16640068

www.elsevier.com / locate / econbase

Single-peaked preferences, endowments and

population-monotonicity

*

Bernardo Moreno

´ ´ ´ ´

Departamento de Teorıa e Historia Economica, Universidad de Malaga, Campus El Ejido s /n, 29071 Malaga, Spain

Received 5 May 1999; received in revised form 4 September 2001; accepted 24 September 2001

Abstract

We consider economies with a finite set of agents with single-peaked preferences and with private endowments of an infinitely divisible commodity. We look for solutions that distribute fairly across agents the gains made possible by redistributions of the commodity under the consideration of changes in the number of agents. We show that the uniform rule is the only selection from the weak no-envy and Pareto efficient solution satisfying one-sided population-monotonicity.  2002 Elsevier Science B.V. All rights reserved.

Keywords: Endowments; Population-monotonicity; Uniform rule

JEL classification: D6

1. Introduction

We consider economies with a finite set of agents with single-peaked preferences and with private endowments of an infinitely divisible commodity. We look for solutions that distribute fairly across agents the gains made possible by redistributions of the commodity. The model in which there are no private endowments but there is a social endowment to be allocated has been analyzed, among others,

`

by Sprumont (1991) and Thomson (1994a,b, 1995a). Barbera et al. (1997) characterize the class of rules that are strategy-proof and Pareto efficient and satisfy an individual monotonicity property. Thomson (1995b) presents economies in which there are private and social endowments in a unified framework of open economies. The model in which there are private endowments is also studied in Klaus et al. (1997, 1998). Among other desirable properties, they focus on envy-freeness and

population-monotonicity. Klaus et al. (1997) use an envy-freeness condition in which an agent

compares either the net allotment change of another agent or the feasible part of the agent’s allotment

*Tel.: 134-95-213-1247; fax:134-95-213-1299. E-mail address: [email protected] (B. Moreno).

0165-1765 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 ( 0 1 ) 0 0 5 7 6 - 6

change. We use a weaker envy-freeness condition in which an agent only compares redistributions of other agents which are feasible for him (Klaus et al. (1997) refer to this condition as weak no-envy).

1

We show that the uniform rule is the only selection from the weak no-envy and Pareto efficient solution satisfying one-sided population-monotonicity (Theorem 1 below). Klaus et al. (1997) characterize the uniform rule by means of Pareto efficiency, envy-freeness, and one-sided

population-monotonicity using the fact that the uniform rule is the only selection from the envy-free and Pareto efficient solution which satisfies endowment-peak-only. However, it can be shown by means of an

example that it is not possible to use weak envy-freeness in the characterization of the uniform rule as

2

the only weak no-envy and peaks-only rule .

This paper proceeds as follows. In Section 2, we introduce the model and in Section 3, we present the results.

2. The model

We follow Thomson (1995a) and Moreno (1996). There is an infinite population of ‘potential

˜

agents’, indexed by the positive integers, :. Let N denote the class of non-empty finite subsets of :. Each agent i[: is equipped with a continuous preference relation R defined over R _{and an}

i 1

endowment v [R _{. Let P denote the strict preference relation associated with R , and I the}

i 1 i i i

indifference relation. These preference relations are single-peaked: for each R , there is a number,_i

9

9

9

9

denoted by p(R )[R _{, such that for all z ,z [}R _{if z ,z <p(R ), or p(R )<z ,z , then z P z .}

i 1 i i 1 i i i i i i i i i

The preference relation R can be described in terms of the function r :R _{<h`j</sub>→R <sub><h`jdefined}

i i 1 1

as follows: given z_i<p(R ), r (z )_{i i i} >p(R ) and z I r (z ) if such a number exists, and r (z )_{i i i i i i i} 5 ` otherwise; given z<sub>i</sub>>p(R ), r (z )<sub>i i i</sub> <p(R ) and z I r (z ) if such a number exists, and r (z )<sub>i i i i i i i</sub> 50 otherwise. Let r (`)5lim r (z ). LetR be the class of single-peaked preference relations onR _{. For}

i _z→` i i 1

i

N

˜

all N[N, let R denote the cartesian product of N copies ofu u R, indexed by the members of N.

N

Similarly, let R _{denote the cartesian product of N copies ofu u} R _{. We write R 5(R ) ,}

1 1 N i i[N

N N _˜

p(R )5( p(R )) , and v 5(v) . An economy is a pair e5(R ,v )[R 3R _{. For all N[N,}

N i i[N N i i[N N N 1

N

let j denote the set of admissible economies.

N N

˜

For all N[N and all e5(R ,v )[j , let Z(e)5h(z ) [R _{:o z 5o vj denote the set of}

N N i i[N 1 N i N i

˜

feasible allocations of e. A solution is a mapping w which associates with every N[N and

N

ˆ

e5(R ,NvN)[j , a non-empty subset of Z(e). The intersection of twow solutions and w is denoted

ˆ ww.

N

˜

For all N[N and all e5(R ,NvN)[j , the allocation z[Z(e) is Pareto efficient for e if there is no

ˆ ˆ ˆ

z[Z(e) with z R z for all i_{i i i} [N, and z P z for some i_{i i i} [N. Let P(e) be the set of these allocations.

N

˜

For all N[N and all e5(R ,_Nv_N)[j , the allocation z[Z(e) is individually rational for e if z R_{i i}v_i

for all i[N. Let I(e) be the set of these allocations. The allocation z[Z(e) is envy-free in final consumptions for e if for all i, j[N, z R z (Foley, 1967).i i j

1

Moreno (1996) shows that there is no population-monotonic selection from the weak no-envy and Pareto efficient solution. However, he provides a solution, the proportional-sacrifice solution, that it is a selection from the individually rational and Pareto efficient solution satisfying population-monotonicity under a minor additional restriction on preferences.

2

Note that the intersection of the sets of envy-free and individually rational allocations may be empty. However, we can ask how to distribute fairly across agents the gains made possible by redistributions. We will therefore be looking for notions of equitable redistributions.

N

˜

Definition 1. For all N[N and e5(R ,NvN)[j , the allocation z[Z(e) is obtained through an

3

envy-free redistribution if for no i, j[N, such that v 1(v 2z )[R _{, we have v 1(v 2z )P z .}

i j j 1 i j j i i

Let F(e) be the set of allocations obtained through envy-free redistributions.

N

˜

For all N[N and e5(R ,NvN)[j , the set of Pareto efficient allocations obtained through envy-free

4

redistributions is non-empty. Indeed, the uniform allocation, introduced next, always exists, and

satisfies weak no-envy and Pareto efficiency.

N

˜

Definition 2. For all N[N and e5(R ,NvN)[j , the allocation z[Z(e) is the uniform allocation of e if there is l[R _{such that (i) when o p(R )>o v, then for all i[N, z 5minhl1v, p(R )j,}

1 N i N i i i i

and (ii) when oNp(R )i <oNvi then for all i[N, zi5maxhvi2l, p(R )i j.

Let U(e) denote the uniform allocation of e.

3. One-sided population-monotonicity

We now consider changes in the number of agents. We require the arrival of new agents to affect all agents present before the change in the same direction. A general version of this requirement was proposed and studied by Thomson (1983a,b). Chun (1986) considered a quasi-linear model of cost allocation and proposed the condition that all agents be affected in the same direction by the arrival of additional agents.

However, population-monotonicity is a strong requirement when combined with weak no-envy and

Pareto efficiency. In fact, there is no selection from the weak no-envy and Pareto efficient solution

satisfying population-monotonicity. The next step is to weaken the population-monotonicity con-dition. The following requirement is an adaptation of Thomson (1995a) to our context.

Definition 3. A solution is one-sided population-monotonic if for all N,N9[: with N9,N, and all

N N9

e5(R ,NvN)[j and e9 5(R ,N9vN9)[j if either (i)oNp(R )i <oNvi andoN9p(R )i <oN9vi or (ii)

o_Np(R )_i >o_Nv_i and o_N9p(R )_i >o_N9v_i, then for all i[N9, w_i(e)R_iw_i(e9), or for all i[N9,

wi(e9)Riwi(e).

When we impose one-sided population-monotonicity on selections from the no-envy and Pareto

efficient solution, the unique solution that remains acceptable is the uniform rule, as we show in the

5

next theorem .

3

The concept of envy-freeness in terms of allotment changes, called fair net trade, was formulated by Kolm (1972) and Schmeidler and Vind (1972) in the more general context of exchange economies.

4

In the sequel, when we say weak no-envy allocations, we refer to allocations obtained through envy-free redistributions.

5

Theorem 1. The uniform rule is the only selection from the weak no-envy and Pareto efficient

solution satisfying one-sided population-monotonicity.

Proof. It is clear that the uniform rule is weak envy-free and Pareto efficient and satisfies one-sided

population-monotonicity. Now we show that the uniform rule is the only one satisfying the above

properties. We will make the proof in two steps.

N

˜

Step 1. Let N[N and e5(R ,v )[j be such that o p(R )>o v. Let w7_{FP be a one-sided}

N N N i N i

population-monotonic solution. We prove thatw(e)5U(e). Let z5w(e). Since w7_{P this implies that} for all i[N, z_i<p(R ). The proof follows from the four claims below:_i

Claim 1.1. (Fig. 1) For all j[N such that v_j>p(R ), z_{j j}5p(R ). Suppose not. Then, for some j_j [N

such that v_j>p(R ), z_{j j},p(R ). By feasibility and efficiency, for some k_j [N, v_k,z_k<p(R ). Since_k ˜

w#F, z R_{j j}v_j1(z_k2v_k). Let N9[N be a set of agents such that uN9u5u uN and N>N9 5 [. Let

b:N→N9 be a bijection. Let R_{b(i )}5R for all i_i [N\hjj, let R_{b( j )}[R be such that p(R_{b( j )})5p(R ),_j

and v_{b( j )}1(z_i2v_i)P_{b( j ) j}z , and let v_{b(i )}5v_i, for all i[N. Let N0 5N<N9. Let e0 5(R_N99,v_N0)[

N0

j . Let z0 5w(e0). Note that o_Np(R )_i 1o_N9p(Rb(i ))>oNvi1oNvb(i ), and sincew#P, for all i[N

99

99

z_i <p(R ), and z_{i b(i )}<p(R_{b(i )}). Since w#F, v_{b(i )}5v_i for all i[N\hjj, and p(R_{b(i )})5p(R ) for all_i

99

99

i[N\hjj, we have z_{b(i )}5z_i for all i[N\hjj. Sincew#FP,v_{b( j )}5v_j, and p(R_{b( j )})5p(R ), we have_j

99

99

99

99

99

z 5z . Since w#F, z 5z , and v 1(z 2v)P z , we have z ±_{z . Thus, in the change}

b( j ) j b( j ) j b( j ) i i b( j ) j b( j ) j

99

99

from e to e0, if zb( j ).z (zj b( j ),z ) agent j gains (loses) and at least one ij [N\hjj loses (gains) in violation of one-sided population-monotonicity.

Claim 1.2. (Fig. 1) For all k[N such that v_k<p(R ), z_{k k}>v_k. Suppose not. Then, for some k[N

such thatvk<p(R ), zk k,vk. By feasibility and efficiency, for some i[N\hkj, vi,zi<p(R ). Sincei

w#F, z R_{k k}v_k1(z_i2v_i). Note that this is possible only if v_k1(z_i2v_i)>r (z ). Let us now_{k k}

introduce an additional agent, agent n11. Let v_k5v_n11 and let R_n11[R be such that p(R )_k 5

N<hn11j p(R_n11) and r_n11(v_n11).v_n111(z_i2v_i)1(v_k2z ). Let e_k 9 5(R ,R_{N n11},v_N,v_n11)[j be the

9

9

resulting economy and z9 5w(e9). Sincew#FP,v_k5v_n11 and p(R )k 5p(Rn11), we have zk5zn11.

9

9

9

9

Note that z_j 5z for all j_j [N\hij, z_n115z and z_{k i} 5z_i1(v_n112z_n11) is a feasible allocation. Since

w#F and rn11(vn11).vn111(zi2vi)1(vk2z ), the above allocation is not obtained through ank

9

envy-free redistribution. Indeed, if v_n11.z_n11.z , agent k gains and by one-sided population-_k

9

9

monotonicity, all agents gain. Therefore, zi >z . By feasibility, zi i ,zi1(vn112z ) and sincek

r_n11(v_n11).v_n111(z_i2v_i)1(v_k2z ), agent n_k 11 envies the redistribution of agent i, in violation

9

9

ofw#F. Thus, in the change from e to e9, if zn11>vn11 (zn11,z ) agent k gains (loses) and at leastk

one i[N\hkj loses (gains), in violation of one-sided population-monotonicity.

Claim 1.3. For all i, j[N such that vi<p(R ),i vj<p(R ) and p(R )j i 2vi>p(R )j 2vj, we have

z_i2v_i>z_j2v_j. Suppose not. Then, for some i, j[N such that v_i<p(R ),_i v_j<p(R ), and p(R )_{j i} 2

vi>p(R )j 2vj, we have zi2vi,zj2vj. Sincew#P, it is clear that vi1(zj2vj)P z , in violationi i

of w#F.

Claim 1.4. (Fig. 2) For all i, j[N such that vi<p(R ),i vj<p(R ) and p(R )j i 2vi>p(R )j 2vj, we have z_i2v_i.z_j2v_j if and only if z_j5p(R ). Suppose not. Then, for some i, j_j [N such that vi<p(R ),i vj<p(R ), and p(R )j i 2vi>p(R )j 2vj, we have zi2vi.zj2vj and zj,p(R ). Sincej

w#F, z R_{j j}v_j1(z_i2v_i). Note that this is possible only ifv_j1(z_i2v_i)>r (z ). Let us now introduce_{j j}

two additional agents, agents n11 and n12. Let v_j5v_n11 and let R_n11[R be such that

p(R )_j 5p(R_n11), vn111(zi2vi)Pn11 jz . Let Rn12[R and vn12 be such that vn122p(Rn12)5zj2

N<hn11j<hn12j

v_j. Let e9 5(R ,R_{N n11},R_n12,v_N,v_n11,v_n12)[j be the resulting economy. Note that

9

9

9

p(R ) for all i_i [N, z_n11<p(R_n11), and z_n12<p(R_n12). Sincew#F, v_j5v_n11 and p(R )_j 5p(R_n11),

9

9

9

9

9

we have zj 5zn11. By Claim 1.1, we have zn125p(Rn12). Note that zi5z for all ii [N, zn115zj

9

9

9

and z_n125p(R_n12) is a feasible allocation. Since w#F, z_n115z , and_j v_n111(z_i2v_i)P_{n11 j}z , we

9

9

9

have z ±_{z . Thus, in the change from e to e9, if z .z (z ,z ) agent j gains (loses) and at}

n11 j n11 j n11 j

least one i[N\hjj loses (gains), in violation of one-sided population-monotonicity.

N

˜

Step 2. Let N[N and e5(R ,_Nv_N)[j be such that o_Np(R )_i <o_Nv_i. Let w#FP be a one-sided population-monotonic solution. We prove thatw(e)5U(e). Let z5w(e). Since w#P this implies that

for all i[N, z_i>p(R ). The proof follows from the four claims below:_i

Fig. 3. Characterization of the uniform rule on the basis of one-sided population-monotonicity (Theorem 1, Claim 2.1).

Claim 2.1. (Fig. 3) For all j[N such that v_j<p(R ), z_{j j}5p(R ). Suppose not. Then, for some j_j [N

such that v_j<p(R ), z_{j j}.p(R ). By feasibility and efficiency, for some k_j [N, v_k.z_k>p(R ). Since_k ˜

w#F, for all v 1(z 2v )[R _{, z Rv 1(z 2v ). Let N9[N be a set of agents such that}

j k k 1 j j j k k

uN9u5u uN and N>N9 5 [. Let b:N→N9 be a bijection. Let R_{b(i )}5R and_i v_{b(i )}5v_i for all i[N\hjj, and let R_{b( j )}[R andv_{b( j )} be such that p(R_{b( j )})2v_{b( j )}5p(R )_j 2v_j, andv_{b( j )}1(z_i2v_i)P_{b( j )}v_{b( j )}1

N0

(z_j2v_j). Let N0 5N<N9. Let e0 5(R ,_N0v_N0)[j . Let z0 5w(e0). Note that o_Np(R )_i 1

99

99

o_N9p(Rb(i ))<oNvi1oNvb(i ), and since w#P, for all i[N zi >p(R ), and zi b(i )>p(Rb(i )). Since

99

99

w#F, v_{b(i )}5v_i for all i[N\hjj, and p(R_{b(i )})5p(R ) for all i_i [N\hjj, we have z_{b(i )}5z_i for all

99

99

i[N\hjj. Since w#FP, and p(R_{b( j )})2v_{b( j )}5p(R )_j 2v_j, we have z_{b( j )}2v_{b( j )}5z_j 2v_j. Since

99

99

99

w#F, z 2v 5z 2v, and v 1(z 2v)P v 1(z 2v), we have z 2v ±_{z 2}

b( j ) b( j ) j j b( j ) i i b( j ) b( j ) j j b( j ) b( j ) j

99

99

v_j. Thus, in the change from e to e0, if z_{b( j )}2v_{b( j )},z_j2v_j (z_{b( j )}2v_{b( j )}.z_j2v_j) agent j gains (loses) and at least one i[N\hjj loses (gains) in violation of one-sided population-monotonicity.

Claim 2.2. (Fig. 4) For all k[N such that vk>p(R ), zk k<vk. Suppose not. Then, for some k[N

such thatv_k>p(R ), z_{k k}.v_k. By feasibility and efficiency, for some i[N\hkj, v_i.z_i>p(R ). Since_i w#F, for allv 1(z 2v)[R _{, z R v 1(z 2v). Let us now introduce an additional agent, agent}

j i i 1 k k k i i

n11. Let R_n11[R and v_n11 be such that v_n112p(R_n11)5v_k2p(R ) and r_{k n11}(v_n11),v_n111

N<hn11j

(z_i2v_i)1(v_k2z ). Let e_k 9 5(R ,R_{N n11},v_N,v_n11)[j be the resulting economy and z9 5w(e9).

9

9

Since w#FP, and v_n112p(R_n11)5v_k2p(R ), we have z_{k k}2p(R )_k 5z_n112p(R_n11). Note that

9

9

9

9

z_j 5z for all j_j [N\hij, z_n112vn115zk2vk and zi 5zi1(vn112zn11) is a feasible allocation. Since w#F and r_n11(v_n11),v_n111(z_i2v_i)1(v_k2z ), the above allocation is not obtained_k

9

through an envy-free redistribution. Indeed, if v_n11,zn11,vn111(zk2vk), agent k gains and by

9

9

one-sided population-monotonicity, all agents gain. Therefore, z_i <z . By feasibility, z_{i i} .z_i1

9

(v_n112zn11) and since rn11(vn11).vn111(zi2vi)1(vk2z ), agent nk 11 envies the

redistribu-9

tion of agent i, in violation ofw#F. Thus, in the change from e to e9, if z_n11<v_n11 (z_n11.v_n111 (z_k2v_k)) agent k gains (loses) and at least one i[N\hkj loses (gains), in violation of one-sided

population-monotonicity.

Claim 2.3. For all i, j[N such that v_i>p(R ),_i v_j>p(R ) and_j v_i2p(R )_i >v_j2p(R ), we have_j v_i2z_i>v_j2z . Suppose not. Then, for some i, j_j [N such that v_i>p(R ),_i v_j>p(R ), and_j v_i2

p(R )_i >v_j2p(R ), we have_j v_i2z_i,v_j2z . Since_j w#P, it is clear that v_i1(z_j2v_j)P z , in_{i i} violation of w#F.

Claim 2.4. (Fig. 5) For all i, j[N such that v_i>p(R ),_i v_j>p(R ) and_j v_i2p(R )_i >v_j2p(R ), we_j

have v_i2z_i.v_j2z if and only if z_{j j}5p(R ). Suppose not. Then, for some i, j_j [N such that v_i>p(R ),_i v_j>p(R ), and_j v_i2p(R )_i >v_j2p(R ), we have_j v_i2z_i.v_j2z and z_{j j}.p(R ). Since_j w#F, for all v 1(z 2v)[R _{, z Rv 1(z 2v). Let us now introduce two additional agents,}

j i i 1 j j j i i

agents n11 and n12. Let R_n11[R and vn11 be such that vn112p(Rn11)5vj2p(R ), andj

v_n111(z_i2v_i)P_n11v_n111(z_j2v_j). Let R_n12[R andv_n12 be such that p(R_n12)2v_n125v_j2z ._j

N<hn11j<hn12j

Let e9 5(R ,R_{N n11},R_n12,v_N,v_n11,v_n12)[j be the resulting economy. Note that

9

9

9

p(R ) for all i_i [N, z_n11>p(R_n11), and z_n12>p(R_n12). Since w#F, and v_n112p(R_n11)5v_j2

9

9

9

9

p(R ), we have_j v_j2z_j 5v_n112z_n11. By Claim 2.1, we have z_n125p(R_n12). Note that z_i 5z for all_i

9

9

9

i[N, v_n112z_n115v_j2z and z_{j n12}5p(R_n12) is a feasible allocation. Since w#F, v_j2z_j 5

9

9

v_n112z_n11, and v_n111(z_i2v_i)P_n11v_n111(z_j2v_j), we have v_n112z_n115v_j2z . Thus, in the_j

9

9

change from e to e9, if vn112zn11.vj2z (vj n112zn11,vj2z ) agent j gains (loses) and at leastj

one i[N\hjj loses (gains), in violation of one-sided population-monotonicity. h

Note that here, in contrast to the problem of fair allocation (Thomson, 1995a), we have not used

replication-invariance to characterize the uniform rule as the only one-sided population-monotonic

selection from the weak no-envy and Pareto efficient solution. By Theorem 1, since the uniform rule is replication-invariant, it is clear that if a subsolution of the weak no-envy and Pareto efficient solution satisfies one-sided population-monotonicity, then it is replication-invariant. In the problem of fair division (Thomson, 1995a), the counterpart of the above statement is not true.

Acknowledgements

´ ´ ´

I would like to thank Pablo Amoros, Salvador Barbera, Carmen Bevia, Luis Corchon, Carmen ´ ¨

Herrero, Cristina Mazon, Jorg Naeve and William Thomson for their very helpful comments. I thank the referee and the associate editor, whose comments led to an improvement of the paper. Financial

´

support from the Instituto Valenciano de Investigaciones Economicas is gratefully acknowledged. All remaining errors are mine.

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