Locally efficient and strategy-proof allocation mechanisms in exchange economies

In this paper, we investigate whether efficiency and strategy-proofness of allocation mechanisms defined on a “local” preference set imply dictatorship. Although there is an extensive literature on the characterization of efficient and strategy-proof allocation mechanisms defined on the whole preference set, little attention has been given to a local characterization even in two-agent economies. This paper presents three results. First, we point out that locally efficient, strategy-proof, and nondictatorial allocation mechanisms exists even in two-agent economies, when boundary allocations can be efficient. Second, excluding such exceptional cases, we show that in economies where the number of goods equals or exceeds the number of agents, any efficient and strategy-proof allocation mechanism defined on any local preference domain is alternately dictatorial, that is, it always allocates the total amount of goods to some single agent, even if the receivers vary. Third, we clarify that the local characterization is generally an open question even with allocation conditions such as the minimum consumption guarantee, and show that efficiency and strategy-proofness are incompatible with allocation conditions when all agents have the same local preference set.


Introduction
Following the seminal work of Hurwicz (1972), the manipulability and efficiency of allocation mechanisms in pure exchange economies have been studied intensively.In particular, many studies have been stimulated by the work of Zhou (1991), who 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.Although the dictatorship result has been strengthened and extended in various directions, little attention has been paid to local characterization.Hence, in this paper, we aim to investigate this issue, responding to the following research question: Does the dictatorship result hold if an allocation mechanism is efficient and strategy-proof only on a local preference domain?To ask the question positively, does local efficiency and strategy-proofness of allocation mechanisms allow a nondictatorial allocation?This would be a natural question to ask not only from a theoretical perspective but also from a practical perspective because it is quite likely that a planner will have some information about the agents' preferences, which allows him/her to narrow the focus to a local preference set rather than the whole preference set.Zhou's (1991) dictatorship result in two-agent economies has been strengthened by being proven in the domain of restricted preferences.See Schummer (1997), Ju (2003), Hashimoto (2008), andMomi (2013a). 1 Note that these studies restricted the types of preferences, and we will narrow the domain of classical preferences to a neighborhood of a given preference in this paper.
First, we emphasize that there exist Pareto-efficient, strategy-proof, and nondictatorial allocation mechanisms defined on a local preference domain even in the case of two-agent economies.2This is in sharp contrast to the abovementioned dictatorship result on the whole preference domain.In an economy with two agents and two goods, consider allocation where one agent's consumption is on the boundary of the agent's consumption set.For many reasonable preferences, the allocation will be Pareto efficient despite an inequality of marginal rates of substitution.The mechanism that always assigns this allocation is strategy-proof and nondictatorial.Furthermore, it is locally efficient because the allocation keeps being Pareto efficient for preferences in a neighborhood.In the Appendix, we provide an illustration of such an example.Although the example should be noted as a positive result, it is exceptional in that it ignores what Pareto efficiency usually imposes on allocation mechanisms, that is, the equality of the marginal rates of substitution of goods among agents.To exclude such cases, we assume that the upper contour set of preferences at any positive consumption is a strictly convex set included in the interior of the consumption set.
Recently, the dictatorship result of Zhou (1991) has been extended to many-agent economies.As pointed out by Satterthwaite and Sonnenschein (1981) and Kato and Ohseto (2002), there exist Pareto-efficient, strategy-proof, and nondictatorial allocation mechanisms in many-agent economies, but these are typically alternating dictatorships.Momi (2017) proved the alternating dictatorship result in many-agent economies under the condition that the number of goods equals or exceeds the number of agents, and subsequently proved the result without such a condition (Momi 2020).
In this paper, we extend Momi's (2017) approach and show that if the number of goods equals or exceeds the number of agents, any Pareto-efficient and strategyproof allocation mechanism defined on any local preference domain is alternately dictatorial.We leave examining the issue without the condition on the numbers of agents and goods to future research.
It is important to note that we cannot directly extend the previous studies on the whole preference set to the local characterization problem.The difficulty is that most studies have used specific, carefully constructed preference profiles in their proofs, and an arbitrarily given local preference set would not include such a specific preference profile.In particular, most of the abovementioned studies of two-agent economies used a preference profile where both agents have the same preference.Although Hashimoto (2008) and Momi (2013a) proved the local characterization in two-agent economies with Cobb-Douglas preferences, an arbitrarily given local preference set of classical preferences might not include any Cobb-Douglas preferences. 3Replacing the Cobb-Douglas preferences used in Momi (2017) with preferences in an arbitrarily given preference set is one of the difficulties we overcome in this paper, as we will discuss later.We cannot apply the approach of Momi (2020) for the same reason.In the proof, Momi (2020) replaced all but two of the agents' preferences with a common preference, and such replacements are not possible in an arbitrarily given local preference set.If a local preference set includes such a specific preference profile, a local characterization result might be obtained as a straight extension.We will observe this in establishing the local incompatibility of Pareto efficiency and strategy-proofness of allocation mechanisms when agents' individual local preference sets are the same.Hurwicz (1972) originally proved that, in two-agent economies, Pareto-efficient and strategy-proof allocation mechanisms are incompatible with the individual rationality condition, where agents possess their initial endowments and a mechanism is assumed to allocate consumption that benefits all agents.The incompatibility result has been extended to many-agent economies with various welfare lower bound (WLB) condiitions.Serizawa (2002) showed the incompatibility with the individual rationality condition.Serizawa and Weymark (2003) showed the incompatibility with a minimum consumption guarantee condition, where the consumption of each agent is assumed to be away from zero with some minimum distance.Momi (2013b) relaxed the minimum consumption guarantee condition to a simple positive consumption condition, where a mechanism is assumed to allocate positive consumption to all agents. 4Again, the proofs of these papers relied on a preference profile where all agents have the same preference, which would not be included in an arbitrarily given local preference set.Therefore, it is an open question whether such incompatibility results hold locally.In this paper, we extend Momi's (2013b) proof and show that Pareto efficiency and T. Momi strategy-proofness are incompatible with the positive consumption condition when all agents have the same local preference set Note that the alternating dictatorship, where one agent receives all goods and the others receive zero consumption, violates the WLB conditions.Therefore, the alternating dictatorship result in this paper establishes the incompatibility of Pareto efficiency and strategy-proofness with the WLB conditions for any local preference set when the number of agents does not exceed the number of goods.However, it remains an open question whether a Pareto-efficient and strategy-proof allocation mechanism defined on an arbitrarily given local preference set can be compatible with the WLB conditions when the number of agents exceeds the number of goods.
The remainder of the paper is organized as follows.Section 2 describes the model and results.Section 3 reviews the approach by Momi (2017) and explains the difficulty that we face when a mechanism is defined only locally.Section 4 presents some technical results.Sections 5 and 6 provide the proofs of the main results.The Appendix contains an example of the positive result and the proofs of all lemmas and corollaries from Sects. 4 and 5.

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 is R L + .A consumption bundle for agent i ∈ N is a vector The total endowment of goods for the economy is = ( 1 , . . ., L ) ∈ R L ++ .An allocation is a vector x = x 1 , . . ., x N ∈ R L N + .Thus, the set of feasible allocations for the economy with N agents and L goods is X = x ∈ R L N + : i∈N x i ≤ .A preference R is a complete, reflexive, and transitive binary relation on R L + .The corresponding strict preference P R and indifference I R are defined in the usual way.Given a preference R and a consumption bundle x Rx , and the lower contour set of ++ , and strictly monotonic on R L ++ , then UC(x; R) ⊂ R L ++ for any x ∈ R L ++ and the boundary ∂R L + is an indifference set.Note that the definition of strict convexity differs slightly from that of Zhou (1991), where UC(x; R), x ∈ R L ++ might have an intersection with the boundary.As mentioned in the introduction, a mechanism that assigns a boundary allocation could be locally Pareto efficient, strategy-proof and nondictatorial.Therefore, the assumption that keeps the upper contour sets away from the boundary is crucial in this paper.
A preference R is homothetic if, for any x and x in R L + and any t > 0, x Rx implies that (t x)R(t x ).A preference R is smooth if for any x ∈ R L ++ , there exists a unique vector p ∈ S L−1 x = 1 such that p is the normal of a supporting hyperplane to UC(x; R) at x.We call the vector p the gradient vector of R at x, and write p = p(R, x).Note that if R is smooth, strictly convex on R L ++ , and strictly monotonic on R L ++ , then the gradient vector is positive in the positive orthant: Let R denote the set of preferences that are continuous, strictly convex on R L ++ , strictly monotonic on R L ++ , smooth, and homothetic.A preference profile is an We also write R −{i, j} to denote the subprofile obtained by removing R i and R j from R.
A social choice function f : R N → X assigns a feasible allocation to each preference profile in R N .For a preference profile R ∈ R N , the outcome chosen can be written as , where f i (R) is the consumption bundle allocated to agent i by f .We let B ⊂ R N be a subset of R N .In this paper, we deal with a case where a social choice function is defined on B, or it satisfies desirable properties only on B.
The following theree deficinitons are standard.
We say that a social choice function is alternately dictatorial if it always allocates the total endowment to some single agent though not always the same agent.Note that under an alternately dictatorial social choice function, the identity of the receiver of the total endowment may vary depending on preference profiles.
Taking the same approach as previous studies, including Serizawa (2002) and Momi (2013b), we follow Kannai (1970) and introduce the Kannai metric into R to discuss continuity.For x ∈ R L + \ 0, we use [x] to denote the ray starting from the origin and passing through x: [x] = y ∈ R L + : y = t x, t ≥ 0 .We define 1 ≡ (1, . . ., 1) ∈ R L + 123 so that [1] denotes the principal diagonal of R L + .Using these definitions, the Kannai metric d(R, R ) for continuous and monotonic preferences R and R is defined as where • denotes the Euclid norm in R L .With the Kannai metric, R is a metric space.See Kannai (1970) for details. 6e write B ( R) ⊂ R to denote the open ball of preferences in R, with center R and radius > 0: We often write B i ⊂ R to denote an open ball of agent i's preferences without a specified center or radius and B = N i=1 B i to denote the product of such open balls over agents.
We define various open balls in a similar manner.For example, we write Our main result is as follows.
Theorem 1 Assume that L ≥ N .If a social choice function f : R N → X is Pareto efficient and strategy-proof on a product set of open balls B = N i=1 B i , then f is alternately dictatorial on B.
Alternating dictatorship violates the individual rationality condition of Serizawa (2002), the minimum consumption guarantee condition of Serizawa and Weymark (2003), and the positive consumption condition of Momi (2013b).Therefore, the incompatibility of the Pareto-efficient and strategy-proof mechanism with these WLB conditions is ensured by this theorem for the case where L ≥ N .If we assume that all agents have the same local preference sets, we can show the incompatibility without the condition on the numbers of agents and goods.In this paper, we prove the incompatibility with the positive consumption condition.
Theorem 2 If a social choice function f : R N → X is Pareto efficient and strategyproof on a product set of open balls B = N i=1 B i , where B i = B for all i, then there exist some j and R ∈ B such that f j (R ) = 0.

Preliminary results
Momi (2017) proved the alternating dictatorship result for a social choice function f defined on the whole domain R N ; when L ≥ N , any Pareto-efficient and strategyproof social choice function f : R N → X is alternately dictatorial.In this section, referring to this result, we explain the difficulties faced in the case of local preference domains.
We consider a social choice function f that is Pareto efficient and strategy-proof on a product set of open balls B = N i=1 B i .As in Momi (2017), we define the option set as follows.For agent i, when the other agents' preferences R−i ∈ B −i ≡ j =i B j are fixed, we define the option set, G i ( R−i ) ⊂ R L + , as the union of the agent's consumption bundles given by f over his/her preferences in B i : The key feature of the option set is that, because of the strategy-proofness on B, Because the boundary of the consumption set ∂R L + is an indifference set of each agent's preferences, each agent's consumption assigned by a Pareto-efficient social choice function f is not on the boundary except for the origin. 7Then, because of the Pareto efficiency, at least one agent i has positive consumption f i (R) ∈ R L ++ , and the agent's gradient vector at the consumption is well defined and p R i , f i (R) ∈ S L−1 ++ .The Pareto efficiency also implies that all agents who are assigned positive consumption have the same gradient vector.We call the gradient vector the price vector at the allocation and write p(R, f ).Thus, the price vector p(R, f ) ∈ S L−1 ++ is well defined for the economy even if the gradient vector of an agent who is assigned zero consumption is not well defined.
On the other hand, for a preference R ∈ R and a price vector p ∈ S L−1 ++ , we define the consumption-direction vector g(R, p) ∈ S L−1 ++ as the normalized consumption vector where the gradient vector of R is p.It should be noted that the consumptiondirection vector is well defined because the preference is homothetic and strictly convex in R L ++ .The proof of the next lemma is provided in the Appendix.
Thus, g R i , p(R, f ) is agent i's consumption-direction vector at the preference profile R under f .Note that this is well defined even for an agent who is assigned zero consumption.To simplify notation, we write assigned by f should be on the ray g i (R, f ) , and we can write Momi (2017) focused on a preference profile R ∈ B where the consumptiondirection vectors are independent.The role of this independence should be clear.Because consumption vectors f i ( R), i = 1, . . ., N , are on the rays g i (R, f ) , i = 1, . . ., N , respectively, and they sum to the total endowment , the consumption vectors should be determined uniquely if the consumption-direction vectors are Fig. 1 The option set independent.Momi (2017) showed that in a neighborhood of f i ( R), where the consumption-direction vectors are independent at R, the option set G i ( R−i ) is the L − 1-dimensional smooth surface of a strictly convex set, as drawn in Fig. 1(i).
The role of this strict convexity and smoothness is clear.If the option set satisfies such properties, f i R i , R−i , which is the most preferred consumption bundle in the option set with respect to R i , is a continuous function of R i .Based on these topological properties of the option set, we can prove the following proposition.See Momi (2017, Proposition 6) for the proof.

Proposition 1 Suppose that f is a social choice function that is Pareto efficient and strategy-proof on a product set of open balls
This proposition ensures the alternating dictatorship at a preference profile where the consumption-direction vectors are independent.If the consumption-direction vectors are independent at a preference profile, independence holds for a preference profile in a neighborhood because of the continuity of f , and the alternating dictatorship also holds in the neighborhood.However, in general, it is difficult to know whether the independence of the consumption-direction vectors holds for a given preference profile.It depends not only on the preference profile R but also on the price vector p(R, f ) determined by the social choice function f , the behavior of which is unknown.Without the independence of the consumption directions, the option set might not be either strictly convex or smooth, and hence f might not even be a continuous function.Figure 1 (ii) depicts an example of such an option set.
A key to overcome this difficulty is that there exists a preference profile R * ∈ R N that ensures the independence of the consumption-direction for any price vector.Momi (2017) constructed such a preference profile R * using Cobb-Douglas utility functions.Then, through preference exchanges between two preference profiles, the alternating dictatorship result at R * was extended to any preference profile.
However, in a small local domain B, we cannot expect the existence of such a preference profile that ensures the independence of the consumption-direction vectors for any price vectors.This is the difficulty that we face in this paper.In an arbitrarily given domain B, we have to find a preference profile that satisfies the condition of Proposition 1.

Technical results
As mentioned in the previous section, we must deal with the case where g i (R, f ), i = 1, . . ., N , are dependent.In the next section, starting from such a preference profile, we construct a preference profile in any neighborhood where the independence of the consumption-direction vectors holds.In this section, we present some technical results that we use for the proof.Throughout this section, we assume that the social choice function f is Pareto efficient and strategy-proof on a product set B and we deal with preference profiles in B, although we do not repeat them in each lemma.Proofs of all lemmas and corollaries are provided in the Appendix.
For a preference R ∈ R and a consumption bundle 8 It is well known that if an agent receives x at a preference profile R, strategy-proofness implies that this agent receives the same consumption bundle x when his/her preference is subject to an MMT at x.Note that R and R share the same price vector at x.As shown in Momi (2013b, Lemma 4), for a preference R ∈ R and a consumption bundle x ∈ R L ++ , there exists a preference that is an MMT of R at x in any neighborhood of R. Because As mentioned in the previous section, this might not be a unique intersection, and then f i •, R −i might not be a continuous function of R i .Therefore, we define F R i ; G i R −i as the intersection between the upper contour set of R i at f i (R) and the option set:

union of open balls with radius δ and center
Lemma 2 For any δ > 0, there exists If the consumption f i (R) is the unique intersection between the upper contour set and the option set, that is, if The next lemma considers the case where the other agents' preferences change.
Lemma 3 Suppose that Ri is an MMT of R i at xi and the gradient vector at xi is p.For any ε > 0, there exists Lemma 3 implies that if Ri is an MMT of R i at xi with price vector p, and if , the intersection between the upper contour set of Ri and the option set, is close to xi , and hence The next lemma implies that if the consumption-direction vectors are independent among some agents, then the independence holds after slight changes in their preferences and the price vector.Furthermore, if the total endowments are assigned among the agents, then the positive consumption receivers remain such receivers after the changes.
(1) g i R i , p , i = 1, . . ., K , are independent for any R i ∈ B ¯ Ri and p ∈ In the proof of the theorem, we change the agents' preferences slightly and increase the number of agents whose consumption-direction vectors are independent.In the process, we have to change the preferences of an agent who receives positive consumption.When agent i's consumption is positive, we exchange the agent's preference R i with a preference in a neighborhood of an MMT of R i at the consumption.Then, the price changes only slightly, as shown in Lemma 2.
The next lemma shows that if the consumption-direction vectors are independent among some agents who are not assigned zero consumption or the total consumption, then we can change their preferences slightly such that another agent receives neither zero nor the total consumption and the price vector is sufficiently close to the original price vector.Lemma 5 Suppose that g i R, f , i = 1, . . ., K , where K < N , are independent and f i R / ∈ {0, } for any i = 1, . . ., K , at R = R1 , . . ., R N .For any > and ε > 0, there exist R 1 , . . ., The next corollary is an immediate consequence of Lemma 5.
Corollary 1 Suppose that g i ( R, f ), i = 1, . . ., K , where K < N , are independent and f i ( R) / ∈ {0, } for any i = 1, . . ., K , at R = R1 , . . ., R N .There exists j ≥ K + 1 such that for any > 0 and ε > 0, some R 1 , . . ., The next lemma relaxes the condition of Lemma 5.If the consumption-direction vectors are independent among some agents and one of them is assigned neither zero nor the total consumption, then we can find a slight change in their preferences such that another agent receives positive consumption and the price vector is sufficiently close to the original price vector.
Finally, the next corollary is to Lemma 6 as Corollary 1 is to Lemma 5.

Proof of Theorem 1
We first explain how we use constant elasticity substitution (CES) utility functions to achieve the independence of the consumption-direction vectors and then prove the theorem.All proofs of the lemmas are in the Appendix.In this section, we consider preferences represented by CES utility functions.Note that the indifference curve of CES utility functions intersects with the boundary of the consumption set, and hence the preferences are not elements of R. It should not cause a confusion that we sometimes extend definitions in Sect. 2 to such preferences.Assume that a preference profile ( R1 , . . ., R N ) ∈ R N and a price vector p ∈ S L ++ are given.We write ḡi = g( Ri , p) to denote agent i's consumption-direction vector for the price p and the preference Ri .
(1) CES utility function: We explain the CES utility function that we use.Let U α,ρ : R L + → R denote the CES utility function with parameters ρ < 1 and α = (α 1 , . . ., α L ) ∈ S L−1 ++ : Abusing notation, we let U α,ρ denote not only the utility function but also the preference represented by the utility function.It is straightforward to calculate the gradient vector of the utility function U α,ρ at a consumption bundle x ∈ R L + and to observe that it is parallel to α where y z denotes that vectors y ∈ R L and z ∈ R L are parallel.Therefore, we can calculate the parameter α of the CES utility function such that the gradient vector of the represented preference at a given consumption bundle x equals p ∈ S L ++ by solving with respect to α for fixed ρ, x, and p.In addition, we can calculate the consumptiondirection vector of a given CES utility function for a price p ∈ S L ++ by solving (3) with respect to x for fixed values of α, ρ and p.
We set α i as the parameter such that the gradient vector at ḡi is parallel to p: 10 Then, the preferences U α i ,ρ and Ri have the same gradient vector p at ḡi .Note that α i thus depends on ρ.
It is well known that the CES utility function defined by (1) converges to a Leontief utility function as ρ → −∞.Therefore, with a sufficiently small ρ, in a neighborhood of ḡi , any consumption bundle except ḡi itself in the upper contour set of U α i ,ρ at ḡi is strictly preferred to ḡi with respect to Ri .We fix ρ such that U α i ,ρ satisfies this property for any i = 1, . . ., N .We write the CES utility function as U i with the fixed ρ and the α i determined by the ρ, as in the previous paragraph.
As mentioned above, for a given price vector p = ( p 1 , . . ., p L ), the consumptiondirection vector g U i , p ∈ S L−1 ++ of the CES preference is determined by , (4) 9 The gradient vector of a utility function should not be confused with the gradient vector of a preference that we defined in Sect. 2. Whereas ∂U α,ρ ∂ x (x) is the gradient vector of the utility function U α,ρ , its normalization is the gradient vector of the preference U α,ρ represented by the utility function. 10In fact, α i is obtained as the normalization of by solving (3) with respect to x.Therefore, the consumption-direction vectors among agents are independent for any price p ∈ S L−1 ++ if and only if the vectors 1/α i 1 1/(ρ−1) , . . ., 1/α i L 1/(ρ−1) , i = 1, . . ., L, are independent.
We define the CES utility function U γ i with γ i as the parameter by Then, for the preferences U γ i , i = 1, . . ., N , the consumption-direction vectors 1) , are independent for any p as long as 0 < δ i ≤ δ.The independence of the consumption-direction vectors holds for preference profiles in a neighborhood of U γ 1 , . . ., U γ N .Formally, this is stated in the next lemma.
(2) Preference construction: Although the CES utility functions satisfy the independence of the consumption-direction vectors for any price, they are not in a neighborhood of the original preferences Ri , i = 1, . . ., N .We construct a preference that is close to Ri and is represented by U γ i in a neighborhood of ḡi .See Momi (2017) for more details of the preference construction.Figure 2 depicts the preferences Ri and U γ i .We let t > 0 be a sufficiently small parameter and define C as the set of consumption bundles x that are connected to ḡi in UC ḡi ; U γ i LC ḡi + t p; Ri .That is, C is the crescent-shaped set in Fig. 2 between the indifference sets I ḡi ; U γ i and I ḡi + t p; Ri .The indifference set of the CES preference U γ i intersects the boundary of the consumption set, whereas that of Ri is away from the boundary.Therefore, they intersect again, although this is not depicted in Fig. 2, and the crescent-shaped set C is not equal to UC ḡi ; U γ i LC ḡi + t p; Ri .We define where co(Y ) denotes the convex hull of a subset Y ⊂ R L .Therefore, this is the convex hull of the upper contour set of Ri at ḡi + t p added to the crescent-shaped set C. The set C depends on the CES utility function U γ i , and U γ i depends on the parameter δ i .Because this parameter plays a role, we write it explicitly as A i δ i (t).As the convex set A i δ i (t) is not strictly convex, it cannot be an upper contour set of a preference in R. To modify A i δ i (t) into a strictly convex set, we prepare a strictly convex set as follows.Setting the parameter t = 1 in C, we let C denote the set of consumption bundles x that are connected to ḡi in UC ḡi ; U γ i LC ḡi + p; Ri .That is, C is the crescent-shaped set between the indifference sets I ḡi ; U γ i and I ḡi + p; Ri analogous to C. We define Note that Di is a strictly convex set; it does not intersect the boundary of the consumption set, but its surface is not smooth.To make its surface smooth, we let D i δ i denote the union of closed balls with a sufficiently small radius ε included in Di : Momi (2017, Lemma 6) for the proof that this makes a smooth surface.We explicitly write the index δ i as in the case of A i δ i (t).Using D i δ i , we modify A i δ i (t) into a strictly convex set.We let s > 0 be a sufficiently small parameter and define where L(x) is the half line starting from x and extending in the direction of the vector p: is the boundary of a strictly convex set. 11We let R i δ i ,t,s ∈ R denote the preference that has B i δ i (t, s) as its indifference set.
Note that, as long as t > 0, the boundaries of A i δ i (t), D i δ i (t), and B i δ i (t, s) coincide in a neighborhood of ḡi , which is defined by the indifference set of the CES preference.That is, the indifference set of R i δ i ,t,s equals to that of U γ i in a neighborhood of ḡi .Hence, R i δ i ,t,s inherits the properties of U γ i .In particular, independence of the consumption-direction vectors holds for prices in a neighborhood of p.In addition, note that, for s > 0 and t > 0, R i 0,s,t , where δ i = 0, is an MMT of Ri at ḡi .
The independence of the consumption-direction vectors holds for preferences in neighborhoods of R i δ i ,t,s , i = 1, . . ., N .
Lemma 10 Fix s > 0 and t > 0. There exist ε > 0 and functions i : (0, δ] → R ++ , i = 1, . . ., N , such that g R i , p , i = 1, . . ., N , are independent for any p ∈ B ε( p) Combining the preference construction with the results in the previous section, we prove the theorem.In the proof, we repeatedly find an agent who receives neither zero consumption nor the total consumption when increasing the number of agents whose consumption-direction vectors are independent.Then, finally, we make all agents have independent consumption-direction vectors, which contradicts Proposition 1.

Proof of Theorem 1
We let L ≥ N and the social choice function f be Pareto efficient and strategy-proof on B = N i=1 B i .We suppose that an agent j has neither zero nor the total consumption, f j ( R) / ∈ {0, }, at a preference profile R = R1 , . . ., R N ∈ B, and we show a contradiction.If g i R, f , i = 1, . . ., N , are independent, this immediately contradicts Proposition 1.We consider the case where these consumptiondirection vectors are dependent.
We write p and ḡi to denote the price vector and agent i's consumption-direction vector at R, respectively: p = p R, f and ḡi = g i R, f = g Ri , p R, f .
Furthermore, by Lemma 10, we have ε and i : We set ε sufficiently small such that ε < ε. we also set the function i such that its value is sufficiently small and B i (δ i ) (R i δ i ,t,s ) ∈ B ¯ Ri for any 0 < δ i ≤ δi .From now on, parameters other than δ i are fixed and we write R i δ i = R i δ i ,t,s .Figure 3 depicts the neighborhoods of Ri and p.We will replace each preference Ri with R i δ i and then find a new preference in B i (δ i ) (R i δ i ) while keeping the price vector in B ε( p) so that the independence of the consumption vectors will be ensured by (8).
For the operation to keep the price vector in B ε( p), we prepare a function φ as follows.For each i = 1, . . ., N , R i 0 (that is R i δ i where δ i = 0) is an MMT of Ri at ḡi .Applying Lemma 3 to these preferences, we have a positive function φi : ε → φi ε for each i = 1, . . ., N , which maps ε to φ in Lemma 3: We select a positive function φ such that φ ε < min φ1 ε , . . ., φ N ε , ε for any ε .We write Without loss of generality, we assume that f 1 R / ∈ {0, }.Starting from this agent, we will increase the number of agents whose consumption-direction vectors are independent in the following steps, and finally reach a situation contradicting Proposition 1.
Step 1: First, we modify agent 1's preference for the later steps and show that there exists another agent who receives neither zero consumption nor the total consumption.
Replacing R1 with R 1 δ 1 of any parameter 0 < δ 1 < δ1 , we have f 1 R 1 δ 1 , R−i / ∈ {0, } because of the strategy-proofness of f , and there exists another agent receiving neither zero consumption nor the total consumption because of the Pareto efficiency of f .
Although the identity of the receiver might vary as δ 1 changes, there exists an agent j ≥ 2 such that for any δ 1 , there exists If there exists no such agent, then for each j ≥ 2 there exists δ 1 j such that for any δ for any parameter δ 1 .Without loss of generality, we assume that agent 2 is such an agent.We let We take a value of δ 1 that is sufficiently small so that p R 1 Step 2: Second, we modify agent 2's preference, and then apply Corollary 2 to find another agent j ≥ 3 who receives neither zero consumption nor the total consumption.Keeping in mind that agent 2 now receives neither zero nor the total consumption at the preference profile R implies that the price vector at the preference profile is in B φ (N −1) (ε) ( p).Also note that φ (N −1) (ε) ≤ φ2 φ (N −2) (ε) by the definitions of φ and φi .Therefore, we have . Therefore, we set δ 2 to be sufficiently small such that p R 2 δ 2 ; Of course, this implies that the price vector at the preference profile We can apply Corollary 2 to agents 1 and 2 because the price vector in as in ( 8).Corollary 2 implies the existence of a preference subprofile R 1 (1) , R 2 (1) arbitrarily close to R 1 δ 1 , R 2 δ 2 such that the price vector is arbitrarily close to p and another agent j ≥ 3 receives neither zero consumption nor the total consumption at the preference profile . We set ¯ i , i = 1, 2 to be sufficiently small such that ¯ i < i δ i and apply Corollary 2. Without loss of generality, we asume agent 3 is the receiver of positive consumption.As a result, we have R 1 ( p) and agent 3 receives neither zero nor the total consumption at the preference profile R 1 (1) , R 2 (1) , R−{1,2} .♦ Step 3: In this step, we modify agent 3's preference, and then apply Corollary 2 to find another agent j ≥ 4 who receives neither zero nor the total consumption.Because agent 3 now receives neither zero consumption nor the total consumption, we replace R3 with R 3 0 .By (9), we have p R 3 as we observed for agent 2.
We replace R 3 0 with R 3 δ 3 , where δ 3 is sufficiently small such that p R 3 as we observed for agent 2.
We can apply Corollary 2 to agents 1,2, and Corollary 2 implies the existence of a preference subprofile R 1 , f and another agent j ≥ 4 receives neither zero nor the total consumption at the preference profile R 1 (2) , R 2 (2) , R 3 (2) , R−{1,2,3} .We set ¯ 3 sufficiently small such that ¯ 3 < 3 δ 3 and apply Corollary 2. Without loss of generality, we assume that agent 4 receives neither zero consumption nor the total consumption.As a result, we have R 1 and agent 4 receives neither zero conusmption nor the total consumption at the preference profile Step 4: We modify agent 4's preference and then apply Corollary 2 to find another agent j ≥ 5 who receives neither zero consumption nor the total consumption.We replace R4 with R 4 δ 4 , where δ 4 is sufficiently small such that p R 4 Applying Corollary 2 to agents 1, . . ., 4, we obtain a preference subprofile R 1 where ¯ 4 satisfies ¯ 4 < 4 δ 4 such that the price vector is in B φ (N −4) (ε) ( p) and an agent j ≥ 5 receives neither zero nor the total consumption at the preference profile R 1 (3) , . . ., R 4 (3) , R−{1,...,4} .We repeat this process.Finally, in Step N − 1, we have a preference profile R 1 ( p) and agent N receives neither zero consumption nor the total consumption at the preference profile where δ N is sufficiently small, we still have the price vector in B ε( p) and agent N receives neither zero consumption nor the total consumption at the preference profile R 1

Proof of Theorem 2
To prove the theorem, we reconstruct Momi's (2013b) proof in a local preference set.Proposition 2 in Momi (2013b), which plays a key role in the proof, implies that if two preferences, R and R, and two consumption bundles, x and x, are as drawn in Fig. 3, then there exists a preference R that is an MMT of R at x and of R at x.We reconstruct this proposition in a local preference set.We show that if R and R are sufficiently close, then R can be selected to be close to R and R.
Proposition 2 For any R, R ∈ R, and any x, x ∈ R L ++ , if x ∈ P I (x; R) [ x]; R , then there exists a preference R ∈ R, that is an MMT of R at x and of R at x. Furthermore, we can select a R that is arbitrarily close to R and R if R and R are sufficiently close.
Proof The first statement is proved by Momi (2013b), so we need only show the second statement.Also note that as shown by Momi (2013b), the general case drawn in Fig. 4 (i) turns into Fig. 4 (ii), which is the case we have to consider.
We let Ř denote the homothetic, monotonic, and continuous preference whose indifference set is defined by ∂ UC(x; R) UC x; R .It is clear that, following the construction of R in Momi (2013b), we can construct a R arbitrarily close to Ř with respect to the Kannai metric.It is clear from the definition of the Kannai metric that the distance between R and Ř or between R and Ř is less than the distance between R and R. Therefore, as R and R are closer, we can select R closer to R and R.
We can follow the proof by Momi (2013b) using preferences in the local set B to show the incompatibility of Pareto efficiency, strategy-proofness and the positive consumption on the local preference set.When all agents have the same preference, all agents should be allocated positive portions of : f i (R) = λ i with some 0 < λ i < 1 for R = (R, . . ., R).We pick two different preferences R and R in B that are sufficiently close, and consider the allocations given by f at R = (R, . . ., R) and R = R, . . ., R .

We let
and we let , which is the upperright part of the consumption set partitioned by A(x; R) and let , which is the lower-left part.Without loss of generality, we assume that f 1 (R) ≤ f 1 ( R). See the Edgeworth box described in Fig. 5, where the consumption of agent 1 is measured from the lowerleft vertex and the sum of the consumptions of the other agents is measured from the upper-right vertex.We pick  Momi (2013b), agent 1's consumption should be on A f 1 (R) ; R (resp.A f 1 R ; R ) when other agents' preferences are R (resp.R) and agent 1's preference is changed.Let R and R be agent 1's preferences in x is sufficiently close to f 1 R , and hence x1 is sufficiently close to f 1 ( R).Then, we can select such a R in B.
We let Ř ∈ B be a preference that is an MMT of R at The following discussion is the same as that in Momi (2013b).We observe that the consumption allocated to agent 1 should not be changed when the preferences of agents other than agent 1 are changed to Ř from the profile R, R −1 or from R, R−1 .
Because f is Pareto efficient and allocates positive consumption, at the profile R, R −1 , agent 1 receives x1 and each of the other agents i = 2, . . ., N , receives a positive portion of − x1 : λi − x1 , i = 2, . . ., N , where 0 < λi < 1 and λi = 1.Note that because we have chosen a x1 that is not parallel to , the vectors x1 and − x1 are independent.Now, let us change agent 2's preference to Ř from R. We write the new profile as R, Ř, R −{1,2} where agent 1's preference is R, agent 2's is Ř and the other agents' preferences are R.
Given that Ř is an MMT of R at − x1 , this is also the case at agent 2's consumption.Therefore agent 2's consumption should not be changed and nor should his/her gradient vector.Because of the Pareto efficiency, all agents' gradient vectors at their consumptions should be the same.Hence, all agents have the same gradient vector at both profiles R, R −1 and R, Ř, R −{1,2} .Because the preferences are homothetic, the equality of the gradient vectors implies that agents' consumptions at both profiles should be parallel.That is, at the new profile, agent 1's consumption is parallel to x1 and the other agents' consumptions are parallel to − x1 .Because of the Pareto efficiency, the consumptions at the new profile should sum up to the total endowment .Then, agent 1's consumption should remain x1 .By applying the discussion repeatedly until all preferences except agent 1's are changed to Ř, we finally obtain that f 1 R, Ř−1 = x1 , where Ř−1 = Ř, . . ., Ř ∈ R N −1 .The discussions are the same for the profile R, R−1 and we obtain Remember our choice of x1 and x1 .From the construction, x is strictly preferred to x1 with respect to any preference for agent 1 and x1 can be chosen arbitrarily close to x .Therefore x1 could have been chosen to be preferred to x1 with respect to agent 1's preference R.This violates the strategy-proofness of f .This ends the proof of Theorem 2.

Concluding remarks
In this paper, we focused on the local characterization of Pareto-efficient and strategyproof social choice functions.We proved that Pareto-efficiency and strategy-proofness of a social choice function defined in a local preference set imply alternating dictator-ship if the number of goods equals or exceeds the number of agents.In addition, we proved that Pareto efficiency and strategy-proofness of a social choice function defined in a local preference set are incompatible with the positive consumption condition if agents' individual preference sets are the same,.
We comment on open questions left for future research.The most interesting question is whether local Pareto efficiency and strategy-proofness of a social choice function imply alternating dictatorship without the condition on the numbers of agents and goods.Although Momi (2020) proved the result on the whole preference domain, it seems difficult to extend the proof to the case of a local preference domain because Momi's proof changes all agents' preferences, except for those of two agents, to a common preference.Reflecting on the proof, the question may become slightly easier if all agents have the same local preference set.Without the condition on the numbers of agents and goods, it is also an open question whether Pareto-efficient and strategyproof social choice functions defined on an arbitrarily given local preference set are incompatible with allocation conditions, such as the rationality, minimum consumption guarantee, and positive consumption conditions.
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A.1 An example
Consider an economy with two agents i = 1, 2, and two goods l = 1, 2. Let U i (x 1 , x 2 ) be agent i's utility function defined on the consumption set R 2 + as We identify utility functions and the preferences represented by the utility functions.Let = (2, 2) be the total endowment of the economy and define the allocation x = x1 , x2 where x1 = (0, 1) and x2 = (2, 1).Each agent's marginal rate of substitution (MRS) at the consumption bundle xi is calculated as which is the set of consumption indifferent to xi with respect to U i on the possible allocation set X .Observe that x, that is on the boundary of X , is Pareto efficient even though the MRSs are not equal between agents 1 and 2.
Consider a social choice function f that always allocates x for any preferences.It is clear that f is strategy-proof and nondictatorial.It is also clear that for preference profiles in a neighborhood of U 1 , U 2 , the allocation x remains Pareto efficient because the inequality MRS 1 < MRS 2 holds.
Although the agent 1's preference represented by U 1 is not homothetic, it can be modified to be so without affecting the result.We pick up the indifference set of U 1 at x1 and construct a new homothetic preference by scaling this indifference set.

A.2 Proofs of lemmas and corollaries
Proof of Lemma 1 Let R ∈ R and p ∈ S L−1 ++ be given.We pick up a consumption bundle x ∈ R L ++ and the corresponding indifference set Ī = I ( x, R).It is clear from the strict convexity of R that the consumption bundle x ∈ Ī on the indifference set where the gradient vector is p: p(R, x ) = p, is determined uniquely.Furthermore, by the homotheticity of R, the gradient vectors are the same on any consumption bundles on a ray starting from the origin.In particular, the gradient vector at a consumption bundle x ∈ R L ++ is p if and only if x is on the ray [x ].Therefore, the consumptiondirection vector g(R, p) ∈ S L−1 ++ , which is the normalization of these consumption bundles, is uniquely determined as x x .To prove the continuity, we fix R ∈ R and p ∈ S L−1 ++ .We want to show that for any > 0, there exists a δ > 0 such that if R satisfies d(R, R) ≤ δ, then In contrast, we suppose that there exists an > 0 such that for any δ > 0 there exists some R satisfying d R, R < δ, and g(R, p) − g R, p ≥ , and show a contradiction.
We suppose that a sufficiently small δ and R satisfy the condition.We fix a hyperplane H parallel to p ⊥ as in Fig. 7.
We define a as the intersection between g R,

Proof of Lemma 2
We fix δ arbitrarily.We let { n } ∞ n=0 be a decreasing sequence of scalars converging to 0: n < n for n > n and n → 0 as n → ∞.We assume that for any n, there exist some R i , and we show a contradiction. Because , and hence this intersection converges to some set in the indifference set where n k < n k for k > k because the set of feasible allocations is compact.We let x = lim k→∞ x (n k ) denote the limit.Thus, we have Momi (2017, Lemma 3).
Thus, we have x ∈ F R i ; G i R −i and this contradicts that

Proof of Lemma 3
We fix ε arbitrarily.We suppose that for any φ , there exists By taking a sufficiently small value of φ , we have p R i , R −i , f arbitrarily close to p.Then, the ray f i R i , R −i is arbitrarily close to xi .Then, for any is arbitrarily close to the ray xi , and hence, the ray xi should be arbitrarily close to xi .This is a contradiction.
Proof of Lemma 4 Note that g i R, f = g R i , p R, f , i = 1, . . ., K , and they are independent.As p → p R, f , we have g Ri , p → g Ri , p R, f .Therefore, there exists ε such that g Ri , p , i = 1, . . ., K , are independent for any Therefore, there exists p such that g R i , p , i = 1, . . ., K , are independent for any R i ∈ B p Ri , i = 1, . . ., K .We define as the minimum of p as p moves over B ¯ ( p( R, f )): = min p∈B ¯ p R, f p .It is clear that and ε satisfy Lemma 4 (1).We consider R i ∈ B Ri , i = 1, . . ., K , and p(R, f ) ∈ B ε p R, f .Then, converges to g i R, f as R i and p(R, f ) converge to Ri and p R, f , respectively.Therefore the scalars α i (R, f ), i = 1, . . ., K satisfying K i=1 α i (R, f )g i (R, f ) = , are determined uniquely and α i (R, f ) converges to ᾱi , for any i = 1, . . ., K , as

Proof of Lemma 5
We select and ε arbitrarily.If f j R = 0 for some j ≥ K + 1, the lemma holds.Therefore, we suppose that f j R = 0 for any j ≥ K + 1.We let ¯ > 0 and ε > 0 be scalars that support Lemma 4 with respect to R. We define ˜ = min{ , ¯ } and ε = min{ε, ε}.Note that, because of Lemma 4, at any R such , are all positive and independent.We let Ri ∈ B ˜ Ri be an MMT of Ri at f i R for each i = 1, . . ., K .
Case 1 We consider the case where some agent j ≥ K + 1 receives positive consumption after replacements of Ri with Ri for some agents i = 1, . . ., K , that is, Without loss of generality, by relabeling the consumer indexes if necessary, we assume that f j R1 , . . ., R S , R S+1 , . . ., R N = 0 for some j ≥ K + 1 where S is the smallest number of agents whose preference Ri should be replaced with Ri so that some agent j ≥ K + 1 has positive consumption.We show that the lemma holds at the preference profile R1 , .
Starting from the preference profile R, we replace Ri with Ri and then replace Ri with a preference R i in a neighborhood of Ri for i = 1, . . ., K .If no agents j ≥ K +1 have positive consumption, then we reach a situation contradicting Proposition 1.
This replacement of preferences should be done so that the independence of the consumption vectors is maintained.For this purpose, we prepare a function φ i as follows.For each i = 1, . . ., K , we consider a function φi : ε → φi ε that maps ε to φ in Lemma 3 with respect to Ri and Ri .Furthermore we define a function φ i : ε → φ i ε as φ i ε = min φi ε , ε .Then, of course, for any ε and any R −i , if f i Ri , R −i = 0 and p Ri , R −i , f ∈ B φ i (ε) p R, f , then p Ri ; G i R −i ∈ B ε p R, f because of Lemma 3.
Step 1 We replace R1 with R1 .This replacement does not change the price vector.We let 1 be a sufficiently small scalar such that B 1 R1 ⊂ B ˜ R1 and for any R 1 ∈ B 1 R1 .Note that f 1 ( R) is the unique intersection of UC f 1 R ; R1 and G 1 R −1 .Therefore, if R 1 is sufficiently close to R1 , then f R 1 , R−1 and p R 1 ; G 1 R−1 are sufficiently close to f 1 R and p R, f , respectively, as discussed after Lemma 2. Thus, a value of 1 satisfying the condition exists.
The lemma holds if f j R 1 , R−1 = 0 for some j ≥ K + 1 with some R 1 ∈ B 1 R1 .Therefore, we suppose that f j R 1 , R−1 = 0 for any j ≥ K + 1 and any R 1 ∈ B 1 R1 .Then, f i R 1 , R−1 , i = 1, . . ., K , are all positive and independent because Step 2 We replace R 2 with R2 .By (10) and Lemma 3, we have for any R 1 ∈ B 1 R1 .We let 2 be sufficiently small such that B 2 R2 ⊂ B ˜ R2 for any R 2 ∈ B 2 R2 and any R 1 ∈ B 1 R1 .Note that a sufficiently small value of 2 ensures (12) because of (11).

Proof of Lemma 6
We select and ε arbitrarily.If f j R = 0 for some j ≥ K +1, the lemma holds.Therefore, we assume that f j R = 0 for j ≥ K + 1.When an agent

Fig. 3
Fig. 3 Neighborhoods of Ri , R i δ i , and p

−
and x1 is not parallel to .Next, let x be the intersection of A f 1 (R) ; R and the segment x1 , and pick x1 ∈ A f 1 (R) ; R in the neighborhood of x so that x1 ∈ A x ; R − .As we observed in Proposition 1 of − x1 and of R at − x1 .Observe that our choice of x1 and x1 ensures the condition in Proposition 2: − x1 ∈ P I − x1 ; R − x1 ; R .Furthermore, because R and R are sufficiently close, we can select such a Ř in B.

Fig. 6
Fig. 6 Edgeworth Box of the example

Fig. 7
Fig. 7 Continuity of the consumption-direction vector p and H , b as the intersection between [g(R, p)] and H , b as the intersection between [g(R, p)] and I a; R , c as the intersection between [1] and I (b; R), and c as the intersection between [1] and I a; R .We define e as the intersection between I a; R and I (b; R), and c as the intersection between I b; R and [1].Because g(R, p) is away from g R, p by at least the distance , b is away from b by some distance, which we write as : b − b = .Note that, because the preferences are homothetic, c − c : c = b − b : b .Therefore, c − c = c b .We consider the Kannai metric between R and R: d R, R = max x∈R L +I (x;R) [1]−I (x; R) [1] 1+ x 2 .Letting e be the x in the definition of the Kannai metric, we haved R, R ≥ I (e; R) [1] − I e; R [1] 1+ e 2 = c − c 1+ e 2 .Letting b be the x in the definition of the Kannai metric, we haved R, R ≥ I (b; R) [1] − I b; R [1] 1+ b 2 = c − c 1+ b 2 .The condition d R, R < δ with a sufficiently small δ implies that c − c < 1+ e 2 δ and c − c < 1+ b 2 δ, that is, both c and c are sufficiently close to c.However, this contradicts to that c and c have the distance c − c = c b between them.
and we show a contradiction.Because p Ri ; G i R −i is not included in B ε ( p), there exists xi ∈ F i Ri , R −i , they are indifferent with respect to Ri .Therefore, we have xi Ri f RK .All we have to show is that the price vector at the preference profile is in B ε 1 , R S , . . ., R N = f i R for any i = 1, . . ., K , and any R i ∈ { Ri , Ri }, i = 1, . . ., S−1.In particular, we have f S R1 , . . ., R S−1 , R S , . . ., R N = S , . . ., R N , f = p R, f .S+1 , . . ., R N , f = p R, f as desired.