contiguous and are ranked just below a ¯.14 By (∗), o∈U (Rν¯ ,¯a) πjo ν ) < o∈U (Rν¯ ,¯a) πko j j P ν¯ . By (¯ Thus, τ¯ + (¯ ν )|A| < o∈U (Rν¯ ,¯a) πko ν )-sd-efficiency of π ν¯ , for each o ∈ U (Rjν¯ , a ¯) \ {¯ a}, j P ν ¯ ν ¯ ν ¯ ν ¯ πko < (¯ ν ). Since |U (Rj , a ¯) \ {¯ a}| ≤ |A| − 1, we have o∈U (Rν¯ ,¯a) πko < πk¯a + (¯ ν )(|A| − 1). j P P ν¯ − (¯ ν¯ ≤ ν¯ ν¯ − (¯ Altogether, τ¯ < πk¯ ν ). Since πk¯ ¯ < o∈U (Rν¯ ,¯a) πko ν ), o∈U (Rν¯ ,¯ a) πko , we have τ a a i

i

in violation of (∗). We conclude the proof by the same argument as in (i).

References [1] Birkhoff G (1946) Three observations on linear algebra. Revi. Univ. Nac. Tucuman ser A 5: 147-151 [2] Black D (1958) The theory of committees and elections. Cambridge University Press, Cambridge [3] Bogomolnaia A, Heo EJ (2012) Probabilistic assignment of objects: characterizing the serial rule. J Econ Theory 147, 2072-2082. [4] Bogomolnaia A, Moulin H (2001) A new solution to the random assignment problem. J Econ Theory 100: 295-328 [5] Foley D (1967) Resource allocation and the public sector. Yale Econ Essays 7: 45-98 [6] Hashimoto, T., Hirata, D., 2011. Characterizations of the probabilistic serial mechanism. mimeo. ¨ [7] Hashimoto T, Hirata D, Kesten O, Kurino M, Unver U (2013) Two axiomatic approaches to the probabilistic serial mechanism. Theor Econ, forthcoming [8] Heo EJ (2012) Random assignment problem with multiple demands: a generalization of the serial rule and a characterization. Mimeo ¨ (2012) A characterization of the extended serial correspondence. [9] Heo EJ, Yılmaz O Mimeo [10] Kandori M, Kojima F, Yasuda Y (2010) Tiers, preference similarity, and the limits on stable partners. Mimeo [11] Katta AK, Sethuraman J (2006) A solution to the random assignment problem on the full preference domain. J Econ Theory 131: 231-250 14

For the detail, see Case 2 in the proof of Theorem 1.

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[12] Kasajima Y (2012) Probabilistic assignment of indivisible goods with single-peaked preferences. Soc Choice and Welfare, forthcoming ¨ [13] Kesten O, Kurino M, Unver U (2011) Fair and efficient assignment via the probabilistic serial mechanism. Mimeo [14] Kesten O (2010) School choice with consent. Q J Econ 125: 1297-1348 [15] Liu Q, Pycia M (2012) Ordinal efficiency, fairness, and incentives in large markets. Mimeo [16] Moulin H (1980) On strategy-proofness and single peakedness. Pub Choice 35: 437-455 [17] Sprumont Y (1991) The division problem with single-peaked preferences: a characterization of the uniform allocation rule. Econometrica 59: 509-519 [18] Thomson W (2010a) Consistency for the probabilistic assignment of objects. Mimeo [19] Thomson W (2010b) Strategy-proof allocation rules. Mimeo [20] Von Neumann J (1953) A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Kuhn HW, Tucker AW (ed) Contributions to the theory of games, vol. II, Princeton University Press, Princeton, pp 5-12

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