Discussion Paper No. 947

FAIR REALLOCATION IN ECONOMIES
WITH
SINGLE-PEAKED PREFERENCES

Kazuhiko Hashimoto
Takuma Wakayama

September 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Fair Reallocation in Economies
with Single-Peaked Preferences
Kazuhiko Hashimoto∗

Takuma Wakayama†

September 2, 2015

Abstract
We consider the problem of fairly reallocating the individual endowmentsof a perfectly divisible good among agents with single-peaked preferences. We
provide a new concept of fairness, called position-wise envy-freeness, that iscompatible with individual rationality. This new concept requires that eachdemander (i.e., agent whose most preferred amount is strictly greater thanhis endowment) should not envy another demander who does not receive his
endowment and that each supplier (i.e., agent whose most preferred amount isstrictly less than his endowment) should not envy another supplier who does
not receive his endowment. We establish that a rule is eﬃcient, individually
rational, strategy-proof, and position-wise envy-free if and only if it is the“gradual uniform rule,” which is an extension of the well-known uniform rule.
Keywords: Envy-freeness; Individual rationality; Uniform rule; Single-peaked
preferences; Strategy-proofness.
JEL codes: D71; D63.

∗

Institute of Social and Economic Research, Osaka University 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, JAPAN; k-hashimoto@iser.osaka-u.ac.jp†
Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto
612-8577, JAPAN; wakayama@econ.ryukoku.ac.jp1

1

Introduction

We consider the problem of fairly reallocating the individual endowments of a perfectly divisible good among agents with single-peaked preferences.1 This situation may occur particularly when the existing allocation is unsatisfactory owing tochanges in preferences over time.2
Envy-freeness is the most standard concept of fairness in the literature. It statesthat every agent prefers his assignment to everybody else’s. However, this concept
is quite demanding in our setting, because the set of individually rational (no agent
prefers his endowment to his own assignment) and envy-free allocations might beempty. This motivates us to search for fairness concepts that are compatible withindividual rationality.
We propose a new notion of fairness called position-wise envy-freeness. In our
setting, we categorize “traders” (i.e., agents whose most preferred amounts are notequal to their endowments) into two positions: an agent is a “demander” (“supplier”)
if his endowment is strictly less (greater) than his most preferred amount. The notionof position-wise envy-freeness then states that each demander (supplier) prefers his
own assignment to another demander’s (supplier’s) assignment that is not equal tohis endowment. That is, we weaken envy-freeness by only requiring each agent not toenvy another agent who is in the same position and does not receive his endowment.
The existence of individually rational and position-wise envy-free allocations, ofcourse, is guaranteed.
We next construct a position-wise envy-free and individually rational rule. We
call it the “gradual uniform rule.” This rule is an extension of the uniform rule (Benassy, 1982), which is the best-known rule in this problem. Furthermore, we establish that the gradual uniform rule is the only one satisfying eﬃciency (no Pareto
improvement is possible) and strategy-proofness (no one can gain by preference misrepresentation) in addition to individual rationality and position-wise envy-freeness.
There is another way to extend the notion of envy-freeness to the case of economies
with individual endowments, which is to define envy-freeness in terms of changes
in allocations rather than in terms of final allocations. This concept is called envyfreeness on net trade, which requires that no agent prefers another agent’s net trade1

This “reallocation problem,” first analyzed by Klaus et al. (1997, 1998a, 1998b), is a naturalextension of the problem of fairly allocating a social endowment of a perfectly divisible goodamong agents having single-peaked preferences (Sprumont, 1991). For comprehensive surveysof the literature on private good economies in which agents have single-peaked preferences, see
Klaus (1998) and Thomson (2014).
2
See Klaus et al. (1997, 1998a, 1998b) and Klaus (1998) for interpretations of this problem.

2

to his own.3 It is also compatible with individual rationality. Klaus et al. (1997,
1998a, 1998b) apply it to our setting and construct an envy-free on net trade rulecalled “uniform reallocation rule.” The uniform reallocation rule, which is anotherextension of the uniform rule, is the only one satisfying eﬃciency, strategy-proofness,and envy-freeness on net trade (Klaus et al., 1998b). However, the uniform reallocation rule is not position-wise envy-free. This suggests that whenever we insist
on fairness criteria for final allocations rather than for changes in allocations, thegradual uniform rule is more favorable than the uniform reallocation rule.
In Section 2, we set up the model and introduce basic properties of allocations
and rules. In Section 3, we spell out a new concept of fairness. In Section 4, we
introduce the three main rules and show the main result.

2

Preliminaries

Let N = {1, . . . , n} be the set of agents. There is one perfectly divisible good.
Each agent i ∈ N owns an individual endowment ei ∈ R+ of the good. Let e ≡∑(e1 , . . . , en ) ∈ Rn+ be the profile of individual endowments, and let E ≡ i∈N ei .
Each agent i ∈ N has a single-peaked preference relation Ri on [0, E]: there is apoint p(Ri ) ∈ [0, E] such that for each pair {xi , yi } ⊂ [0, E], if either yi < xi ≤ p(Ri )or p(Ri ) ≤ xi < yi , then xi Pi yi , where Pi is the asymmetric part of Ri . The pointp(Ri ) is called the peak of Ri . Let p(R) ≡ (p(R1 ), . . . , p(Rn )) be the profile ofpeaks. We denote the set of all single-peaked preferences defined on [0, E] by R.
An element R ≡ (R1 , . . . , Rn ) of R n is called a preference profile. Given R ∈ R n
and i ∈ N , let R−i ≡ (Rj )j̸=i . The set of feasible allocations is{X≡(x1 , . . . , xn ) ∈ Rn+ :∑}xi = E.

i∈N

A rule is a function f : R n → X assigning to each preference profile R ∈ R n
a feasible allocation f (R) ≡ (f1 (R), . . . , fn (R)) ∈ X, where fi (R) means agent i’s
assignment at R.
The following properties of allocations and rules are standard in the literature.
• Eﬃciency: An allocation x ∈ X is eﬃcient for R ∈ R n if there is no x′ ∈ Xsuch that for each i ∈ N , x′i Ri xi , and for some j ∈ N , x′j Pj xj . A rule f is3

The notion of envy-freeness in terms of changes in allocations is first formulated by Schmeidler
and Vind (1972) in the context of pure exchange economies.

3

eﬃcient if for each R ∈ R n , f (R) is eﬃcient for R.
• Individual rationality: An allocation x ∈ X is individually rational for
R ∈ R n if for each i ∈ N , xi Ri ei . A rule f is individually rational if for eachR ∈ R n , f (R) is individually rational for R.
• Strategy-proofness: For each R ∈ R n , each i ∈ N , and each Ri′ ∈ R,
fi (R) Ri fi (Ri′ , R−i ).
Eﬃciency requires that there be no other feasible allocation such that someone
can be made better oﬀ without anyone else being made worse oﬀ. Individual rationality requires that no agent prefers his individual endowment to his own assignment.
Strategy-proofness requires that no agent can ever benefit from misrepresenting his
preferences.

3

Position-Wise Envy-Freeness

The following property of fairness is central to the mechanism design literature.
• Envy-freeness: An allocation x ∈ X is envy-free for R ∈ R n if for each pair{i, j} ⊆ N , xi Ri xj . A rule f is envy-free if for each R ∈ R n , f (R) is envy-freefor R.
Envy-freeness, which is first introduced by Foley (1967), requires that no agent
prefers another agent’s assignment to his own. Unfortunately, in our setting, theset of individually rational and envy-free allocations might be empty. This observation motivates us to seek meaningful notions of fairness that are compatible withindividual rationality.
We now introduce a new concept of fairness that is compatible with individual
rationality. In doing so, we call an agent a trader if his endowment is not equal tohis peak. We now categorize traders into two “positions”: an agent is a demander(supplier) if his endowment is strictly less (greater) than his peak. Given R ∈ R n ,let N d (R) ≡ {i ∈ N : p(Ri ) > ei } and N s (R) ≡ {i ∈ N : p(Ri ) < ei } be the sets ofdemanders and suppliers at R, respectively.
• Position-wise envy-freeness: An allocation x ∈ X is position-wise envyfree for R ∈ R n if for each pair {i, j} ⊆ N , if xj ̸= ej and either {i, j} ⊆ N d (R)or {i, j} ⊆ N s (R), then xi Ri xj . A rule f is position-wise envy-free if for eachR ∈ R n , f (R) is position-wise envy-free for R.

4

Position-wise envy-freeness states that each demander (supplier) prefers his own
assignment to another demander’s (supplier’s) assignment that is not equal to his
individual endowment. Hence, we weaken the notion of envy-freeness by allowing
agents to envy (i) agents not in the same position and (ii) agents receiving theirendowments.
It should be noted that an individually rational and position-wise envy-free allocation always exists. A typical example of such an allocation is the profile ofindividual endowments.
Remark 1. Klaus et al. (1998b) consider the following notion of fairness that iscompatible with individual rationality:• Envy-freeness on net trade: For each R ∈ R n and each pair {i, j} ⊆ N ,fi (R) Ri max{ei + (fj (R) − ej ), 0}.
It requires that no agent prefers another agent’s net trade to his own. Although boththis notion and our notion incorporate individual endowments into envy-freeness,they are based on significantly diﬀerent concepts: the former is related to fairness
criteria for final allocations, and the latter, to fairness criteria for changes in allocations.4♢

4

Results

The following rule is the best-known one in the social endowment setting.5
Uniform rule, U : For each R ∈ R n and each i ∈ N ,Ui (R) =where λ ∈ R+ solves∑min{p(R ), λ}iifmax{p(R ), λ} ifi∑∑j∈N p(Rj ) ≥ E,j∈N p(Rj ) ≤ E,

j∈N Uj (R) = E.

4

As mentioned in Introduction, envy-freeness on net trade is based on the idea of Schmeidler
and Vind (1972).
5
The uniform rule is introduced by Benassy (1982). Sprumont (1991) first provides an axiomaticcharacterization of the uniform rule in the social endowment setting. Subsequently, there have been
many studies on characterizing the uniform rule in the social endowment setting. See, for example,
Ching (1994), Thomson (1994a, 1994b, 1995, 1997), Chun (2006), Serizawa (2006), Mizobuchi and
Serizawa (2006), Sakai and Wakayama (2012), and Wakayama (2015). See also Thomson (2014)
for a survey of the characterizations of the uniform rule. Sönmez (1994) and Barberà et al. (1997)
describe an algorithm that computes the allocation of the uniform rule.

5

The uniform rule is eﬃcient, strategy-proof, and envy-free. Moreover, it is the
only rule satisfying these three properties (Sprumont, 1991). However, this ruleviolates individual rationality.6 In this sense, the uniform rule is not suitable for use
in our setting.
The following rule extends the uniform rule to account for individual endowments.
Uniform reallocation rule, U r : For each R ∈ R n and each i ∈ N ,Uir (R) =where λ ∈ R+ solvesmin{p(R ), e + λ}
i

i

if

max{p(R ), e − λ} if
i
i

∑

∑
∑

j∈N p(Rj ) ≥ E,j∈N p(Rj ) ≤ E,rj∈N Uj (R) = E.

The uniform reallocation rule is the most studied in our setting.7 It is well known
that the uniform reallocation rule is the only one satisfying eﬃciency, strategyproofness, and envy-freeness on net trade (Klaus et al., 1998b). Furthermore, it isindividually rational. However, as we show below, this rule violates position-wise
envy-freeness.
Proposition 1. The uniform reallocation rule is not position-wise envy-free.
Proof. Without loss of generality, we assume n = 3. Let e = (1, 5, 10) and R ∈ R 3
be such that p(R) = (6, 15, 8). Then, U r (R) = (2, 6, 8). Hence, {1, 2} ⊆ N d (R) andU2r (R) > e2 , but U2r (R) = 6 P1 2 = U1r (R). This means that the uniform reallocation
rule is not position-wise envy-free.
We extend the uniform rule to satisfy both individual rationality and positionwise envy-freeness.8
Gradual uniform rule, G: For each R ∈ R n and each i ∈ N ,{
Gi (R) =where λ ∈ R+ solves∑
min{p(Ri ), max{ei , λ}} ifp(Rj ) ≥ E,∑j∈N
max{p(Ri ), min{ei , λ}} ifj∈N p(Rj ) ≤ E,∑

j∈N Gj (R) = E.

6

As mentioned above, envy-freeness is incompatible with individual rationality.
See Klaus et al. (1997, 1998a, 1998b), Klaus (1998, 2001), Moreno (2002), and Bonifacio (2015).
8
A similar functional form appears in the social endowment setting under constraints. See
Bergantiños et al. (2015).
7

6

The gradual uniform rule G is an extension of the uniform rule U in the sensethat if for each pair {i, j} ⊆ N , ei = ej , then U (R) = G(R).
The following example demonstrates the computation of the gradual uniform
allocation for the case where the sum of agents’ peaks is strictly greater than thesum of individual endowments.
Example 1. Let N = {1, 2, 3, 4} and e = (1, 7, 3, 10). Let R ∈ R 4 be such thatp(R) = (7, 13, 4, 4). Therefore, the sum of the peaks is strictly greater than the sumof the individual endowments. Then, the gradual uniform allocation of this problem
can be determined as follows. Agent 4 receives his peak. We now have to divide theamount agent 4 supplies (i.e., the six units of the good) among the demanders. We
increase the amounts of the demanders until we have assigned the amount agent 4
supplies as follows. We first increase the amount of the demander with the smallest
individual endowment (i.e., agent 1) until it reaches either his peak or the second
smallest individual endowment among the demanders (i.e., agent 3’s individual endowment). We then obtain “temporary” allocation (3, 7, 3, 4). There remain thefour units of the good. We next increase the amounts of agents 1 and 3 equally until one of them reaches either his peak or the third smallest individual endowment
among demanders (i.e., agent 2’s individual endowment). We then obtain “temporary” allocation (4, 7, 4, 4). Now, agent 3 receives his peak. We still have to allocate
the two unit of the good. Again, we increase the amount of agent 1 until it reaches
either his peak or the third smallest individual endowment among demanders. We
then obtain allocation (6, 7, 4, 4) and complete this process. Note that this example
yields λ = 6. We note further that allocation (6, 7, 4, 4) is individually rational andposition-wise envy-free.
■
From Example 1, we observe that there exist diﬀerences between the gradualuniform rule and the uniform reallocation rule. According to the latter rule, we
obtain allocation (3.5, 9.5, 4, 4). However, this allocation is not position-wise envyfree, because both agent 1 and agent 3 are demanders, and agent 1 envies agent 3
who does not receive his endowment. In contrast, the gradual uniform rule selects a
position-wise envy-free allocation. In fact, for every preference profile, the gradualuniform rule selects a position-wise envy-free allocation.
Proposition 2. The gradual uniform rule is position-wise envy-free.
Proof. Let R ∈ R n . In order to prove position-wise envy-freeness of G, we haveto show that for each pair {i, j} ⊆ N , if Gj (R) ̸= ej and either {i, j} ⊆ N d (R) or7

{i, j} ⊆ N s (R), thenGi (R) Ri Gj (R).

(1)∑
Let i ∈ N . Consider the case j∈N p(Rj ) ≥ E (the other case is similar and weomit the details). We distinguish two cases.
Case 1: i ∈/ N d (R). Then, Gi (R) = p(Ri ). This implies (1).
Case 2: i ∈ N d (R). Let j ∈ N d (R) be such that Gj (R) ̸= ej . Then, ej < λ.
Otherwise, Gj (R) ≡ min{p(Rj ), max{ej , λ}} = min{p(Rj ), ej }. Since p(Rj ) > ej ,we have Gj (R) = ej , a contradiction. Therefore,Gj (R) ≡ min{p(Rj ), max{ej , λ}} = min{p(Rj ), λ}.

(2)

By the definition of G, there are two subcases.
Subcase 2.1: p(Ri ) ≤ max{ei , λ}. Then, Gi (R) = p(Ri ). Hence, (1) istrivially true.
Subcase 2.2: p(Ri ) > max{ei , λ}. By (2),Gj (R) = min{p(Rj ), λ} ≤ λ ≤ max{ei , λ} < p(Ri ).
Since Gi (R) = max{ei , λ}, this implies (1).
The gradual uniform rule satisfies not only individual rationality and positionwise envy-freeness but also eﬃciency and strategy-proofness. Furthermore, we showthat it is the only rule satisfying these properties.
Theorem 1. The gradual uniform rule is the only rule that satisfies eﬃciency,
individual rationality, strategy-proofness, and position-wise envy-freeness.
Proof. It is easy to see that the gradual uniform rule G is eﬃcient, individually
rational, and strategy-proof. By Proposition 2, G is position-wise envy-free.
To prove the remaining part of the theorem, let f be a rule satisfying the four
properties. Let R ∈ R n . We show that for each i ∈ N , fi (R) = Gi (R). Consider∑
the case j∈N p(Rj ) ≥ E (the other case is similar and we omit the details).
Claim 1. For each i ̸∈ N d (R), fi (R) = p(Ri ) = Gi (R).
Proof of Claim 1. Let i ̸∈ N d (R). Then, p(Ri ) ≤ ei . By the definition of G, p(Ri ) =Gi (R) is obvious. We prove only fi (R) = p(Ri ). Suppose, by contradiction, that8fi (R) ̸= p(Ri ). It then follows from eﬃciency that fi (R) < p(Ri ). Since p(Ri ) ≤ ei ,we can take Ri′ ∈ R such that p(Ri′ ) = p(Ri ) and ei Pi′ fi (R). By individual
rationality and eﬃciency, fi (R) < fi (Ri′ , R−i ) ≤ p(Ri ). Then, fi (Ri′ , R−i ) Pi fi (R),which contradicts strategy-proofness. Hence, fi (R) = p(Ri ).
If for each i ∈ N d (R), fi (R) ≥ Gi (R), then Claim 1 and feasibility together
imply that for each i ∈ N d (R), fi (R) = Gi (R). This, together with Claim 1, yields
that for each i ∈ N , fi (R) = Gi (R). Therefore, in what follows, suppose that thereis j ∈ N d (R) with fj (R) < Gj (R). Without loss of generality, we assume
f1 (R) = min{fj (R) : fj (R) < Gj (R)}.

(3)

Claim 1 ensures that 1 ∈ N d (R), that is,p(R1 ) > e1 .

(4)Claim 2. f1 (R) < λ.
Proof of Claim 2. Suppose, by contradiction, that f1 (R) ≥ λ. By (3), f1 (R) <G1 (R) ≡ min{p(R1 ), max{e1 , λ}}. It then follows that f1 (R) < max{e1 , λ}. If
max{e1 , λ} = λ, then λ > f1 (R) ≥ λ, a contradiction. If max{e1 , λ} = e1 , then (4)implies that f1 (R) < e1 < p(R1 ), which contradicts individual rationality. Hence,
f1 (R) < λ.
Claim 3. For each i ∈ N d (R), if f1 (R) > fi (R), then fi (R) = p(Ri ).
Proof of Claim 3. Suppose, by contradiction, that there is i ∈ N d (R) such that
f1 (R) > fi (R) and fi (R) ̸= p(Ri ). Then, (3) implies that fi (R) ≥ Gi (R). Thus, by
Claim 2,
λ > f1 (R) > fi (R) ≥ Gi (R) ≡ min{p(Ri ), max{ei , λ}}.

(5)

If min{p(Ri ), max{ei , λ}} = p(Ri ), then fi (R) ≥ p(Ri ). Since fi (R) ̸= p(Ri ),fi (R) > p(Ri ), which contradicts eﬃciency. If min{p(Ri ), max{ei , λ}} = max{ei , λ},then fi (R) ≥ max{ei , λ} ≥ λ. This, together with (5), implies that λ > λ, a contradiction.
Claim 4. There is h ∈ N d (R) such that fh (R) > eh and fh (R) > f1 (R).

9

Proof of Claim 4. Note that by (3), f1 (R) < G1 (R). Thus, by feasibility, there ish ∈ N such that fh (R) > Gh (R) ≡ min{p(Rh ), max{eh , λ}}. By Claim 1, h ∈N d (R). We also assert that min{p(Rh ), max{eh , λ}} = max{eh , λ}, that is, fh (R) >max{eh , λ}; otherwise, fh (R) > p(Rh ), which contradicts eﬃciency. Hence, fh (R) >eh . It also follows from Claim 2 that fh (R) > max{eh , λ} ≥ λ > f1 (R).
Notice that since G is eﬃcient, (3) implies that E ≥ p(R1 ) ≥ G1 (R) > f1 (R).
Therefore, we can take R̂1 ∈ R such thatp(R̂1 ) = p(R1 ) and E P̂1 f1 (R).

(6)

Claim 5. For each i ∈ N , fi (R̂1 , R−1 ) ≤ fi (R).
Proof of Claim 5. Let i ∈ N . We divide the argument into three cases.
Case 1: i = 1. By eﬃciency, f1 (R) ≤ p(R1 ) = p(R̂1 ) and f1 (R̂1 , R−1 ) ≤p(R1 ) = p(R̂1 ). If f1 (R) < f1 (R̂1 , R−1 ), then f1 (R̂1 , R−1 ) P1 f1 (R), which contradicts strategy-proofness. If f1 (R̂1 , R−1 ) < f1 (R), then f1 (R) P̂1 f1 (R̂1 , R−1 ), which
contradicts strategy-proofness. Hence, f1 (R̂1 , R−1 ) = f1 (R).
Case 2: i ̸∈ N d (R). By the same way as Claim 1, we have fi (R̂1 , R−1 ) = p(Ri ).
Hence, fi (R̂1 , R−1 ) = fi (R).
Case 3: i ∈ N d (R). Then,p(Ri ) > ei .

(7)Now, there are three subcases.
Subcase 3.1: f1 (R) > fi (R). By Claim 3, fi (R) = p(Ri ). Then, by eﬃciency,fi (R̂1 , R−1 ) ≤ p(Ri ) = fi (R).
Subcase 3.2: f1 (R) = fi (R). Suppose, by contradiction, that fi (R̂1 , R−1 ) >fi (R). Then, fi (R̂1 , R−1 ) ̸= ei ; otherwise, by (7), p(Ri ) > ei = fi (R̂1 , R−1 ) > fi (R),which contradicts individual rationality. By Case 1, fi (R̂1 , R−1 ) > fi (R) = f1 (R) =f1 (R̂1 , R−1 ). By eﬃciency, p(R̂1 ) = p(R1 ) > f1 (R) = f1 (R̂1 , R−1 ). Then, either(i) p(R̂1 ) ≥ fi (R̂1 , R−1 ) > f1 (R̂1 , R−1 ) or(ii) E ≥ fi (R̂1 , R−1 ) > p(R̂1 ) > f1 (R̂1 , R−1 ).

10In either case, by (6),fi (R̂1 , R−1 ) P̂1 f1 (R̂1 , R−1 ).

(8)

Since fi (R̂1 , R−1 ) ̸= ei and {1, i} ⊆ N d (R̂1 , R−1 ), (8) implies a contradiction toposition-wise envy-freeness. Hence, fi (R̂1 , R−1 ) ≤ fi (R).
Subcase 3.3: f1 (R) < fi (R). By position-wise envy-freeness, either(i) fi (R̂1 , R−1 ) = ei or(ii) fi (R̂1 , R−1 ) ≤ f1 (R̂1 , R−1 ).
If (i) holds, then fi (R̂1 , R−1 ) = ei ≤ fi (R); otherwise, by (7), p(Ri ) > ei =fi (R̂1 , R−1 ) > fi (R), which contradicts individual rationality. If (ii) holds, then,by Case 1, fi (R̂1 , R−1 ) ≤ f1 (R̂1 , R−1 ) = f1 (R) < fi (R). Hence, fi (R̂1 , R−1 ) ≤fi (R).
Claim 6. There is h ∈ N d (R) such that fh (R̂1 , R−1 ) < fh (R).
Proof of Claim 6. By Claim 4, there is h ∈ N d (R) such that fh (R) > eh andfh (R) > f1 (R). Then, we show that fh (R̂1 , R−1 ) < fh (R). Otherwise, since f1 (R) =f1 (R̂1 , R−1 ) (Case 1 of Claim 5), fh (R̂1 , R−1 ) ≥ fh (R) > f1 (R) = f1 (R̂1 , R−1 ). It
then follows from (6) thatfh (R̂1 , R−1 ) P̂1 f1 (R̂1 , R−1 ).

(9)Since fh (R̂1 , R−1 ) ≥ fh (R) > eh and {1, h} ⊆ N d (R̂1 , R−1 ), (9) implies a contradiction to position-wise envy-freeness.
By Claim 5, for each i ∈ N , fi (R̂1 , R−1 ) ≤ fi (R). By Claim 6, for at least one
agent h ∈ N , fh (R̂1 , R−1 ) < fh (R). These implyE=∑

fj (R̂1 , R−1 ) <j∈N∑

fj (R) = E,j∈N

which is a contradiction. Therefore, for each i ∈ N , fi (R) ≥ Gi (R). It thus follows
from feasibility that for each i ∈ N , fi (R) = Gi (R).
Remark 2. Our theorem does not use the full force of strategy-proofness. In fact,
Theorem 1 holds even if strategy-proofness is relaxed with “strategy-proofness for11

same tops” (Sakai and Wakayama, 2012).9 From this, we can easily obtain anothercharacterization of the gradual uniform rule by replacing strategy-proofness with“peak-only.”10♢
Before concluding this section, we establish the tightness of Theorem 1. The
uniform reallocation rule is an example of a rule satisfying all properties but notposition-wise envy-freeness. The uniform rule is an example of a rule satisfying allproperties but not individual rationality. The endowment rule (that always assigns
to any agent his endowment) is an example of a rule satisfying all properties but
not eﬃciency. The following example illustrates a rule satisfying all properties but
not strategy-proofness.
Example 2. Let n = 3 and e = (1, 2, 1). Let R̂ ∈ R 3 be such that 1 Iˆ1 3 andp(R̂) = (2, 3, 0), where Iˆ1 denote the indiﬀerence part of R̂1 . Define f ∗ as follows:for each R ∈ R 3 ,(1, 3, 0) if R = R̂,∗f (R) =
G(R)otherwise.
Then, f ∗ is eﬃcient, individually rational, and position-wise envy-free. However, f ∗is not strategy-proof. To see this, let R1 ∈ R be such that p(R1 ) = 2 and 3 P1 1.
Then, f ∗ (R1 , R̂−1 ) = G(R1 , R̂−1 ) = (2, 2, 0) and thus, f1∗ (R1 , R̂−1 ) = 2 P̂1 1 =
f1∗ (R̂), a violation of strategy-proofness.
■

9

The notion of “strategy-proofness for same tops” states that an agent cannot gain if he truthfully reports his peak.
10
The notion of “peak-only” states that the amount assigned to agents depends only on theirpeaks.

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