Discussion Paper No. 628

MAXIMAL DOMAINFOR STRATEGY-PROOF RULESIN ALLOTMENT ECONOMIES*

Hideyuki Mizobuchi
andShigehiro Serizawa

February 2005

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan*Forthcoming in Social Choice and WelfareMaximal Domain for Strategy-Proof Rules
in Allotment Economies
1

Hideyuki Mizobuchi
Department of Economics, University of British Columbia
Vancouver, B.C. Canada V6T 1Z1
E-mail: mizobuch@interchange.ubc.ca
andShigehiro Serizawa
Institute of Social and Economic Research, Osaka University
6-1, Mihogaoka, Ibaraki, Osaka 567-0047, Japan
E-mail: serizawa@iser.osaka-u.ac.jp

January 19, 2005

We consider the problem of allocating an amount of a perfectly divisible good amonga group of n agents. We study how large a preference domain can be to allow for the
existence of strategy-proof, symmetric, and eﬃcient allocation rules when the amount of
the good is a variable. This question is qualified by an additional requirement that adomain should include a minimally rich domain. We first characterize the uniform rule
(Bennasy, 1982) as the unique strategy-proof, symmetric, and eﬃcient rule on a minimallyrich domain when the amount of the good is fixed. Then, exploiting this characterization,
we establish the following: There is a unique maximal domain that includes a minimallyrich domain and allows for the existence of strategy-proof, symmetric, and eﬃcient rules
when the amount of good is a variable. It is the single-plateaued domain.

1

The authors thank Professors Miyake, M., and Thomson, W. and two anonymous referee for theirhelpful and detailed suggestions. They also thank Asano, T. and Suzuki, M. for their useful comments.
The authors completed this article when Serizawa visited the Department of Economics at the Universityof British Columbia. He appreciates the good research environment of the department.

1. INTRODUCTION
We consider the problem of allocating an endowment of a perfectly divisible goodamong a group of n agents. An allotment economy is a pair of a preference profile and an
amount of the good. A “rule” chooses a feasible allocation for each allotment economy; itis formally defined as a function from the class of allotment economies to the set of allocations. When the endowment is fixed, an allotment economy is described by a preference
profile alone. A class of preference profiles are called a “preference domain.” Since preferences are only privately known, the choices of rules must be based on preferences agentsannounce. Thus, agents might strategically misrepresent their preferences to obtain outcomes they prefer. As a result, the decision that the rule should make for agents’ true
preferences may not be realized. Thus, the condition called “strategy-proofness” is often
imposed on rules to guarantee that agents have the incentive to reveal their true preferences. It says that no agent is better oﬀ by misrepresenting his preferences, no matter
what his true preference is and no matter what preferences others announce. “Symmetry”
and Pareto eﬃciency are also often imposed on rules. “Symmetry” is the distributional
requirement that two agents with the same preference should be given indiﬀerent amountsof the good.
If a rule satisfies strategy-proofness on a preference domain, it also satisfies strategyproofness on any subdomain. Therefore, the larger the domain on which rules are required
to satisfy strategy-proofness, the stronger the requirement is. For instance, in the modelof public alternatives, it is well-known that strategy-proofness is so strong on the universal domain (the class of all preferences on the set of the public alternatives) that only
dictatorships can satisfy strategy-proofness.2 Similarly, the requirements of eﬃciency andsymmetry are stronger on larger domains. Thus, the smaller a domain is, the more rules
satisfying the three requirements potentially exist. On the other hand, if the domain is
too large, the three requirements may become so strong that there exist no rules satisfying
them. In this paper, we identify (i) minimal domains on which the three requirements
imply uniqueness of the rule, and (ii) maximal domains on which rules satisfying the threerequirements exist.
Many authors have analyzed strategy-proof rules on the class of “single-peaked” preferences. A preferences is “single-peaked” if more is preferred to less up to some level, called
the “peak”, and less is preferred to more beyond the peak. The single-peaked domain
is the set of all single-peaked preferences. By fixing the endowment, Sprumont (1991)
and Ching (1994) establish that there exists a unique rule satisfying strategy-proofness,symmetry, and eﬃciency on the single-peaked domain, the “uniform rule”.3 The “uniformrule” allocates the total endowment to agents as follows. When the sum of agents’ peaks
is greater than or equal to the total endowment, an agent gets his peak if that level isless than the common upper bound; otherwise he receives the common bound; and thecommon bound is chosen so as to satisfy the feasibility. When the sum of agents’ peaks
is less than the total endowment, the opposite principle is applied, that is, an agent gets
his peak if that level is more than the common lower bound; and so on. The results by
Sprumont (1991) and Ching (1994) motivate us to investigate the following two questions.
2This is the so-called Gibbard-Satterthwaite Theorem. (Gibbard, 1973, and Satterthwaite, 1975).
First, Sprumont (1991) establishes that the uniform rule is the unique rule satisfying strategyproofness, “anonymity”, and (Pareto) eﬃciency on the single-peaked domain. “Anonymity” is the distributional requirement that the names of agents should not matter. Later, Ching (1994) strengthens
Sprumont’s (1991) characterization by replacing anonymity with the weaker requirement of symmetry.
3

1

The first question is how much we can shrink the domain of single-peaked preferences
while preserving the uniform rule as the unique rule satisfying strategy-proofness, symmetry, and eﬃciency. A “minimally rich domain” is a small subset of the single-peakeddomain satisfying the following two conditions: (i) for each consumption level, there exists only one preference whose peak coincides with the consumption level; (ii) given two
distinct consumption levels, say x and y, there exists at least one preference whose peakis between x and y such that x is preferred to y. The “symmetric”4 domain is an example of the minimally rich domain. Since any minimally rich domain is much smaller
than the single-peaked domain, strategy-proofness on a minimally rich domain is weaker
than strategy-proofness on the single-peaked domain, and so are eﬃciency and symmetry.
Therefore, potentially there exist more rules satisfying the weaker requirements. However,
we show that the uniform rule is still the unique rule satisfying the three properties ona minimally rich domain. We establish that a rule on a minimally rich domain satisfies
strategy-proofness, symmetry, and eﬃciency if and only if it is the uniform rule. Following Sprumont (1991) and Ching (1994), we first fix the endowment and establish thischaracterization.
The second question is how much we can enlarge the preference domain while allowing
for the existence of strategy-proof, symmetric, and eﬃcient rules. Ching and Serizawa(1998) also study the same question. Ching and Serizawa (1998) show that the “singleplateaued” domain is the unique maximal preference domain that includes the singlepeaked domain and allows for the existence of strategy-proof, symmetric, and eﬃcient
rules. “Single-plateaued” preferences are variants of single-peaked preferences for which
the sets of most preferred consumption levels are intervals.5 The single-plateaued domain
is the set of all single-plateaued preferences. The setting of Ching and Serizawa (1998) is
diﬀerent from Sprumont (1991) and Ching (1994). Ching and Serizawa (1998) consider
the situation in which rules have the amount of the good to be allocated as a variable
and each economy is represented by a pair of one preference profile and the endowmentof the good. We adapt the same setting as Ching and Serizawa (1998) in studying thesecond question, and establish a similar result. However, our result is stronger than theirs
for the following reason. Ching and Serizawa (1998) require that the maximal domainshould include the single-peaked domain. Here, we only require the maximal domain to
include a minimally rich domain. Since any minimally rich domain is much smaller than
the single-peaked domain, our requirement is weaker than Ching and Serizawa’s (1998).
Therefore, potentially there may exist a domain which includes a minimally rich domain
and allows for the existence of strategy-proof, symmetric, and eﬃcient rules, but is larger
than or diﬀerent from the single-plateaued domain. However, we establish that the singleplateaued domain is still the unique maximal domain including a minimally rich domain
for strategy-proofness, symmetry, and eﬃciency. Our result implies Ching and Serizawa’s(1998) result as a corollary.
Recently, Massó and Neme (2001) succeed in obtaining a maximal domain result when
the amount of the good to be allocated is fixed. Since they adopt “strong symmetry”6 as
4A single-peaked preference is “symmetric” if its utility representation is symmetric around the peak.
Symmetric preferences are also called “quadratic” preferences in Border and Jordan (1983) since they are
represented by quadratic utility functions.
5
It is studied by Moulin (1984) and Berga (1998).
6
Strong symmetry means that agents with the same preferences are given the same amount of the
good.

2

a property of rules in place of symmetry in our result, and since they require the maximal
domain to include the single-peaked domain, our result (Theorem 2) is independent of
theirs.
Besides Ching and Serizawa (1998) and Massó and Neme (2001), there are several
articles that study the maximal domain for strategy-proofness. Massó and Neme (2004)
and Ching and Serizawa (2003) study the maximal domain for strategy-proof rules inallotment economies by assuming the continuity of rules and establish diﬀerent maximal
domain results.7 Barberà, Sonnenschein, and Zhou (1991) and Serizawa (1995) study
models of public alternatives and give the maximal domain where the class of rules called“the generalized median voter schemes” satisfies strategy-proofness. While they exclude
rules other than the generalized median voter schemes, we do not restrict the rules apriori, but obtain the maximal domain by just imposing properties on rules. Berga andSerizawa (2000) also investigate the two questions that are parallel to ours in a modelof public alternatives. Recently, Ehlers (2002) studies maximal domains for coalitional
strategy-proofness in an indivisible goods model.
This paper is organized as follows. Section 2 explains the model and the main results.
Section 3 is devoted to the proof of the results of Section 2. Section 4 raises open questions.
2. THE MODEL AND THE RESULTS
We consider the problem of allocating one perfectly divisible private good among a
finite number of agents. Let N = {1, · · · , n} be the set of agents, where n ≥ 2. Let
M ∈ R++ be the amount of the good. Each agent i ∈ N is equipped with a preference
relation Ri on R+ ∪ {∞}. Let Pi be the strict preference relation associated with Ri ,and Ii the indiﬀerence preference relation. We assume that preferences are continuous,that is, for all x ∈ R+ ∪ {∞}, the sets {y ∈ R+ ∪ {∞} : yRi x} and {y ∈ R+ ∪ {∞} :xRi y} are closed. Let RC be the class of all continuous preferences. Given Ri ∈ RC ,let p(Ri ) = {x ∈ R+ ∪ {∞}|∀y ∈ R+ , xRi y} be the set of preferred consumptions for
Ri . Let p(Ri ) = inf p(Ri ) and p(Ri ) = sup p(Ri ). When p(Ri ) is a singleton, we slightly
abuse notation and use p(Ri ) to denote its single element. A preference profile is alist R = (R1 , · · · , Rn ) ∈ RnC . When we emphasize the role of agent i ∈ N, we write
R = (Ri, R−i ) where R−i = (Rj )j∈N\{i} . An economy is a pair of (R, M) ∈ RnC × R++ .
When P
the amount of the good is M, an allocation is a vector z = (z1, · · · , zn ) ∈ Rn+
such that i∈N zi = M. Let Z(M) be the set of all allocations for the economy (R, M).
A rule associates to each economy (R, M) an allocation z ∈ Z(M). It can be regarded asa recommendation for each economy. A domain is a subset R of RC . A rule is a function
ϕ : Rn × R++ → Rn+ such that for all (R, M) ∈ Rn × R++ , ϕ(R, M) ∈ Z(M). For alli ∈ N, ϕi (R, M) represents the amount of the good allocated to agent i. When we wantto emphasize the domain of a rule, we call it a rule on Rn × R++
Agents are assumed to be equipped with the following preferences.
Definition 1. A preference R0 ∈ RC is single-peaked if p(R0 ) is a singleton, andfor all x, y ∈ R+ ∪ {∞}, we have xP0 y whenever y < x ≤ p(R0 ) or p(R0 ) ≤ x < y.
Let RS ⊆ RC be the domain of all single-peaked preferences. We call it the singlepeaked domain.
7

Massó and Neme (2004) are only concerned with tops-only rules and Ching and Serizawa (2003) use
unanimity instead of eﬃciency.

3

Definition 2. A single peaked preference R0 ∈ RC is symmetric if for all x, y ∈ R+ ,xP0 y if and only if |x − p(R0 )| < |y − p(R0 )|.
Let RM ⊆ RC be the domain of all symmetric preferences. We call it the symmetricdomain.
Definition 3. A preference R0 ∈ RC is single-plateaued if p(R0 ) is an interval[p(R0 ), p(R0 )]; and for all x, y ∈ R+ ∪ {∞} with [x < y ≤ p(R0 )] or [p(R0 ) ≤ x < y], wehave xP0 y.
Let RP ⊆ RC be the domain of all single-plateaued preferences. We call it the singleplateaued domain.
Rules are required to satisfy the following three properties. The first one is thestrongest incentive compatibility. No one can ever benefit by misrepresenting his preferences.
Definition 4. A rule ϕ on Rn × R++ is strategy-proof if for all (R, M) ∈ Rn × R++ ,all i ∈ N, and all Ri0 ∈ R, ϕi (R, M)Ri ϕi (Ri0 , R−i , M).
The second property is a distributional requirement. If two agents have the same
preference, their allocations are indiﬀerent to each other.
Definition 5. A rule ϕ on Rn × R++ is symmetric if for all (R, M) ∈ Rn × R++
and all i, j ∈ N such that Ri = Rj , ϕi (R, M)Ii ϕj (R, M).
The third property is the standard eﬃciency requirement.
Definition 6. A rule ϕ on Rn × R++ is eﬃcient if for all (R, M) ∈ Rn × R++ , there
is no z ∈ Z(M) such that for all i ∈ N, zi Ri ϕi (R, M) and for some j ∈ N, zj Pj ϕj (R, M).
The fact below is derived directly from the definition of eﬃciency (Definition 6). It isuseful in the proofs of Lemmas 1, 4, and Theorem 1.
Fact. Let R ⊆ RS ⊆ RC . If a rule ϕ on Rn × R++ is eﬃcient, then for all (R, M) ∈Rn ×P
R++ , the following properties hold.
(i) If i∈NP
p(Ri ) ≤ M, then for all j ∈ N, p(Rj ) ≤ ϕj (R).
If M ≤ i∈N p(Ri ), then for all j ∈ N, ϕj (R) ≤ p(Rj ).
P(ii) If there exists an agent i ∈ I such that p(Ri ) < ϕi (R), then j∈NPp(Rj ) < M.
If there exists an agent i ∈ I such that ϕi (R) < p(Ri ), then M < j∈N p(Rj ).
(iii) If there exists an agent i ∈ I such that ϕi (R) < p(Ri ), then for all agent j ∈ I,
ϕj (R) ≤ p(Rj ).
If there exists an agent i ∈ I such that p(Ri ) < ϕi (R), then for all agent j ∈ I,
p(Rj ) ≤ ϕj (R).
2. 1. Characterization
Sprumont (1991) and Ching (1994) show that the following rule, the uniform rule, isthe only one satisfying all three properties of strategy-proofness, symmetry, and eﬃciency
on the single-peaked domain, RnS .
Definition 7. The uniform rule U = (U1 , · · · , Un ) is defined as follows. For all(R, M) ∈ RnS × R++ and all i ∈ N,½
P
max{p(Ri ), λ(R, M)} if
p(Rj ) ≤ M
Ui (R)=
min{p(Ri ), λ(R, M)} otherwise,4

where λ(R, M) solves

P

8j∈N Uj (R, M) = M.

How much can we weaken the assumption of strategy-proofness, symmetry, and eﬃciency on the single-peaked domain, while preserving the uniform rule as the unique rulesatisfying strategy-proofness, symmetry, and eﬃciency? By the definition of strategyproofness, if a rule satisfies strategy-proofness on a domain, it also satisfies strategyproofness on any subdomain. Therefore the smaller domain (in the sense of inclusion) on
which we assume a rule satisfies strategy-proofness, the less demanding we are. Similarly,
the conditions of eﬃciency and symmetry are weaker on the smaller domains. We characterize the uniform rule on the following minimally rich domains.9 Since any minimally richdomain is much smaller than the single-peaked domain, our characterization is stronger
than Sprumont (1991) and Ching (1994).
Definition 8. A domain R ⊆ RC is minimally rich if(1) R ⊆ RS(2) for all x ∈ R+ , there exists a unique preference R0 ∈ R such that p(R0 ) = x, and(3) for all x, y ∈ R+ such that x 6= y, there exists R0 ∈ R such that xP0 y and p(R0 ) ∈(min{x, y}, max{x, y}).
We denote a generic minimally rich domain by RR . The symmetric domain is an
example of minimally rich domain. The domain R0 defined below is one of many otherminimally rich domains.
R0 = {R0 ∈ RS : for all x, y ∈ [0, M] such that x < p(R0 ) < y, xI0 y if and only if|x − p(R0 )| = 2|y − p(R0 )|}.
Now we establish a characterization of the uniform rule on minimally rich domains.
Sprumont (1991) and Ching (1994) characterize the uniform rule when M is fixed. Theirresults also imply the characterization when M is not fixed. Therefore, we first show ourcharacterization when M is fixed. The characterization when M is not fixed follows.
Given M ∈ R++ , we define RS (M), RM (M), and RR (M) as the set of preferences
obtained by restricting on [0, M] all preferences in RS , RM , and RR respectively. We
denote a generic element of RS (M) by Ri (M). When M is fixed, we consider that agent
i’s preference is defined only on [0, M], an economy is represented by a list R(M) =(R1 (M), · · · , Rn (M)) ∈ R(M)n , and that when a domain is a subset R(M) of RS (M),a rule is a function ϕ(·, M) : R(M)n → Z(M). When we want to emphasize the domainof a rule, we call it a rule on R(M)n . Strategy-proofness, symmetry, and eﬃciency are
similarly defined in this setting. Thus, we omit their definitions.
Theorem 1. Let M ∈ R++ . A rule ϕ(·, M) on a minimally rich domain RR (M)nis strategy-proof, symmetric, and eﬃcient if and only if ϕ(R(M), M) = U(R(M), M) for
all R(M) ∈ RR (M)n .
Theorem 1 says that when M is fixed, the uniform rule is the unique rule satisfying
strategy-proofness, symmetry, and eﬃciency on a minimally rich domain. The characterization when M is not fixed is a corollary of Theorem 1.
Corollary 1. A rule ϕ on a minimally rich domain RnR ×R++ is strategy-proof,
symmetric, and eﬃcient if and only if ϕ(R, M) = U (R, M) for all ( R, M) ∈ RnR × R++ .
8

The uniform rule can be regarded as a system of equations. Therefore, λ(R, M ) is endogenously
determined and depends on a profile of agents’ preferences and the total amount of the good to beallocated. Sönmez (1994) establishes the algorithm to find λ(R, M ).
9Minimally rich domains are firstly studied by Berga and Serizawa (2000).

5Since the symmetric domain is a minimally rich domain, the next corollary is also
obtained.
Corollary 2. A rule ϕ on the symmetric domain RnM × R++ is strategy-proof,
symmetric, and eﬃcient if and only if ϕ(R, M) = U(R, M) for all (R, M) ∈ RnM × R++ .
2.2. Maximal Domain
We proceed to the next question; how much larger can a preference domain be whileallowing for the existence of strategy-proof, symmetric, and eﬃcient rules? To answer
this question precisely, we need the following notion.
Definition 9. A domain Rm is a maximal domain for a list of properties if (i)Rm ⊆ RC , (ii) there is a rule on Rnm × R++ satisfying the properties; and (iii) there is
no rule satisfying the same properties on any RnA × R++ such that Rnm ( RnA ⊆ RnC .
Note that a maximal domain for a list of properties may not be unique. However, Chingand Serizawa (1998) show that the single-plateaued domain is the unique maximal domain
including the single-peaked domain for strategy-proofness, symmetry, and eﬃciency. Any
domain that includes the single-peaked domain also includes a minimally rich domain.
Thus the set of domains that includes the single-peaked domain is contained by the set of
domains including a minimally rich domain. Theorem 2 below says that even among themore candidate, the single-plateaued domain still remains the unique maximal domain.
Theorem 2 strengthens Ching and Serizawa’s result (1998).
Theorem 2. The single-plateaued domain is the unique maximal domain that includes
a minimally rich domain for strategy-proofness, symmetry, and eﬃciency.
Note that we obtain Ching and Serizawa’s result (1998) as a corollary of Theorem 2.
Corollary 3 (Ching and Serizawa 1998). The single-plateaued domain is the uniquemaximal domain including the single-peaked domain for strategy-proofness, symmetry, andeﬃciency.
Since the symmetric domain is a minimally rich domain, we also have the following
corollary.
Corollary 4. The single-plateaued domain is the unique maximal domain including
the symmetric domain for strategy-proofness, symmetry, and eﬃciency.
3. PROOFS
3. 1. Proofs for Theorem 1
For the proofs, we introduce the notion of “the option set.” In the model of publicalternatives, Berga and Serizawa (2000) employed the notion in their proof of characterization. Similarly to them, we prove that option sets are convex in allotment economies.
Our proof of Theorem 1 is similar to Ching’s proof (1994) in structure. However, because
a minimally rich domain is smaller than the single-peaked domain, his procedure cannot
be applied throughout our proof. The convexity of option sets plays an important role inovercoming that diﬃculty, relevant in Lemma 5. Although our proofs of Lemmas 1, 2, 3,
and Theorem 1 are similar to the proofs of Sprumont (1991) and Ching (1994), we state
all proofs for completeness.
Before stating all the previous results, let us simplify notation. In this subsection, M isfixed. Thus each economy is represented by a profile of preferences. To simplify notation,6

we denote RS (M), RR (M), and RM (M) by RS , RR , and RM respectively throughout
this subsection. We denote a generic element of R(M) by R instead of R(M). For thesame reason, ϕ(R, M) and U(R, M) are replaced by ϕ(R) and U (R) respectively.
Ching (1994) shows that strategy-proof rules on the single-peaked domain satisfy theproperty of “own peak monotonicity.” It says that if the peak of an agent decreases, hisshare does not increase. Similarly to him, we can show that the uniform rule is own peakmonotonic on any minimally rich domain.
Lemma 1 (“Own Peak Monotonicity”). If a rule ϕ on RnR is eﬃcient and strategyproof, then for all R ∈ RnR , all i ∈ N, and all Ri0 ∈ RR such that p(Ri0 ) ≤ p(Ri ),ϕi (Ri0 , R−i ) ≤ ϕi (R).
Proof. Let R ∈ RnR , i ∈ N and Ri0 ∈ RR be such that p(Ri0 ) ≤ p(Ri ). By contradiction,
suppose that ϕi (R) < P
ϕi (Ri0 , R−i ).
First suppose that j∈N p(Rj ) ≤ M. By Fact (i), for all j ∈ N, p(Rj ) ≤ ϕj (R). Sincep(Ri0 ) ≤ p(Ri ), p(Ri0 ) ≤ p(Ri ) ≤ ϕi (R) < ϕi (Ri0 , R−i ). Thus ϕi (R)Pi0 ϕi (Ri0 , R−i ). So agenti manipulates ϕ at (Ri0 ,PR−i ) via Ri , contradicting strategy-proofness.
Next, suppose that j∈N p(Rj ) > M. There are two subcases.
P
Subcase 1. M ≤ p(Ri0 ) + j6=i p(Rj ).
By Fact (i), ϕi (Ri0 , R−i ) ≤ p(Ri0 ). Since ϕi (R) < ϕi (Ri0 , R−i ) and p(Ri0 ) ≤ p(Ri ),ϕi (R) < ϕi (Ri0 , R−i ) ≤ p(Ri0 ) ≤ p(Ri ). Thus ϕi (Ri0 , R−i )Pi ϕi (R). Therefore agent i manipulates ϕ at R via Ri0 , contradicting strategy-proofness.
P
Subcase 2. M > p(Ri0 ) + j6=i p(Rj ).
PP
By Fact (i),Pfor all j ∈ N\{i}, ϕj (R) ≤ p(R
).
Thus
ϕ
(R)
≤
j
j
j6
=
i
j6=i p(Rj ).
P
00
Since p(Ri ) + j6=i p(Rj ) < M, p(Ri ) < M − j6=i ϕj (R) = ϕi (R). Since ϕi (R) <ϕi (Ri0 , R−i ), p(Ri0 ) < ϕi (R) < ϕi (Ri0 , R−i ). Thus ϕi (R)Pi0 ϕi (Ri0 , R−i ). Therefore agent i
manipulates ϕ at (Ri0 , R−i ) via Ri , contradicting strategy-proofness.
We obtain contradictions in all cases. Therefore, ϕi (Ri0 , R−i ) ≤ ϕi (R).

Q. E. D.

We introduce the notion of the option set.
n
Definition 11. Given i ∈ N, R−i ∈ Rn−1
R , and a rule ϕ on RR , agent i’s option set
at R−i is the set σ i (R−i ) = {x ∈ [0, M] : ∃Ri ∈ RR such that ϕi (Ri , R−i ) = x}.

The following lemma is well-known and used for the proof of Lemma 3.
Lemma 2. Let a rule ϕ on RnR be strategy-proof. Let R ∈ RnR and x ∈ σ i (R−i ) besuch that for all x0 ∈ σ i (R−i )\{x}, xPi x0 . Then ϕi (R) = x.
Proof. By contradiction, suppose that ϕi (R) 6= x. By definition, there exists Ri0 ∈RR such that ϕi (Ri0 , R−i ) = x. By assumption, ϕi (Ri0 , R−i )Pi ϕi (R). Therefore agent i
manipulates ϕi at R via Ri0 , contradicting strategy-proofness. Therefore, ϕi (R) = x.
Q. E. D.

7

In allotment economies, Sprumont (1991) shows that option sets are closed on the
single-peaked domain. We can prove the same property on a minimally rich domain as
he does.
Lemma 3 (“Closedness of Option Set”). Let a rule ϕ on RnR be strategy-proof. Leti ∈ N, and R−i ∈ Rn−1
R . Then σ i (R−i ) is closed.
Proof. By contradiction, suppose that σ i (R−i ) is not closed. Then, there exists a
sequence {xk }k∈N in σ i (R−i ) such that xk converges to x ∈ R as k goes to infinity andx∈/ σ i (R−i ). Since [0, M] is closed, x ∈ [0, M]. By the definition of minimally rich domain,
there is Ri0 ∈ RR such that p(Ri0 ) = x. Let ϕi (Ri0 , R−i ) = x0 . Since x ∈/ σ i (R−i ) and x 6= x0 ,xPi0 x0 . Moreover, since x = limk→∞ xk , and Ri0 is continuous, there exists k ∈ N such that
xk Pi0 x0 . Since xk ∈ σ i (R−i ), there exists Ri00 ∈ RR such that ϕi (Ri00 , R−i ) = xk . Thus agenti manipulates ϕ at (Ri0 , R−i ) via Ri00 , contradicting strategy-proofness. Therefore, σ i (R−i )is closed.
Q. E. D.
In the model of public alternatives, Berga and Serizawa (2000) show that option setsare convex on any minimally rich domain. We prove the convexity of the option set inallotment economies.
Lemma 4 (“Convexity of Option Set”). Let ϕ be a strategy-proof and eﬃcient ruleϕ on RnR . Let i ∈ N, and R−i ∈ Rn−1
R . Then the set σ i (R−i ) is convex.
Proof. Suppose, by contradiction, that σ i (R−i ) is not convex. Then there exist x <z < y such that x, y ∈ σ i (R−i ) and z ∈/ σ i (R−i ). But the convexity of [0, M] guarantees
000
00[x, y] ⊆ [0, M]. Let x = sup{x : x ∈ σ i (R−i ) and x00 < z} and y 0 = inf{y 00 : y 00 ∈σ i (R−i ) and y 00 > z}. By the closedness of σ i (R−i ) (Lemma 3), x0 , y 0 ∈ σ i (R−i ). By
the definition of a minimally rich domain, there exist Ri∗ , Ri∗∗ ∈ RR such that x0 Pi∗ y 0 ,p(Ri∗ ) ∈ (x0 , y 0 ), y 0 Pi∗∗ x0 and p(Ri∗∗ ) ∈ (x0 , y 0 ). Since x0 Pi∗ x00 for all x00 ∈ σ i (R−i )\{x0 } andy 0 Pi∗∗ y 00 for all y 00 ∈ σ i (R−i )\{y 0 }, it follows from Lemma 2 that ϕi (Ri∗ , R−i ) = x0 and
ϕi (Ri∗∗ , R−i ) = y 0 .
P∗Since ϕi (Ri∗ , R−i ) < p(Ri∗ ), Fact (ii) guarantees M <
p(R
)
+
i
j6=i p(Rj ). SinceP
p(Ri∗∗ ) < ϕi (Ri∗∗ , R−i ), Fact (ii) also guarantees p(Ri∗∗ ) + j6=i p(Rj ) < M. From these
two inequalities, we obtain p(Ri∗∗ ) < p(Ri∗ ). Then, by own peak monotonicity (Lemma1), ϕi (Ri∗∗ , R−i ) ≤ ϕi (Ri∗ , R−i ). However, since x0 < y 0 , this is a contradiction. Therefore,
σ i (R−i ) is convex.
Q. E. D.
Ching (1994) establishes a property called “uncompromisingness” on the single-peakeddomain. It says that if an agent’s original share is too much (less) to him, he cannot change
it by revealing any other preferences for which it is too much (less). Lemma 5 below
says that the same conclusion holds on a minimally rich domain. The structure of ourproof of Lemma 5 is also similar to Ching (1994). However, Ching’s proof does not work
here since preferences employed in his proof may not be in a minimally rich domain. We
exploit the convexity of option sets (Lemma 4) to overcome this diﬃculty. Note that theuniform rule is uncompromising on any minimally rich domain.
Lemma 5 (“Uncompromisingness”). If a rule ϕ on RnR is eﬃcient and strategy-proof,
the following property holds. For all R ∈ RnR , all i ∈ N, and all Ri0 ∈ RR ,8

if p(Ri ) < ϕi (R) and p(Ri0 ) ≤ ϕi (R), then ϕi (Ri0 , R−i ) = ϕi (R);
if p(Ri ) > ϕi (R) and p(Ri0 ) ≥ ϕi (R), then ϕi (Ri0 , R−i ) = ϕi (R).
Proof. We consider only the case when there exist R ∈ RnR , i ∈ N, and Ri0 ∈ RR suchthat p(Ri ) < ϕi (R) and p(Ri0 ) ≤ ϕi (R). The other case can be treated symmetrically.
By contradiction, suppose that ϕi (Ri0 , R−i ) 6= ϕi (R). There are two cases.
Case 1. ϕi (R) < ϕi (Ri0 , R−i ).
Since p(Ri0 ) ≤ ϕi (R), pi (Ri0 ) ≤ ϕi (R) < ϕi (Ri0 , R−i ). Therefore, ϕi (R)Pi0 ϕi (Ri0 , R−i ).
Agent i can manipulate ϕ at (Ri0 , R−i ) via Ri , contradicting strategy-proofness.
Case 2. ϕi (Ri0 , R−i ) < ϕi (R).
If p(Ri ) ≤ ϕi (Ri0 , R−i ) < ϕi (R), then ϕi (Ri0 , R−i )Pi ϕi (R), contradicting strategyproofness. Thus, let p(Ri ) > ϕi (Ri0 , R−i ). By the convexity of the option set σ i (R−i )(Lemma 4), there exists Ri00 ∈ RR such that ϕi (Ri00 , R−i ) = p(Ri ). Then, ϕi (Ri00 , R−i )Pi ϕi (R),contradicting strategy-proofness.
We obtain contradictions in both cases. Therefore, ϕi (Ri0 , R−i ) = ϕi (R).
Q. E. D.
Ching (1994) proves his characterization by using the properties of own peak monotonicity and uncompromisingness on the single-peaked domain. We have established the same
properties on a minimally rich domain. Using these properties, we can prove Theorem 1
as Ching (1994) does.
Proof of Theorem 1 .
It is trivial that the uniform rule satisfies strategy-proofness, symmetry, and eﬃciency
on a minimally rich domain. To prove the uniqueness, let ϕ be a strategy-proof, symmetry,and eﬃcient rule on a minimally rich domain RnR . We show that for all R ∈ RnR , ϕ(R) =U(R).
Let R ∈ RnR . There are three cases.
P
Case 1.
i∈N p(Ri ) = M.
By Fact (i), for all i ∈ N, ϕi (R) = p(Ri ). By the definition of the uniform rule, for
all i ∈ N, Ui (R) = p(Ri ). Thus, since for all i ∈ N, ϕi (R) = Ui (R), it follows that
ϕ(R) = U(R).
P
Case 2.
i∈N p(Ri ) < M.
Without loss of generality, we rename the agents so that p(R1 ) ≤ · · · ≤ p(Rn ). There
are two cases.
Case 2-1. R1 = · · · = Rn .
Then by Fact (i) and symmetry, for all i ∈ N, ϕi (R) = M/n. By the definition of theuniform rule, for all i ∈ N, Ui (R) = M/n. Therefore, ϕ(R) = U(R).
Case 2-2. R 6= (R1 , · · · , R1 ).
This means that at least one agent has a diﬀerent preference from other agents. By
contradiction, suppose that ϕ(R) 6= U(R).
Step 1. By Fact (i) and feasibility, there exists k ∈ N such that p(Rk ) ≤ ϕk (R) <Uk (R). Let Rk0 ∈ RR be such that p(Rk0 ) = p(R1 ). Suppose that agent k changes
his preference from Rk to Rk0 . Since p(Rk0 ) ≤ p(Rk ), by the own peak monotonicityof ϕ (Lemma 1), ϕk (Rk0 , R−k ) ≤ ϕk (R). Since p(Rk ) < Uk (R) and p(Rk0 ) ≤ Uk (R),the uncompromisingness of the uniform rule implies that Uk (Rk0 , R−k ) = Uk (R). Thus
ϕk (Rk0 , R−k ) < Uk (Rk0 , R−k ). Then if (Rk0 , R−k ) = (R1 , · · · , R1 ), by Case 2-1, for all9

i ∈ N, ϕi (Rk0 , R−k ) = Ui (Rk0 , R−k ) = M/n. Therefore, ϕk (Rk0 , R−k ) = Uk (Rk0 , R−k ). Thisis a contradiction. If (Rk0 , R−k ) 6= (R1 , · · · , R1 ), we proceed
Pto Step 2.
00
Step 2. Since ϕk (Rk , R−k ) P< Uk (Rk , R−k ), and since j∈N p(Rj ) < M and p(Rk0 ) ≤0p(Rk ) together imply p(Rk ) + j6=k p(Rj ) < M, it follows from Fact (i) and feasibility
that there exists l ∈ N\{k} such that p(Rl ) ≤ Ul (Rk0 , R−k ) < ϕl (Rk0 , R−k ). Let Rl0 ∈ RR0
be such that p(Rl0 ) = p(R1 ), Rkl= (Rk0 , Rl0 ), and R−kl = (Ri )i∈N\{k,l} . Suppose that agent
l changes his preference from Rl to Rl0 . Since p(Rl0 ) ≤ p(Rl ), the own peak monotonicity
0of the uniform rule implies that Ul (Rkl
, R−kl ) ≤ Ul (Rk0 , R−k ). Since p(Rl ) < ϕl (Rk0 , R−k )000
and p(Rl ) ≤ ϕl (Rk , R−k ), by the uncompromisingness of ϕ (Lemma 5), ϕl (Rkl
, R−kl ) =0000ϕl (Rk , R−k ). Thus Ul (Rkl , R−kl ) < ϕl (Rkl , R−kl ). Then if (Rkl , R−kl ) = (R1 , · · · , R1 ), by000Case 2-1, for all i ∈ N, ϕi (Rkl
, R−kl ) = Ui (Rkl
, R−kl ) = M/n. Therefore Ul (Rkl
, R−kl ) =00ϕl (Rkl , R−kl ). This is a contradiction. If (Rkl , R−kl ) 6= (R1 , · · · , R1 ), we go back to Step 1.
Rk or Rl might be R1 . But both of them are not R1 . Therefore, repeating Step 1 andStep 2, we replace the preference of at least one new agent by R1 . Since there is only
a finite number of agents, we can reach the contradiction at Step 1 or Step 2. Hence,ϕ(R) = U(R).
P
Case 3. M < i∈N p(Ri ).
The same reasoning as in Case 2 can be applied to this case to show ϕ(R) = U(R).
Hence, for all R ∈ RnR , ϕ(R) = U(R).
Q. E. D.
3. 2. Proof of Theorem 2
The basic structure of the proof of Theorem 2 is similar to that of Ching and Serizawa’s (1998) proof. However since Ching and Serizawa (1998) require the domain to
include the single-peaked domain while Theorem 2 only requires the domain to include aminimally rich domain, there are crucial diﬀerences between the two proofs. First, Chingand Serizawa’s (1998) proof depends on Ching’s (1994) characterization on the singlepeaked domain. On the other hand, the proof of Theorem 2 uses Theorem 1 instead.
Secondly, since a minimally rich domain is smaller than the single-peaked domain, thedomain assumed in Theorem 2 may not include some of the single-peaked preferences that
Ching and Serizawa (1998) use in their proof. We overcome this problem by distinguishing
between more cases than Ching and Serizawa (1998) does.
Before starting the proof of Theorem 2, we present two useful lemmas. Lemma 6 below
says that if a rule on a domain including a minimally rich domain satisfies eﬃciency, thensymmetry implies strong symmetry on the minimally rich domain. This result is equivalent
to Lemma 1 in Ching and Serizawa (1998) which was stated for the single-peaked domain.
Lemma 6. Let RR ⊆ RA ⊆ RC . If a rule ϕ on RnA × R++ satisfies symmetry andeﬃciency, then for all (R, M) ∈ RnA × R++ and for all i, j ∈ N such that Ri = Rj ∈RR , ϕi (R, M) = ϕj (R, M).
Proof. Let (R, M) ∈ RnA ×R++ and i, j ∈ N be such that Ri = Rj ∈ RR . By eﬃciency,
ϕi (R, M) ≤ p(Ri ) and ϕj (R, M) ≤ p(Rj ) or ϕi (R, M) ≥ p(Ri ) and ϕj (R, M) ≥ p(Rj ).
Then symmetry and Ri = Rj ∈ RR imply that ϕi (R, M) = ϕj (R, M).
Q. E. D.
We introduce the notion of “convex” preferences.
10Definition 17. A preference R0 ∈ R is convex if p(R0 ) is an interval [p(R0 ), p(R0 )]and for all x, y ∈ R+ such that [y < x < p(R0 )] and [p(R0 ) < x < y], xR0 y.
Let RCONV ⊆ RC be the domain of all convex preferences. We call it the convex
domain.
Lemma 7 below is an extension of Fact (i) in the characterization of eﬃciency. While
Fact (i) pertains to a statement on the single-peaked domain, Lemma 7 pertains to theconvex domain.
Lemma 7.10 Let RCON V ⊆ RA ⊆ RC . If a rule ϕ on RnA × R++ is eﬃcient, then forall (R, M) ∈PRCONV n × R++ , the following properties hold.
(i) If M
P≤ i∈N p(Ri ), forPall j ∈ N, ϕj (R, M) ≤ p(Rj ).
(ii) If Pi∈N p(Ri ) ≤ M ≤ i∈N p(Ri ), for all j ∈ N, p(Rj ) ≤ ϕj (R, M) ≤ p(Rj ).
(iii) If i∈N p(Ri ) ≤ M, for all j ∈ N, p(Rj ) ≤ ϕj (R, M).
P
Proof. First, we prove (i). Suppose M ≤ i∈N p(Ri ). By contradiction, suppose that
there exists agent j ∈ N such that p(Rj ) < ϕj (R, M). By feasibility, there is another agentl ∈ N such that ϕl (R, M) < p(Rl ). Let εj = ϕj (R, M)−p(Rj ), εl = p(Rl )−ϕl (R, M), andε = min{εj , εl }. Let z 0 = (z10 , · · · , zn0 ) be such that zj0 = ϕj (R, M) − ε, zl0 = ϕl (R, M) + ε,and zi0 = ϕi (R, M) for all i ∈ N\{j, l}. Then, z 0 Pareto dominates ϕ(R, M). This is acontradiction.
A symmetric reasoning applies toPprove (iii).
P
Lastly, we prove (ii). Suppose i∈N p(Ri ) ≤ M ≤ i∈N p(Ri ). By contradiction,
supposethat there exists
an agent j ∈ N such that ϕj (R, M) ∈/ [p(Rj ), p(Rj )]. Since
PPi∈N p(Ri ) ≤ M ≤i∈N p(Ri ),Pthere is an allocation z = (z1 , · · · , zn ) such that for alli ∈ N, p(Ri ) ≤ zi ≤ p(Ri ) and i∈N zi = M. Since zj Pj ϕj (R, M) and zi Ri ϕi (R, M) for
all i ∈ N\{j}, z Pareto dominates ϕ(R, M), contradicting eﬃciency.
Q. E. D.
Proof of Theorem 2. Let RR ⊆ RA ⊆ RC . Suppose that a rule ϕ on RnA × R++ isstrategy-proof, symmetric, and eﬃcient. There are two steps.
10

Ching and Serizawa (1998) state a very similar remark on eﬃcient rules on the convex domain,
namely:If a rule P
ϕ on RnCON V is eﬃcient, then for all (R, M ) ∈ RnCONV × R++ , the following properties hold;(i) If M
P≤ i∈N p(Ri ), thenP for all j ∈ N , ϕj (R, M ) ≤ p(Rj ).
(ii) If Pi∈N p(Ri ) ≤ M ≤ i∈N p(Ri ), then for all j ∈ N , p(Rj ) ≤ ϕj (R, M ) ≤ p(Rj ).
(iii) If i∈N p(Ri ) ≤ M , then for all j ∈ N , p(Rj ) ≤ ϕj (R, M ).
However, (i) and (iii) of their remark are false.
To see that (i) is false, we have only to construct an examplePthat there are (R, M ) ∈ RnCON V × R++ ,an eﬃcient allocation z ∈ Z(M ) and j ∈ N such that M ≤ i∈N p(Ri ) and ϕj (R, M ) > p(Rj ). Let
R0 ∈ RCONV be such that x0 < y0 < p(R0 ) < p(R0 ); x0 I0 y 0 for all x0 , y 0 ∈ [x0 , y0 ]; y 0 P0 x0 for all
x0 < y 0 such that x0 , y 0 ∈ (y0 , p(R0 )]; and p(R0 ) − y0 ≤ y0 − x0 . Let ε > 0 be such that x0 < y0 − ε andp(R0 ) < y0 + ε ≤ P
p(R0 ). Let n = 2, M = 2y0 , and R1 = R2 = R0 . Then, the allocation z = (y0 − ε, y0 + ε)
is eﬃcient, M ≤ i∈N p(Ri ), but ϕ2 (R, M ) > p(R2 ).
Similarly, (iii) of Ching and Serizawa’s (1998) remark is false. Although Ching and Serizawa’s (1998)
remark is used in their proof, their proof still works. For our Lemma 7 is enough for their proof. See
Step 2 of the proof of Theorem 2.

11Step 1. First, we show that RA ⊆ RCONV . By contradiction, suppose that there
exists a preference R0 ∈ RA \RCONV . Then there are three points x0 < y0 < z0 such thatx0 P0 y0 and½ z0 P0 y00 . Let
max{x0 ∈ [x0 , y0 ]|x00 I0 x0 } if z0 R0 x0
x∗0 =
and
0
0
0 I0 z0 } otherwise,½ max{x0 0 ∈ [x0 , y0 ]|xmin{z0 ∈ [y0 , z0 ]|z00 I0 x0 } if z0 R0 x0z0∗ =min{z00 ∈ [y0 , z0 ]|z00 I0 z0 } otherwise.
Since R0 is continuous, x∗0 and z0∗ are well-defined. By definition, x0 ≤ x∗0 < y0 < z0∗ ≤z0 , x∗0 I0 z0∗ , and z0∗ P0 x00 for all x00 ∈ (x∗0 , z0∗ ). Let M = nz0∗ . Let R00 ∈ RR be such thatp(R00 ) ∈ (z0∗ , (M − x∗0 )/(n − 1)) and (M − x∗0 )/(n − 1)P00 z0∗ . Let R0 = (R10 , · · · , Rn0 ) be suchthat for all i ∈ N, Ri0 = R00 . By Lemma 6, ϕ(R0 , M) = (z0∗ , · · · , z0∗ ).
Let R1 = R0 . We consider the allocation when agent 1 ∈ N changes his preference
from R10 to R1 . There are two cases.
0, M) = ϕ1 (R0 , M). P
Case 1. ϕ1 (R1 , R−100
In this case, since ϕ1 (R1 , R−1, M) = z0∗ , i6=1 ϕi (R1 , R−1, M) = (n − 1)z0∗ . By Lemma
06, for all i ∈ N\{1}, ϕj (R1 , R−1, M) = z0∗ . Consider the allocation (x∗0 , (M − x∗0 )/(n −1), · · · , (M − x∗0 )/(n − 1)). Since x∗0 I1 z0∗ and (M − x∗0 )/(n − 1)Pj0 z0∗ for all j ∈ N\{1}, the0allocation (x∗0 , (M −x∗0 )/(n−1), · · · , (M −x∗0 )/(n−1)) Pareto dominates ϕ(R1 , R−1, M) =∗∗(z0 , · · · , z0 ), contradicting eﬃciency.
0, M) 6= ϕ1 (R0 , M)(= z0∗ ).
Case 2. ϕ1 (R1 , R−1
There are also four subcases. Let r ∈ R+ be such that rI00 z0∗ and r 6= z0∗ . If there does
not exist such r, we omit Case 2-1 below.
0, M).
Case 2-1. r ≤ ϕ1 (R1 , R−100
0
Let R1 ∈ RR be such that p(R100 ) ≥ ϕ1 (R1 , R−1, M). Corollary 1 tells us that for all00i ∈ N, ϕi (R100 , R−1, M) = Ui (R100 , R−1
, M).
Since
M
=
nz0∗ , z0∗ < p(R100 ), and z0∗ < p(Ri0 ) forP
all i ∈ N, it follows that M < p(R100 ) + i6=1 p(Ri0 ). Thus by the definition of the uniform
000rule, ϕi (R100 , R−1, M) = z0∗ for all i ∈ N. Since ϕ1 (R1 , R−1, M)P100 ϕ1 (R100 , R−1, M), agent 1
00
0
manipulates ϕ at (R1 , R−1 , M) via R1 , contradicting strategy-proofness.
0, M) ∈ (z0∗ , r).
Case 2-2. ϕ1 (R1 , R−10
By the definition of r, ϕ1 (R1 , R−1, M)P10 ϕ1 (R0 , M). Thus agent 1 manipulates ϕ at
0
(R , M) via R1 , contradicting strategy-proofness.
0, M) < z0∗ .
Case 2-3. x∗0 < ϕ1 (R1 , R−10∗0
Since ϕ1 (R , M) = z0 , ϕ1 (R0 , M)P1 ϕ1 (R1 , R−1, M). Thus agent 1 manipulates ϕ at
0
(R1 , R−1
, M) via R10 , contradicting strategy-proofness.
0, M) ≤ x∗0 .
Case 2-4. ϕ1 (R1 , R−10, M). Corollary 1 tells us that
Let R100 ∈ RR be such that p(R100 ) ≤ ϕ1 (R1 , R−1
00
0
00
0
00
∗
ϕ(R1 , R−1 , M) = U(R1 , R−1 ,PM). Since p(R1 ) ≤ x0 and for all i ∈ N, z0∗ < p(Ri0 ) <(M − x∗0 )/(n − 1), p(R100 ) + i6=1 p(Ri0 ) < M. By the definition of the uniform rule,000λ(R100 , R−1, M) ≤ z0∗ P. Otherwise, since for all i ∈ N\{1}, z0∗ < λ(R100 , R−1, M) ≤ Ui (R100 , R−1
, M),
00
0
00
0
it follows that M < i∈N Ui (R1 , R−1 , M) = M. It is a contradiction. Therefore, ϕ1 (R1 , R−1, M) =
000
000
M − (n − 1)p(R0 ) and for all i ∈ N\{1}, ϕi (R1 , R−1 , M) = p(Ri ). Thus, since p(R0 ) ∈00(z0∗ , (M − x∗0 )/(n − 1)), ϕ1 (R100 , R−1, M) ∈ (x∗0 , z0∗ ). Since p(R100 ) ≤ ϕ1 (R1 , R−1, M) <
0000, M), ϕ1 (R1 , R−1, M)P100 ϕ1 (R100 , R−1, M). Agent 1 manipulates ϕ at (R100 , R−1, M)
ϕ1 (R100 , R−1
via R1 , contradicting strategy-proofness.
Since we obtain contradictions in all cases, we know that RA ⊆ RCON V .
12Step 2. We further show that RA ⊆ RP . By contradiction, suppose that there
exists a preference R0 ∈ RA \RP . By Step 1, we know that R0 ∈ RCONV \RP . Suppose,
without loss of generality, that there exist two points x0 , y0 ∈ R++ such that p(R0 ) <x0 < y0 , for all x00 ∈ [p(R0 ), x0 ), x00 P0 x0 , and for all x00 ∈ [x0 , y0 ], x00 I0 x0 . Let M = nx0
and R = (R0 , · · · , R0 ). If ϕ(R, M) 6= (x0 , · · · , x0 ), there exist two agents j and k, suchthat ϕPj (R, M) < x0 < ϕk (R, M). Without loss of generality, let j = 1 and k = 2.
Since i∈N p(Ri ) < M, Lemma 7 guarantees that for all i ∈ N, p(R0 ) ≤ ϕi (R, M).
Since ϕ1 (R, M) ∈ [p(R0 ), x0 ) and x0 < ϕ2 (R, M), it follows that ϕ1 (R, M)P1 x0 and
x0 R1 ϕ2 (R, M). By transitivity, ϕ1 (R, M)P1 ϕ2 (R, M), contradicting symmetry. Therefore
ϕ(R, M) = (x0 , · · · , x0 ).
Let ε > 0 be such that x0 − ε ∈ (p(R0 ), x0 ) and x0 + ε ∈ (x0 , y0 ). Then we consider
the allocation (x0 − ε, x0 + ε, x0 , · · · , x0 ). Since (x0 − ε)P1 x0 and (x0 + ε)R2 x0 , (x0 −ε, x0 + ε, x0 , · · · , x0 ) Pareto dominates ϕ(R, M) = (x0 , · · · , x0 ), contradicting eﬃciency.
Therefore, RA ⊆ RP .
To complete the proof of Theorem 2, we need to identify a rule satisfying strategyproofness, symmetry, and eﬃciency on the single-plateaued domain. Ching and Serizawa(1998) extended the uniform rule to the single-plateaued domain, which satisfies the threerequirements. The extended uniform rule, represented by U = (U 1 , · · · , U n ), is defined
as follows:
For all (R, M) ∈ RnP × R++ and all i ∈ N,
min{p(Ri ), λ(R, M)}
min{p(Ri ) + λ(R, M), p(Ri )}U i (R, M) =
max{p(Ri ), λ(R, M)}

where λ(R, M) solves

Pj∈N U j (R, M) = M.

P
if M ≤ j∈N p(Rj )PPj) < M <j∈N p(Rj∈N p(Rj )Pj∈N p(Rj ) ≤ M

4. CONCLUDING REMARKS
We established that the uniform rule is the unique strategy-proof, symmetric, andeﬃcient rule on a minimally rich domain. Weymark (1999) characterizes the uniform
rule when discontinuous single-peaked preference are admitted. He shows that even if
agents have discontinuous single-peaked preferences, the uniform rule is still the uniquerule satisfying strategy-proofness, anonymity, and eﬃciency. Minimally rich domains canbe generalized so as to contain discontinuous single-peaked preference. Therefore, there
is an open question: Does our characterization of the uniform rule still hold on suchgeneralized minimally rich domains?
We established that there is a unique maximal domain including a minimally richdomain for strategy-proofness, symmetry, and eﬃciency, and it is the single-plateaueddomain. To identify the maximal domain, we defined a rule as a function of a preference
profile and the amount of the good, M. Thus, our result of the maximal domain does not
apply when M is fixed and a rule is a function of the preference profile only. Therefore,
our maximal domain result does not exclude the possibility that when M is fixed, there
exist rules satisfying strategy-proofness, symmetry, and eﬃciency on larger domain than
the single-plateaued domain. Massó and Neme (2001) obtain maximal domain for rules
13for which the amount M is fixed. However, they adopt the stronger version of symmetry,
and assume that domains include the single-peaked domain. Therefore, another open
question is: When the amount of the good is fixed, what domain is a maximal domain
including a minimally rich domain for strategy-proofness, symmetry, and eﬃciency?
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