Efficient and strategy-proof allocation mechanism in
economies with many commodities
Takeshi Momi∗
Department of Economics, Doshisha University
February, 2014
Abstract
We show that in pure exchange economies where the number of commodities
equals or exceeds the number of agents, any Pareto-efficient and strategy-proof
allocation mechanism always allocates the total endowment to some single agent
even if the receivers may vary.
JEL classification: D71
Keywords: Social choice, Strategy-proofness, Pareto-efficiency, Exchange economy
.
1
Introduction
Since the seminal work of Hurwicz (1972), the manipulability and efficiency of allocation
mechanisms in pure exchange economies has been intensively studied. Zhou (1991) established that any Pareto-efficient and strategy-proof allocation mechanism is dictatorial in
exchange economies with two agents having classical (i.e., continuous, strictly monotonic,
and strictly convex) preferences. The dictatorship result in two-agent economies has been
strengthened by proving the result on the domain of restricted preferences.1
Compared to the result in two-agent economies, it has been an open question about
how Pareto-efficient and strategy-proof allocation mechanisms are in economies with many
∗
Address: Department of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan;
Phone: +81-75-251-3647; E-mail: [email protected]
1
See Schummer (1997), Ju (2003), Hashimoto (2008), and Momi (2013a). Nicolò [10], however, showed
a Pareto-efficient, strategy-proof, and non-dictatorial mechanism on the domain of Leontief preferences.
1
agents.2 This is what we want to study in this paper. In many-agent economies, there
actually exist Pareto-efficient, strategy-proof, and no-dictatorial allocation mechanisms.
Satterethwaite and Sonnenschein (1981) constructed such a mechanism, relying on the
reverse dictator’s preference, to select one among the remaining agents, whois allocated
the total endowment. Kato and Ohseto (2002) showed that no reverse dictator is needed
in economies with four and more agents, by constructing a mechanism where all agents
have the opportunities to be allocated the total endowment. A specific feature shared by
all already-known Pareto-efficient and strategy-proof allocation mechanisms is that some
single agent receives the whole amount of commodities even if the receivers may vary.
Such a mechanism is called alternately-dictatorial. The natural question to be asked
is whether there exists a Pareto-efficient, strategy-proof, and non-alternately-dictatorial
allocation mechanism.
Recently, Momi (2013b) showed that any Pareto-efficient and strategy-proof mechanism in three-agent economies is either dictatorial or Satterthwait and Sonnenschein’s
type, that is, alternately-dictatorial. In this paper, we show that in exchange economies
where the number of commodities equals or exceeds the number of agents, any Paretoefficient and strategy-proof allocation mechanism is alternately-dictatorial. We believe
our approach and result would be a foothold toward the general question.
In the rest of this section, we briefly sketch our approach highlighting the role played
by the condition on the numbers of commodities and agents. The tractability of two-agent
economies comes from that an agent’s consumption vector determines the other’s because
they sum up to the total endowment under a Pareto-efficient mechanism. If the number
of agents does not exceed the number of commodities, this tractability is maintained to
some extent in many-agent economies. An agent’s consumption vector determines the
marginal rate of the substitution of his preference at the consumption, which should
be shared by all agents at the allocation given by a Pareto-efficient mechanism. If the
preferences are homothetic, the marginal rate of substitution determines each agent’s
direction of consumption vector. Furthermore, if the directions of consumption vectors
are independent, which is possible only when the number of agents does not exceeed the
number of commodities, then the consumption vectors themselves should be determined
uniquely so that they sum up to the total endowment.
Our approach is a substantial extension of that by Hashimoto (2008) and Momi (2013a,
b). They showed that in two-agent economies, it is possible to exactly grasp how an
2
Some researchers showed incompatibility of Pareto-efficiency and strategy-proofness with allocation
restrictions. Serizawa (2002) showed the incompatibility with the individual rationality restriction. Serizawa and Weymack (2003), Goswami et al. (2013), and Momi (2013b) showed the incompatibility with
positivity restrictions. On the other hand, Barberà and Jackson (1995) characterized strategy-proof
mechanisms satisfying the individual rationality restriction. These restrictions are however so strong to
exclude the dictatorial allocation mechanism.
2
agent’s consumption changes according to a change in his preference under a Paretoefficient and strategy-proof allocation mechanism. Though we cannot explicitly pin down
the consumption change in many-agent economies, we can still show some qualitative
properties such as continuity.
The rest of the paper is organized as follows. Section 2 describes the model and the
main results. Section 3 explains our approach and shows some properties of the allocation
mechanism. Sections 4 and 5 provide the proofs of the results in Section 2.
2
Model and results
We consider an economy with N agents indexed by N = {1, . . . , N } where N ≥ 2, and
L goods indexed by L = {1, . . . , L} where L ≥ 2. The consumption set for each agent
L
L
is R+
. A consumption bundle for agent i ∈ N is a vector xi = (xi1 , . . . , xiL ) ∈ R+
. The
L
total endowment of goods for the economy is Ω = (Ω1 , . . . , ΩL ) ∈ R++ . An allocation is
LN
. The set of feasible allocations for the economy with N
a vector x = (x1 , . . . , xN ) ∈ R+
agents and L goods is thus
LN
X = x ∈ R+
xi ≤ Ω .
i∈N
L
. The
A preference R is a complete, reflexive, and transitive binary relation on R+
corresponding strict preference PR and indifference IR are defined in the usual way. For
L
, xPR x implies that xRx and not x Rx, and xIR x implies that xRx
any x and x in R+
L
and x Rx. Given a preference R and a consumption bundle x ∈ R+
, the upper contour
L set of R at x is UC(x; R) = {x ∈ R+ |x Rx} and the lower contour set of R at x is
L
L LC(x; R) = {x ∈ R+
|xRx }. Further, we let I(x; R) = {x ∈ R+
|x IR x} denote the
L |x PR x} denote the strictly preferred
indifference set of R at x, and P (x; R) = {x ∈ R+
set of R at x.
L
A preference R is continuous if UC(x; R) and LC(x; R) are both closed for any x ∈ R+
.
L
L
if UC(x; R) is strictly convex for any x ∈ R++
.
A preference R is strictly convex on R++
L
L
A preference R is strictly monotonic on R++ if for any x and x in R++ , x > x implies
L
and any t > 0, xRx
that xPR x . A preference R is homothetic if for any x and x in R+
L
implies that (tx)R(tx ). A preference R is smooth if for any x ∈ R++
, there exists a
L−1
L
unique vector p ∈ S++ ≡ {x ∈ R++ | x = 1} such that p is the normal of a supporting
hyperplane to UC(x; R) at x. We call vector p, the gradient vector of R at x
L
L
We call a preference classical when it is continuous on R+
, strictly convex in R++
, and
L
strictly monotonic on R++ , and let RC denote the set of classical preferences. Further,
we let R denote the set of classical, smooth, and homothetic preferences. In this paper,
we prove the results on the restricted domain R and then extend them to RC . To discuss
the continuity in R, we introduce Kannai’s topology into R following Kannai (1970). As
3
mentioned in Momi (2013), any preference R ∈ R is identified by one indifference set in
L
R++
because the other indifference sets of the homothetic preference should be determined
by similarity transformations.
A preference profile is an N-tuple R = (R1 , . . . , RN ) ∈ RN . The subprofile obtained
by removing Ri from R is R−i = (R1 , . . . , Ri−1 , Ri+1 , . . . , RN ). We often write the profile
(R1 , . . . , Ri−1 , R̄i , Ri+1 , . . . , RN ) as (R̄i , R−i ).
A social choice function f : RN → X assigns a feasible allocation to each preference
profile in RN . For a preference profile R ∈ RN , the outcome chosen can be written as
f (R) = (f 1 (R), . . . , f N (R)), where f i (R) is the consumption bundle allocated to agent
i by f .
Definition 1.
A social choice function f is strategy-proof if f i (R)Ri f i (R̄i , R−i ) for
any i ∈ N, any R ∈ RN , and any R̄i ∈ R.
A feasible allocation is Pareto efficient if there is no other feasible allocation that would
benefit someone without worsening anyone else. That is, x ∈ X is Pareto efficient for
preference profile R if there exists no x̄ ∈ X such that x̄i Ri xi for any i ∈ N and x̄j PRj xj
for some j ∈ N. We say that a social choice function is Pareto efficient if it always assigns
a Pareto-efficient allocation.
Definition 2. A social choice function f is Pareto efficient if f (R) is Pareto efficient
for any R ∈ RN .
We say that a social choice function is dictatorial if there exists an agent who is always
allocated the total endowment.
Definition 3. A social choice function f is dictatorial if there exists i ∈ N such that
f i (R) = Ω for any R ∈ RN .
We say that a social choice function is alternately-dictatorial if it always allocates
the total endowment to some single agent. Note the difference from the previous definition: under an alternately-dictatorial social choice function, the receivers of the total
endowment may vary depending on preference profiles.
Definition 4. A social choice function f is alternately-dictatorial if for any R ∈ RN ,
there exists iR ∈ N such that f iR (R) = Ω.
This paper’s main result is as follows.
Theorem. When L ≥ N, a Pareto-efficient and strategy-proof social choice function
f : RN → X is alternately-dictatorial.
4
This is proved on the preference domain R. Let R̄ be a preference domain such that
R ⊂ R̄ ⊂ RC , and let us redefine Definitions 1–4 on R̄.
Corollary. When L ≥ N, a Pareto-efficient and strategy-proof social choice function
f : R̄N → X is alternately-dictatorial.
The proofs of the therem and corollary are given in Sections 4 and 5. As the final
remark in this section, we refer to already-known techniques we use in this paper. For a
L
preference R ∈ R and a consumption bundle x ∈ R+
, a preference R̄ is called a Maskin
monotonic transformation (MMT, hereafter) of R at x if x̄ ∈ UC(x; R̄) and x̄ = x implies
that x̄PR x. It is well known that if an agent receives x at a preference profile R, the
strategy-proofness implies that this agent receives the same commodity bundle x when
his preference is subject to an MMT at x. As shown in Momi (2013b, Lemma 4), for a
L
preference R ∈ R and a consumption bundle x ∈ R++
, there exists a preference that is an
MMT of R at x in any neighborhood of R with respect to the Kannai topology introduced
L
in R. We also use an MMT of two preferences. For any x ∈ R+
\ 0, we let [x] denote the
ray starting from 0 and passing through x. For two preferences R and R̃ in R and two
L
consumption bundles x and x̃ in R++
, if x ∈ P (I(x; R) [x̃]; R̃) holds, then there exists a
preference R̄ ∈ R that is an MMT of R at x and an MMT of R̃ at x̃. See Momi (2013b,
Proposition 2) for details.
3
Properties of the social choice function
We assume a strategy-proof and Pareto-efficient social choice function f . For agent i,
L
when the other agents’ preferences R̄−i are fixed, we define Gi (R̄−i ) ⊂ R+
as the union
of his consumption bundles given by f over his preferences,
f i(Ri , R̄−i ).
Gi (R̄−i ) =
Ri ∈R
Under the strategy-proof social choice function, f i (Ri , R̄−i ) should be the most preferable
consumption in Gi (R̄−i ) with respect to Ri . We approach to the social choice function f
by investigating the shape of the set Gi (R̄−i ). In two trivial cases, Gi (R̄−i ) are given by
single-point sets because of the strategy-proofness. If f i (Ri , R̄−i ) = 0 for some Ri , then
Gi (R̄−i ) = 0. If f i (Ri , R̄−i ) = Ω for some Ri , then Gi (R̄−i ) = Ω. We hereafter investigate
the case where f i (Ri , R̄−i ) = {0, Ω}. Roughly speaking, we show that Gi (R̄−i ) is the L−1
dimensional smooth surface of a convex set.
At an allocation given by a Pareto-efficient social choice function f , any agent shares
the same gradient vector at his consumption as long as it is positive and the gradient
vector is well defined. We call this vector, the price vector at the allocation f (R) and
L−1
write p(R, f ) ∈ S++
.
5
L−1
On the other hand, for a preference R ∈ R and a price vector p ∈ S++
, we let
L−1
g(R, p) ∈ S++ denote the normalized consumption vector where the gradient vector of
R is p. In particular, we call g(Ri , p(R, f )), agent i’s consumption-direction vector at
the preference profile R under f because his consumption f i (R) should be on the ray
[g(Ri , p(R, f ))]. We can write f i (R) = f i (R) g(Ri, p(R, f )).
In this section we focus on preference profile R̄ such that the consumption-direction
vectors are independent. The role of this independency would be clear. Since consumption
vectors f i (R̄), i = 1, . . . , N, are on the ray [g(R̄i, p(R̄, f ))], respectively, and they sum
up to the total endowment Ω, the consumption vectors should be determined uniquely
under the independency. Note that we need L ≥ N for the independency.
Lemma 1. Suppose that a social choice function f is Pareto-efficient and g(R̄i , p(R̄, f )),
i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ). Let R̃ = (R̃1 , . . . , R̃N ) be a
preference profile such that each R̃i has the same gradient vector as R̄i at g(R̄i , p(R̄, f )).
If f j (R̃) = f j (R̄) = 0 holds for an agent j, then f (R̃) = f (R̄).
Proof. Since f j (R̃) = f j (R̄) = 0 and R̃j and R̄j have the same gradient vector at
f j (R̄), we have p(R̄, f ) = p(R̃, f ) and g(R̄i, p(R̄, f )) = g(R̃i , p(R̃, f )) for any i.
Since consumption at preference profiles R̄ and R̃, respectively, sum up to the toN i
N
i
i
tal endowment Ω, we have
f
(
R̄)
=
i=1
i=1 f (R̄) g(R̄ , p(R̄, f )) = Ω and
N i
N
i
i
i=1 f (R̃) =
i=1 f (R̃) g(R̄ , p(R̃, f )) = Ω. Considering the difference be
tween these equations, we have i=j ( f i (R̄) − f i (R̃) )g(R̄i, p(R̄, f )) = 0 because
f j (R̄) = f j (R̃) and g(R̄i, p(R̄, f )) = g(R̃i , p(R̃, f )). Since the consumption direction
vectors are independent, this equation implies that f i (R̄) = f i (R̃) for all i = j.
Thus, we obtain the equality of the two allocations: f (R̄) = f (R̃).
As mentioned, f i (R) should be the most preferable consumption bundle in Gi (R−i )
with respect to Ri . For a while, we rather consider the most preferable consumption
bundle in its closure Gi (R−i ) for tractability. Essential coincidence will be shown later.
Lemma 2. We let f be a strategy-proof social choice function. If x is the most preferable
consumption bundle in Gi (R−i ) with respect to Ri , then x ∈ Gi (R−i ). In particular, if
R̂i is an MMT of Ri at x, then f i (R̂i , R−i ) = x.
Proof. We let R̂i be an MMT of Ri at x and prove that x = f i (R̂i , R−i ). Since R̂i is
an MMT of Ri , UC(x; R̂i ) \ x ⊂ P (x; Ri). Therefore, x is the unique most preferable
consumption bundle in Gi (R−i ) with respect to R̂i .
Since x ∈ Gi (R−i ), there exists x̂ ∈ Gi (R−i ) arbitrarily close to x. Thus, if x is
strictly preferred to f i (R̂i , R−i ) with respect to R̂i , it contradicts the strategy-proofness
of f . Therefore, f i (R̂i , R−i ) ∈ UC(x; R̂i ). Then, f i (R̂i , R−i ) = x contradicts that x is the
6
unique most preferable consumption bundle in Gi (R−i ) with respect to R̂i . Therefore,
x = f i (R̂i , R−i ) ∈ Gi (R−i ).
It is thus clear that if x is the unique most preferable consumption bundle in Gi (R−i )
with respect to Ri , then x = f i (R). We now show that the reverse too holds: f i (R) is
the unique most preferable consumption bundle in Gi (R−i ) with respect to Ri .
Lemma 3. We let f be a Pareto-efficient and strategy-proof social choice function.
We suppose g(R̄i, p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and
f i (R̄) = {0, Ω} for an agent i. Then, f i (R̄) is the unique most preferable consumption
bundle in Gi (R̄−i ) with respect to R̄i .
Proof. Without loss of generality, we prove the statement for agent 1. We write
f (R̄) = x̄ = (x̄1 , . . . , x̄N ) and p̄ = p(R̄, f ). From the definitions, x̄1 is (one of) the
most preferable consumption bundles in G1 (R̄−1 ) with respect to Ri and no consumption
bundle in G1 (R̄−1 ) is strictly preferred to x̄1 with respect to R̄1 .3 We suppose that there
exists a consumption bundle x̃1 in G1 (R̄−1 ) indifferent to x̄1 with respect to R̄1 , and show
a contradiction.
We let R̃1 be an MMT of R̄1 at x̃1 . As in Lemma 2, x̃1 = f 1 (R̃1 , R̄−1). We write
f (R̃1 , R̄−1 ) = x̃ = (x̃1 , . . . , x̃N ) and let p̃ = p((R̃, R̄−1), f ) denote the price vector at this
allocation. See Figure 1.
p̃
R̃1
x2(t)
p̃
tx̃1
R̄2
x̃1
G1 (R̄−1 )
p̃
p̄
x̄1
g(R̂2 , p̃)
R1
R̄1
R̄2
p̄
x̄2
R̂2
Figure 1. Proof of Lemma 3
3
As in the proof of Lemma 2, if there exists x ∈ G1 (R̄−1 ) that is strictly prefered to x̄ with respect
to R̄1 , then there exists x̂ ∈ G1 (R̄−1 ) that is strictly prefered to x̄ with respect to R̄1 .
7
We construct agent 1’s new preferences R1(t) as follows. In addition to R̃1 , which is
an MMT of R̄1 at x̃1 , we pick R1 that is an MMT of R̄1 at x̄1 . Let > 0 be sufficiently
small. For t ∈ (1 − , 1) sufficiently close to 1, we define B(t) as the convex hull of the set
UC(tx̃1 ; R̃1 ) UC(x̄1 ; R1 ):
1
1
1
1
B(t) = co UC(tx̃ ; R̃ ) UC(x̄ ; R ) .
L−1
,
Using B(t), we slightly modify the indifferent set I(x̄1 ; R̄1 ). Note that for any x ∈ S++
1
1
the ray [x] intersects I(x̄ ; R̄ ) and the boundary of B(t), ∂B(t), only once. We fix a
scalar s̄ < 1 sufficiently close to 1 and define
s̄ [x] I(x̄1 ; R̄1 ) + (1 − s̄) [x] ∂B(t) .
I(t) =
L−1
x∈S++
We let R1(t) ∈ R be agent 1’s preference that has I(t) as its indifference set.4 It
is clear from the construction that R1(t) and R̄1 have the same gradient vectors at x̃1
and x̄1 , respectively. Observe that R1(t) is an MMT of R̄1 at x̃1 . Further, observe that
UC(x̄1 ; R1(t) ) LC(x̄1 ; R̄1 ) consists of the point x̄1 and a set in a neighborhood of x̃1 ,
which converges to x̃1 as t converges to 1.
We write f (R1(t) , R̄−1 ) = x(t) = (x1(t) , . . . , xN (t) ). Since R1(t) is an MMT of R̄1 at x̃1 ,
x1(t) = x̃1 .
L
We observe that we can pick a small positive vector α ∈ R++
such that for any
t ∈ (1 − , 1), there exists an agent it ∈ {2, . . . , N } whose consumption xit (t) is strictly
preferred to x̄it + α with respect to R̄it : xit (t) ∈ P (x̄it + α; R̄it ). Contrary to this claim,
L
, x̄i + α ∈ UC(x̄i(t) ; R̄i) holds for any i = 2, . . . , N at
suppose that for any vector α ∈ R++
i
i(t)
is strictly preferred to xi(t)
some t ∈ (1 − , 1). Then, with a sufficiently small α, x̄ +x2
with respect to R̄i for any i = 2, . . . , N by the strict convexity of the preferences. Since
x̄1 +x̃1(t)
is also preferred to x1(t) with respect to R1(t) for any t sufficiently close to 1, this
2
contradicts the Pareto efficiency of x(t) .
For each i = 2, . . . , N, we let Ti denote the set of t such that xi(t) is strictly preferred
to x̄i + α with respect to R̄i : Ti = {t ∈ (1 − , 1)|xi(t) ∈ P (x̄i + α; R̄i)}. Note that there
exists some i such that Ti [t, 1) = ∅ for any t ∈ (1 − , 1). If there does not exist such
i, then for each i = 2, . . . , N, there exists ti such that Ti [ti , 1) = ∅, and there exists
no i satisfying xi(t) ∈ P (x̄i + α, R̄i ) for t ∈ [max{t2 , . . . , tN }, 1), which is a contradiction.
Without loss of generality, we assume that agent 2 is such an agent: T2 [t, 1) = ∅ for any
t ∈ (1 − , 1). From now on, we only consider t in T2 . In particular, we select a sequence
of t converging to 1 in T2 .
We pick R̂2 ∈ R such that (i) R̂2 is an MMT of R̄2 at x̄2 ; (ii) x2(t) ∈ P (x̄2 ; R̂2 ) for
/ H(g(R̄2, p̃), . . . , g(R̄N , p̃)), where H(g(R̄2, p̃), . . . , g(R̄N , p̃))
t ∈ T2 ; and (iii) g(R̂2 , p̃) ∈
4
See Momi (2013) for the same technique to construct an indifference set and obtain the preference
realizing the indifference set.
8
denotes the N −1 dimensional linear space spanned by g(R̄i , p̃)’s, i = 2, . . . , N. What (iii)
requires is that, on the L−1 dimensional indifferent surface I(x̄2 ; R̂2 ), the point where the
gradient vector is p̃ is not in the N−2 dimensinal set I(x̄2 ; R̂2 ) H(g(R̄2, p̃), . . . , g(R̄N , p̃)).
The existence of R̂2 satisfying (i)–(iii) is thus clear.
Observe that f 2 (R̄1 , R̂2 , R̄−{1,2} ) = x̄2 because R̂2 is an MMT of R̄2 at x̄2 , and then
f 1 ((R̄1 , R̂2 , R̄−{1,2} ) = x̄1 as shown in Lemma 1.
We write f (R1(t) , R̂2 , R̄−{1,2} ) = x̂(t) = (x̂1(t) , . . . , x̂N (t) ) and focus on x̂1(t) for t ∈ T2 .
Facing the other agents’ preferences (R̂2 , R̄−{1,2} ), agent 1 can achieve x̄1 by reporting
R̄1 as just mentioned. Therefore, x̂1(t) should be preferred to x̄1 with respect to R1(t) .
On the other hand, x̂1(t) is not strictly preferred to x̄1 with respect to R̄1 . Therefore,
x̂1(t) ∈ UC(x̄1 ; R1(t) ) LC(x̄1 ; R̄1 ). Remember that this set consists of the point x̄1 and
a set in a neighborhood of x̃1 , which converges to x̃1 as t converges to 1. We investigate
these two cases.
We consider the case where x̂1(t) = x̄1 . Since R1(t) and R̄1 have the same gradient vector
at x̄1 and R̂2 and R̄2 have the same gradient vector at x̄2 , this implies that x̂(t) = x̄ as
shown in Lemma 1. In particular, x̂2(t) = x̄2 . Remember that f 2 (R1(t) , R̄2 , R̄−{1,2} ) = x2(t)
and we have chosen R̂2 so that x2(t) ∈ P (x̄2 ; R̂2 ) for any t ∈ T2 . If agent 2 has preference
R̂2 and faces the other agents’ preferences (R1(t) , R̄−{1,2} ), he can be better off by reporting
R̄2 and achieving x2(t) than reporting his true preference R̂1 and achieving x̂1(t) = x̄1 . This
is a contradiction.
(t)
We now consider the case where x̂1 is in the set in a neighborhood of x̃1 that convervges to x̃1 as t converges to 1. In this case, as t converges to 1, the gradient vector of
R1(t) at x̂1(t) converges to p̃, and then, the price vector at x̂(t) and the gradient of R̂2 at
x̂2(t) also converge to p̃.
i
However, note the equation x̂2(t) = x̃2 + i=2 (x̃i − x̂i(t) ) obtained from N
i=1 x̃ = Ω
N i(t)
and i=1 x̂
= Ω. In the right-hand side of this equation, x̃1 − x̂1(t) → 0 as t → 1,
x̃i ∈ [g(R̄i, p̃)] for i ≥ 2, and x̂i(t) converges to a point on [g(R̄i , p̃)] for i ≥ 3. Therefore,
as t → 1, x̂2(t) converges to a point in H(g(R̄2, p̃), . . . , g(R̄N , p̃)), and the gradient vector
of R̂2 at x̂2(t) does not converge to p̃ because of (iii). This contradicts the discussion in
the previous paragraph.
We can prove that f (·, R̄−i ) is a continuous function of Ri in a neighborhood of R̄i .
Proposition 1. We let f be a Pareto-efficient and strategy-proof social choice function.
We suppose g(R̄i, p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and
f i (R̄) = {0, Ω} for an agent i. Then, f (·, R̄−i ) is a continuous function in a neighborhood
of R̄i .
Proof. We only have to prove that if the consumption-direction vectors g(R̄i , p(R̄, f )),
i = 1, . . . , N, are independent at R̄ and f i (R̄) = {0, Ω} for an agent i, then f (·, R̄−i ) is a
9
continuous function at R̄i . Suppose the claim and let Ri be a preference sufficiently close to
R̄i . Then, f (Ri , R̄−i ) is sufficiently close to f (R̄) because of the supposed continuity, and
hence, the consumption-direction vectors of agents at the preference profile (Ri , R̄i) are
still independent and f i (Ri , R̄−i ) = {0, Ω}. Then, f (·, R̄−i) is continuous at Ri because
of the supposed claim.
We first prove that f i (·, R̄−i ) is a continuous function at R̄i . We let {Ri(n) }∞
n=1 be a
i
sequence of preferences converging to R̄ as n → ∞. There exists a convergent subsequence
{f i (Ri(nk ) , R̄−i )}∞
k=1 because of the compactness of the feasible allocation set. We write
f i (Ri(nk ) , R̄−i ) → xi∗ as k → ∞. All we have to show is xi∗ = f i (R̄).
We observe that xi∗ is indifferent to f i (R̄) with respect to R̄i . If xi∗ ∈ P (f i(R̄); R̄i),
then f i (Ri(nk ) , R̄−i ) ∈ P (f i(R̄); R̄i ) for a sufficiently large k. This contradicts strategyproofness. If f i (R̄) ∈ P (xi∗ ; R̄i ), then f i (R̄) ∈ P (f i (Ri(nk ) , R̄−i ); Ri(nk ) ) for a sufficiently
large k because f i(Ri(nk ) , R̄−i ) converges to xi∗ as k → ∞ and P (x; Ri(nk ) ) converges to
P (x; R̄i ) at any consumption x as k → ∞. This again contradicts strategy-proofness.
Thus, xi∗ is indifferent to f i (R̄) with respect to R̄i . On the other hand, xi∗ ∈ G1 (R̄−i )
because f i (Ri(nk ) , R̄−i ) ∈ Gi (R̄−i ) for any k. Then, xi∗ = f i (R̄) by Lemma 3.
We next prove the continuity of f j (·, R̄−i ) at R̄i , j = i. We proved the continuity of
i
f i (·, R̄) at R̄i . That is, if {Ri(n) }∞
n=1 is a sequence of preferences converging to R̄ as n →
∞, then f i (Ri(n) , R̄−i ) converges to f i (R̄). According to this convergence, the gradient
vector of Ri(n) at f i (Ri(n) , R̄−i ) converges to the gradient vector of R̄i at f i (R̄), that is,
p((Ri(n) , R̄−i ), f ) converges to p(R̄, f ), and hence, g(R̄j , p((Ri(n) , R̄−i ), f )) converges to
g(R̄j , p(R̄, f )) for any j = i.
As in the proof of Lemma 1, f j (R) = f j (R) g(Rj , p(R, f )) for any preference
profile R and the consumption summed up over agents equals the total endowment. Thus,
we have two equalities: f i(R̄) + j=i f j (R̄) g(Rj , p(R̄, f )) = Ω and f i (Ri(n) , R̄−i ) +
j
i(n)
, R̄−i ) g(Rj , p((Ri(n) , R̄−i ), f )) = Ω for any n. Considering the difference
j=i f (R
between these equalities, we have
0 = f i (R̄) − f i (Ri(n) , R̄−i )
+
f j (R̄) g(Rj , p(R̄, f )) −
f j (Ri(n) , R̄−i ) g(Rj , p((Ri(n) , R̄−i ), f ))
j=i
i
j=i
i
i(n)
−i
= f (R̄) − f (R , R̄ )
+
( f j (R̄) − f j (Ri(n) , R̄−i ) )g(Rj , p(R̄, f ))
j=i
+
f j (Ri(n) , R̄−i ) (g(Rj , p(R̄, f )) − g(Rj , p((Ri(n) , R̄−i ), f ))).
j=i
As n → ∞, the first and the third elements in the last equation converge to 0. Since
g(R̄j , p(R̄, f )), j = i are independent vectors, f j (Ri(n) , R̄−i ) converges to f j (R̄) for any j = i. This implies that f j (Ri(n) , R̄−i ) converges to f j (R̄) for any j as n → ∞.
10
That is, f j (·, R̄−i ) is continuous at R̄i for any j = i.
L−1
For a price vector p ∈ S++
, we let p⊥ denote the plane perpendicular to p: p⊥ = {x ∈
RL |px = 0}. For subsets A and B in RL , we let A + B = {a + b ∈ RL |a ∈ A, b ∈ B}
and A − B = {a − b ∈ RL |a ∈ A, b ∈ B}. We can show that Gi (R̄−i ) is a surface of a
convex set in the sense that Gi (R̄−i ) is in the “lower-left side” of the spaces separated by
the hyperplane f i (Ri , R̄−i ) + p((Ri , R̄−i ), f )⊥ .
Proposition 2. We let f be a Pareto-efficient and strategy-proof social choice function.
Suppose g(R̄i , p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and f i (R̄) =
{0, Ω} for an agent i. Then, for any Ri in a neighborhood of R̄i , Gi (R̄−i ) ⊂ f i (Ri , R̄−i ) +
L
p((Ri , R̄−i ), f ))⊥ − R+
.
Proof. All we have to prove is that if the consumption-direction vectors g(R̄i , p(R̄, f )),
i = 1, . . . , N, are independent at R̄ and f i(R̄) = {0, Ω} for an agent i, then the statement
of the proposition holds at the preference profile R̄. Suppose this claim. If Ri is in a
neighborhood of R̄i , then f (Ri , R̄−i ) is in a neighborhood of f (R̄) by Proposition 1 and the
price vectors at these two allocations are close. Then, the consumption-direction vectors
of agents at the preference profile (Ri , R̄−i ) are still independent and f i (Ri , R̄−i ) = {0, Ω},
and thus, we have the statement of the proposition at (Ri , R̄−i ) by the supposed claim.
L
We suppose that Gi (R̄−i ) has an intersection with f i (R̄) + p(R̄, f ))⊥ + R++
and show
i(t)
a contradiction. For a parameter t ∈ (0, 1], we let R be agent i’s preference that has
I(t) =
t [x] I(f i(R̄); R̄i ) + (1 − t) [x] (f i (R̄) + p(R̄, f )⊥ )
L−1
x∈S++
as its indifference set. Observe that Ri(1) = R̄i ; Ri(t) is an MMT of Ri(t ) at f i (R̄) for t > t ;
the indifference set I(f i(R̄); Ri(t) ) becomes flatter as t → 0 except in a neighborhood of
the boundary of the consumption set. Clearly, f i (R̄) is an intersection between Gi (R̄−i )
and UC(f i (R̄; Ri(t) ) for any t, and it is the unique intersection for t = 1.
L
, UC(f 1 (R̄); R1(t) ) has
When Gi (R̄−i ) has an intersection with f i (R̄)+p(R̄, f ))⊥ +R++
L
an intersection with Gi (R̄−i ) in f i(R̄)+p(R̄, f ))⊥ +R++
for a small t. Let t be the largest t
L
.
such that UC(f 1 (R̄); R1(t) ) has an intersection with Gi (R̄−i ) in f i(R̄) + p(R̄, f ))⊥ + R++
Then, with respect to Ri(t ) , there exist two most preferable consumption bundles in
Gi (R̄−i ). This contradicts Lemma 3.
If Gi (R̄−i ) is the surface of a strictly convex set, then f i (Ri , R̄−i ) is the unique intersection between Gi (R̄−i ) and the hyperplane f i (Ri , R̄−i ) + p((Ri , R̄−i ), f )⊥ . Though
we cannot insist this, the intersection between Gi (R̄−i ) and the hyperplane f i (Ri , R̄−i ) +
p((Ri , R̄), f )⊥ has less than N − 2 dimensions. We let H({g(Rj , p((Ri , R̄−i ), f ))}j=i )
denote the N − 1 dimensional linear space spanned by g(Rj , p((Ri , R̄−i ), f ))s, j = i.
11
Proposition 3. Suppose g(R̄i, p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N )
and f i (R̄) = {0, Ω} for an agent i. For any Ri in a neighborhood of R̄i ,
Gi (R̄−i ) (f i (Ri , R̄−i ) + p((Ri , R̄−i ), f )⊥ )
⊂ f i (Ri , R̄−i ) + H({g(Rj , p((Ri , R−i ), f ))}j=i) p((Ri , R̄−i ), f )⊥ ,
where H({g(Rj , p(Ri , R−i ), f ))}j=i) p((Ri , R̄−i ), f )⊥ is a linear space of at most N − 2
dimensions.
Proof. As in the proof of Proposition 2, we only have to prove that if the consumptiondirection vectors are independent at R̄ and f i (R̄) = {0, Ω}, then the statement of the
proposition holds at the preference profile R̄.
Suppose x ∈ Gi (R̄−i ) (f i (R̄) + p(R̄, f )⊥ ) is a consumption bundle different from
f i (R̄). We pick an agent i’s preference R̂i whose gradient vector at x is p(R̄, f ). Then,
f i (R̂i , R̄−i ) = x and p((R̂i , R̄), f ) = p(R̄, f ), and hence, g(R̄j , p((R̂i , R̄), f )) = g j (R̄j , p(R̄, f ))
N j i −i
j j
j
for j = i. Thus, we have
j=1 f (R̂ , R̄ ) = x +
j=i α g (R̄ , p(R̄, f )) = Ω with
some parameters αj , j = i. Considering the difference between this equality and the
j
j
equality N
j=1 f (R̄) g(R̄ , p(R̄, f )) = Ω satisfied at the allocation f (R̄), we have
x = f i (R̄) + j=i (αj − f j (R̄) )g(R̄j , p(R̄, f )). This proves that Gi (R̄−i ) (f i(R̄) +
p(R̄, f )⊥ ) ⊂ f i (R̄) + H({g(Rj , p(R, f ))}j=i), and hence, Gi (R̄−i ) (f i (R̄) + p(R̄, f )⊥ ) ⊂
f i (R̄) + H({g(Rj , p(R̄, f ))}j=i) p(R̄, f )⊥ .
Note that p(R̄, f )⊥ is an L − 1 dimensional plane and H({g(Rj , p(R, f ))}j=i) is an
N − 1 dimensional plane in RL . Since p(R̄, f ) and g(R̄j , p(R̄, f )), j = i, are all positive
vectors, the intersection of the two planes has at most N − 2 dimensions.
We show that Gi (R̄−i ) is a smooth surface in the sense that at each point f i (Ri , R̄−i ),
the hyperplane tangent to Gi (R̄−i ) is determined uniquely.
Proposition 4. We let f be a Pareto-efficient and strategy-proof social choice function.
Suppose g(R̄i , p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and f i (R̄) =
{0, Ω} for an agent i. For any Ri in a neighborhood of R̄i , f i (Ri , R̄−i ) + p((Ri , R̄−i ), f )⊥
is the unique hyperplane tangent to Gi (R̄−i ) at f i (Ri , R̄−i ).
Proof. Without loss of generality, we prove the statement for agent 1. As in the proof
of Proposition 2, all we have to prove is that if the consumption-direction vectors are
independent at R̄ and f 1 (R̄) = {0, Ω}, then f 1 (R̄) + p(R̄, f )⊥ is the unique hyperplane
tangent to G1 (R̄−1) at f 1 (R̄).
We suppose g(R̄i, p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and
f 1 (R̄) = {0, Ω}. We write f (R̄) = x̄ = (x̄1 , . . . , x̄N ) and p̄ = p(R̄, f ). As shown in
Proposition 2, x̄1 + p̄⊥ is a hyperplane tangent to G1 (R̄−1 ) at x1 and G1 (R̄−1 ) is in the
lower-left side of this hyperplane. We suppose that G1 (R̄−1 ) has an edge at x̄1 and that
12
there exists another hyperplane x̄1 + p̃⊥ tangent to G1 (R̄−1) at x̄1 with normal vector
L−1
p̃ ∈ S++
, and show a contradiction.
We pick a preference R̃1 ∈ R whose gradient vector at x̄1 is p̃ where x̄1 is the most
preferable consumption bundle in Gi (R̄−1 ) with respect to R̃1 . That is, f 1 (R̃1 , R̄−1 ) = x̄1
and p̃ = p((R̃1 , R̄−1 ), f ). See Figure 2.
p
p̄
G1 (R̄−1 )
x1
x̄1
p
p
p̃
x2
p
y
R̃
1
x2
R̄1
R1
x̄
2
G2 (R̄−2 )
p̄
R2
R̄2
Figure 2. Proof of Proposition 4
Since f 1 (R̄) = {0, Ω}, there exists another agent receiving positive consumption.
Without loss of generality, we assume f 2 (R̄) = 0. We let R2 be agent 2’s preference that
is sufficiently close to R̄2 so that the price vector p = p((R2 , R̄−2 ), f ) is different from
but sufficiently close to p̄.5 Proposition 3 ensures the existence of such a preference. We
write f (R2 , R̄−2) = x = (x1 , . . . , xN ).
We let y = I(x̄1 ; R̄1 ) [x1 ]. That is, y is the point on I(x̄1 ; R̄1 ) where the gradient
vector of R̄1 is p .
We then pick agent 1’s preference R1 such that (i) x̄1 is the most preferable consumption bundle in G1 (R̄−1) with respect to R1 , (ii) the gradient of R1 at x̄1 , which we write as
p , is different from p̄, (iii) y ∈ P (x̄1 ; R1 ), and (iv) g(R1 , p ) and g(R̄i, p ), i = 2, . . . , N,
L−1
be a price
are independent. To observe the existence of such a preference, let p ∈ S++
1
1
−1
vector close to p̄ so that x̄ is the unique intersection of G (R̄ ) and the hyperplane
x̄1 + (p )⊥ , and so that x̄1 and g(R̄i , p ), i = 2, . . . , N are independent. When p is close
to p̄, y is in the upper-right side of the hyperplane x̄1 + (p )⊥ . We let R1 be a preference
5
G2 (R̄−2 ) may also have an edge at x̄2 and f 2 (R2 , R̄−2 ) may be equal to x̄2 .
13
such that its gradient vector at x̄1 is p and y is preferred to x̄1 with respect to R1 . This
R1 satisfies the desired properties. We write f (R1 , R̄−1 ) = x = (x1 , . . . , xN ). Of
course, x1 = x̄1 .
On the other hand, we let R̂2 be agent 2’s preference such that (I) R̂2 is an MMT of
R̄2 at x2 , (II) x2 is the most preferable consumption bundle in G2 (R̄−2 ) with respect
to R̂2 , and (III) the gradient vector of R̂2 at x2 is p . To observe the existence of R̂2 ,
note that when R2 is sufficiently close to R̄2 , x2 + (p )⊥ is sufficiently close to x̄2 +
p̄⊥ . Further, let R2 be a preference such that its gradient vector at x2 is p and the
indifference set I(x2 ; R2 ) is sufficiently close to x2 + (p )⊥ except in a neighborhood of
the boundary of the consumption set. Then, f 2 (R2 , R̄−2 ) = x2 by Proposition 2 and
x2 ∈ P (I(x2 ; R2 ) [x2 ]; R̄2 ). Then, by Proposition 2 in Momi (2013b), there exists a
preference R̂2 that is an MMT of R̄2 at x2 and an MMT of R2 at x2 , and this R̂2 satisfies
the desired properties.
We write f (R̂2 , R̄−2 ) = x̂ = (x̂1 , . . . , x̂N ) and f (R1 , R̂2 , R̄−{1,2} ) = x̌ = (x̌1 , . . . , x̌N ).
Since R̂2 is an MMT of R̄2 at x2 , x̌2 = x2 , and hence, x̌ = x by Lemma 1. On the
other hand, because of (II) and (III), x̂2 = x2 , and hence, x̂ = x . In particular, note
that x̂1 = x1 and x̂1 is on the ray [y].
We first consider the case where x̂1 ∈ P (x̄1 ; R1 ). If agent 1 has preference R1 and
faces other agents’ preference (R̂2 , R̄−{1,2} ), he is better off by reporting R̄1 and achieving
x̂1 than reporting the true preference R1 and achieving x̌1 = x1 = x̄1 . This contradicts
the strategy-proofness of f .
We now consider the case x̂1 ∈
/ UC(x̄1 ; R̄1 ). If agent 1 has preference R̄1 and faces
other agents’ preferences (R̂2 , R̄−{1,2} ), he is better off by reporting R1 and achieving
x̌1 = x1 = x̄1 than reporting his true preference R̄1 and achieving x̂1 . This again
contradicts the strategy-proofness of f .
Since x̂1 ∈ [y] where y ∈ I(x̄1 ; R̄1 ) and y ∈ P (x̄1 ; R1 ), these two cases are all we have
to consider.
Proposition 5. We let f be a Pareto-efficient and strategy-proof social choice function.
Suppose g(R̄i , p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and f i (R̄) =
{0, Ω} for an agent i. In a neighborhood of f i (R̄), Gi (R̄−i ) coincides with Gi (R̄−i ) and
it is an L − 1 dimensional manifold.
Proof. Without loss of generality, we prove the statement for agent 1. We suppose
g(R̄i, p(R̄, f )), i = 1, . . . , N, are independent at R̄ = (R̄1 , . . . , R̄N ) and f 1 (R̄) = {0, Ω}.
We write f (R̄) = x = (x̄1 , . . . , x̄N ) and p(R̄, f ) = p̄.
We consider Cobb–Douglas utility functions uα (x) = xα1 1 · · · xαLL with parameter α =
L−1
and preferences represented by these utility functions. It is esay to
(α1 , . . . , αL ) ∈ S++
observe that the gradient vector of the preference represented by the utility function uα (x)
at a consumption bundle x = (x1 , . . . , xL ) is given by the normalization of ( αx11 , . . . , αxLL ).
14
On the other hand, if a preference represented by a Cobb–Duglas utility function uα (x)
has gradient vector p = (p1 , . . . , pL ) at x = (x1 , . . . , xL ), then the parameter α should be
the normalization of (p1 x1 , . . . , pLxL ).
∗
We let uα (x) be the Cobb–Douglas utility function such that agent 1’s preference
∗
∗
∗
R1α represented by uα (x) has gradient vector p̄ at x̄1 . It is clear that f (R1α , R̄−i ) = x̄
and p(R1∗ , R̄−i ) = p̄. We consider preference R1α represented by a Cobb–Douglas utility
function uα where α is in a neighborhood of α∗ . Then, R1α is, of course, in a neighborhood
∗
of R1α with respect to the topology in R and f 1 (R1α , R̄−i ) is in a neighborhood of x̄1 .
Observe that α = α implies f 1 (R1α , R̄−1) = f (R1α , R̄−1) because the equality of the
consumption f 1 (R1α , R̄−1) = f (R1α , R̄−1) combined with α = α implies that R1α and
R1α have different gradient vectors at the consumption bundle, which is a contradiction
to Proposition 4. Thus, each f 1 (R1α , R̄−1 ) in a neighborhood of x̄1 is identified with the
corresponding parameter α in a neighborhood of α∗ , and hence, in a neighborhood of x̄1 ,
1 1α −1
α f (R , R̄ ) is an L − 1 dimensional manifold.
To end the proof, we prove that α f 1 (R1α , R̄−1 ) coincides with G1 (R̄−1 ) and G1 (R̄−1 )
in a neighborhood of x̄1 . It is clear from the definition that α f (R1α , R̄−1) ⊂ G1 (R̄−1) ⊂
G1 (R̄−1 ). To observe that any element of G1 (R̄−1) in a neighborhood of x̄1 is included in
1 1α −1
1
1
−1
α f (R , R̄ ), note that any ray [y] in a neighborhood of [x̄ ] intersects with G (R̄ ) at
most once because of strategy-proofness, and hence, G1 (R̄−1 ) is at most L−1 dimensional.
4
Proof of Theorem
In the proof, we suppose that there exists a preference profile R̄ = (R1 , . . . , R̄N ) where
at least two agents are allocated positive consumption by a Pareto-efficient and strategyproof social choice function f , and show a contradiction. We first consider the case where
the consumption-direction vectors at R̄ are independent and then consider the case where
they are dependent. Note that we need L ≥ N for the independency of the consumptiondirection vectors.
Part 1. We suppose there exists a preference profile R̄ = (R̄1 , . . . , R̄N ) where consumptiondirection vectors g(R̄i , p(R̄, f )), i = 1, . . . , N, are independent and at least two agents
receive positive consumption. We now show a contradiction.
Without loss of generality, we assume that agents 1 and 2 receive positive consumption.
We write f (R̄) = x̄ = (x̄1 , . . . , x̄N ) and p(R̄, f ) = p̄. Keep in mind that because of the
independency of the consumption-direction vectors, the results in the previous section
hold for agents 1 and 2. Then, even if preferences of agents 1 and 2 are perturbed in a
neighborhood of R̄1 and R̄2 , the independency of the consumption-direction vectors still
holds.
15
We let R̃2 be agent 2’s preference in a neighborhood of R̄2 such that the most preferable consumption bundle in G2 (R̄−1 ) with respect to R̃2 and the gradient vector at the
consumption bundle are different from but sufficiently close to x̄2 and p̄, respectively. We
write f (R̃2 , R̄−2 ) = x̃ = (x̃1 , . . . , x̃N ) and p̃ = p((R̃2 , R̄−2), f ). See Figure 3.
R̄1
p
)
G1 (R̄−1
ȳ(p) = x1
x̄1
G1 (R̃2 , R̄−{1,2}) 1
x̃
R̃2
p̄
p
p̃
p
x̃2
ỹ(p ) = x
1
x2
p̃
p̄
p
x̄2
R̄2
x2
G2 (R̄−2 )
Figure 3. Proof of Theorem
L−1
We now focus on G1 (R̄−1 ) and G1 (R̃2 , R̄−{1,2} ). For a price vector p ∈ S++
in a
1
−1
⊥
neighborhood of p̄, we let ȳ(p) denote the point on G (R̄ ) such that ȳ(p) + p is the
hyperplane tangent to G1 (R̄−1 ) at ȳ(p) and let ỹ(p) denote the point on G1 (R̃2 , R̄−{1,2} )
such that ỹ(p) + p⊥ is the hyperplane tangent to G1 (R̃2 , R̄−{1,2} ) at ỹ(p). As shown in
Proposition 3, the hyperplanes might not be well defined for some p.
We pick a price vector p such that the hyperplanes ȳ(p ) + (p )⊥ and ỹ(p ) + (p )⊥ are
different planes. We observe the existence of such a p . If no such a price exists, then
the hyperplanes ȳ(p) + p⊥ coincides with ỹ(p) + p⊥ for any p in a neighborhod of p̄ as
long as the hyperplanes are well defined. Then G1 (R̄−1 ) coincides with G1 (R̃2 , R̄−{1,2} )
in a neighborhood of x̄1 . This contradicts that with respect to R̄1 , x̄1 is the most preferable consumption bundle in G1 (R̄−1 ) and x̃1 , different from x̄1 , is the most preferable
consumption bundle in G1 (R̃2 , R̄−{1,2} ).
We let R1 and R1 be agent 1’s preferences such that (i) ȳ(p ) is the most preferable
consumption bundle in G1 (R̄−1 ) with respect to R1 , (ii) ỹ(p ) is the most preferable
consumption bundle in G1 (R̃2 , R̄−{1,2} ) with respect to R1 , and (iii) ȳ(p ) ∈ P (ỹ(p ); R1 )
or ỹ(p ) ∈ P (ȳ(p ); R1 ) holds. The existence of such R1 and R1 is straightforward
16
because the hyperplanes ȳ(p ) + (p )⊥ and ỹ(p ) + (p )⊥ , that are tangent to G1 (R̄−1 ) and
G1 (R̃2 , R̄−{1,2} ) at ȳ(p ) and ỹ(p ), respectively, are different. We write f (R1 , R̄−1) = x =
(x1 , . . . , xN ) and f (R1 , R̃r, R̄−{1,3} ) = x = (x1 , . . . , xN ). It is clear that x1 = ȳ 1(p )
and x1 = ỹ 1 (p ) .
We now focus on agent 2’s preference. Note that x2 is the most preferable consumption
bundle in G2 (R1 , R̄−{1,2} ) with respect to R̄2 and the gradient vector of R̄2 at x2 is p .
Further, x2 is the most preferable consumption bundle in G2 (R1 , R̄−{1,2} ) with respect
to R̃2 and the gradient vector of R̃2 at x2 is also p . Now, let R̂2 be agent 2’s preference
such that the gradient vector at x2 is p and the gradient vector at x2 is p , where
p is a price vector sufficiently close to p but different from p . With respect to this
preference R̂2 , the most preferable consumption in G2 (R1 , R̄−{1,2} ) is still x2 and the most
preferable consumption in G2 (R1 , R̄−{1,2} ) is different from x2 but is in a neighborhood
of x2 because of Proposition 2. We let f (R1 , R̂2 , R̄−{1,2} ) = x̂ = (x̂1 , . . . , x̂N ) and
f (R1 , R̂2 , R̄−{1,2} ) = x̌ = (x̌1 , . . . , x̌N ). It is then clear that the price vector at x̂ is p
and x̂ = x because of Lemma 1. The price vector p̂ at x̌ is close to p , hence close to p ,
and then, x̌1 is close to x1 . Thus, x̂1 = x1 ∈ P (x̌1 ; R1 ) or x̌1 ∈ P (x̂1 ; R1 ) still holds.
This contradicts the strategy-proofness of f .
Part 2. We suppose g(R̄i , p(R̄, f )), i = 1, . . . , N, are dependent and there are at least
two agents who receive positive consumption. A contradiction is induced as follows.
We let R∗ = (R1∗ , . . . , RN ∗ ) be a preference profile such that the consumption diL−1
. For
rections g(Ri∗, p), i = 1, . . . , N, are independent for any price vector p ∈ S++
i∗
iᾱ
example, let R be the preference represented by Cobb–Douglas utility function u (x) =
i
i
(x1 )ᾱ1 · · · (xL )ᾱL where the parameter vectors (ᾱ1i , . . . , ᾱLi ), i = 1, . . . , N are independent.
ᾱi
ᾱi
Then, g(Ri∗ , p) is parallel to ( p11 , . . . , pLL ) and consumption directions are always independent. Note that the independency of the consumption directions holds for a preference
profile R in a neighborhood of R∗ because g(Ri , p) is close to g(Ri∗ , p) for any p when Ri
is close to Ri∗ .
We repeat the replacement and perturbation of the preferences of agents who receive
positive consumption as follows. We first pick an agent i who receives positive consump
tion at R̄ and replace his preference with Ri ∗ . Let R denote the new preference profile
after this replacement. Since agent i ’s consumption at R is neither 0 nor Ω, there exists
another agent i who also receives positive consumption at R . We replace this agent’s
preference with Ri ∗ . We then perturb agent i ’s and i ’s preferences in a neighborhood
of Ri ∗ and Ri ∗ . As a result of this perturbation, there should exist a preference profile R where there exists another agent i different from i and i who receives positive
consumption at R . (If such a perturbation does not exist, then f is a Pareto-efficient
and strategy-proof social choice function defined in a neighborhood of (Ri ∗ , Ri ∗ ) in an
economy consisting of agents i and i , allocating positive consumption to both agents at
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(Ri ∗ , Ri ∗ ). This contradicts Part 1.)
We replace agent i ’s preference with Ri ∗ . Let R denote the preference profile
after this replacement. Keep in mind that agent i ’s consumption at R is neither
0 nor Ω. We then perturb these three agents’ preferences in a neighborhood of Ri ∗ ,
Ri ∗ , and Ri ∗ respectively. There exists a forth agent i who has positive consumption
after a perturbation. (If there is no perturbation such that the fourth agent receives
positive consumption, then f is a Pareto-efficient and strategy-proof social choice function
defined in a neighborhood of (Ri ∗ , Ri ∗ , Ri ∗ ) in an economy consisting of agents i , i ,
and i , allocating positive consumption to at least two agents at the preference profile
(Ri ∗ , Ri ∗ , Ri ∗ ). This contradicts Part 1.)
We replace the fourth agent’s preference with Ri ∗ and perturb these four agents’
preferences. Repeating this process, we finally obtain the preference profile R∗ where at
least two agents have positive consumption. This contradicts Part 1.
5
Proof of Corollary
The proof of the corollary is essentially the same as the second part of the proof of the
theorem. We let R̄ = (R̄1 , . . . , R̄N ) ∈ R̄N be a preference profile where at least two
agents receive positive consumption, and show a contradiction.
We let R∗ = (R1∗ , . . . , RN ∗ ) ∈ RN be a preference profile in RN . We repeat the replacement and perturbation of the preferences of agents who receive positive consumption
as follows. We first pick an agent i who receives positive consumption at R̄ and replace
his preference with Ri ∗ . Let R denote the new preference profile after this replacement.
Since agent i ’s consumption at R is neither 0 nor Ω, there exists another agent i who re
ceives positive consumption at R . We replace this agent’s preference with Ri ∗ . We then
perturb agents i ’s and i”’s preferences in R × R. As a result of this perturbation, there
should exist a preference profile R where there exists another agent i different from i
and i receiving positive consumption at R . (If such a perturbation does not exist, then
f is a Pareto-efficient and strategy-proof social choice function defined on R × R in an
economy consisting of agents i and i , allocating positive consumption to both agents at
(Ri ∗ , Ri ∗ ). This contradicts the theorem.) We replace agent i ’s preference with Ri ∗
and further perturb these three agents’ preferences in R × R × R. There exists a fourth
agent i who has positive consumption after a perturbation. We replace his preference
with Ri ∗ and perturb these four agents’ preferences. Repeating this process, we finally
obtain a preference profile in RN where at least two agents have positive consumption.
This contradicts the theorem.
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