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An alternative characterization of the equal-distance rule for

allocation problems with single-peaked preferences

*

Carmen Herrero , Antonio Villar

´ ´

Departamento de Fundamentos del Analisis Economico, Universidad de Alicante, 03071 Alicante, Spain

Received 26 May 1998; received in revised form 28 June 1999; accepted 9 September 1999

Abstract

The main result of this paper is that for allocation problems in which preferences are single-peaked, the equal distance rule is the only rule satisfying symmetry, efficiency, dummy, agenda-independence, and constrained linearity.  2000 Elsevier Science S.A. All rights reserved.

Keywords: Single-peaked preferences, Equal-distance rule

JEL classification: D63

1. Introduction

The problem under consideration is that of allocating a fixed amount of an infinitely divisible good among a group of agents whose preferences are single-peaked. This situation arises, for example, when there is a task to be performed by a group of agents who supply a quantity of labor paid at an hourly rate. If agents’ disutility of labor is concave, each individual has an ideal labor supply, called her peak, and having to work more or less decreases her utility. When the aggregate supply of labor differs from the total amount required to complete the task, some procedure has to be devised to allocate the difference.

The equal-distance rule (Thomson, 1994) is one such procedure. It proposes to divide equally any difference between the aggregate supply of labor and the amount of labor required to accomplish the task, as far as it is compatible with feasibility. The motivation of the rule is that the difference between the effort required and the effort supplied is a common responsibility.

An axiomatic characterization of the equal-distance rule is given by Herrero and Villar (1999). They show that the equal-distance rule is the only rule satisfying efficiency,equal treatment of equals,

*Corresponding author. Tel.: 1_{34-96-5903-618; fax:}1_{34-96-5903-685.}

E-mail address: carmen.herrero@ua.es (C. Herrero)

consistency, agenda-independence, and ar-truncation. Efficiency, equal treatment of equals, and consistency are standard properties [see Sprumont (1991), Ching (1994), and Thomson (1994)].

Agenda-independence requires the allocation proposed by a rule to be the same no matter how the

total amount of work required is allocated, one shot or sequentially. Finally, ar-truncation establishes that any information on preferences below the average rationing experienced by the agents should be ignored.

The purpose of this note is to provide an alternative characterization of the equal-distance rule, by replacing equal treatment of equals,consistency and ar-truncation by symmetry (a weaker form of the

parity principle), dummy (a very weak form of consistency), and a new axiom, constrained linearity. This is nothing else than a weak form of the standard property of linearity. Besides its formal interest, this new characterization provides an additional perspective from which to interpret the equal-distance rule.

The paper is organized as follows. In Section 2 we present the model and the properties. In Section 3 we present the characterization.

2. The model

We follow Thomson (1995). LetN_{be an infinite population of potential agents. For an agent i}[N_,

let R be a preference relation overR _{, and let P be the associated strict preference. The preference}

i 1 i

relation R is single-peaked if there exists a number, p(R )[R _{, such that for all x, x}9[R _,

i i 1 1

x9 ,_x#_{p(R ) or p(R )}#_x,_x9 ⇒_{x P x}9_.

fs

i

d

s

i

dg

i

LetS_{denote the family of single-peaked preference relations over} R_{1. The preferences are drawn}

N

from S_{. Let} 1 _{be the class of all (non-empty) finite subsets of} N_{. Given N}[1_{, let} S _{denote the}

N

Cartesian product of uNu copies of S_{, indexed by the members of N. Similarly,} R_{1 stands for the}

Cartesian product of uNu copies of R _.

1

that ox 5_{T. Let X(e) be the set of feasible allocations of e.}

i

N N

A rule is a mapping F :E→ <R _{, such that for all e}5_{(R,T )}[E _{, F(e) is a feasible allocation for} e.

Next, we consider several properties a rule may fulfill. The first one is a weak form of impartiality: When in a certain problem, all agents have identical preferences, they should all be treated identically.

N

only if each agent consumes no more than her preferred amount if T#o_{p(R ) and no less otherwise}

i

N

it assigns to all agents in N\Q are identical after and before the departure of agents in Q. Note that

1

dummy is a very weak form of consistency (Thomson, 1994). Formally,

N

Consider a group of agents N, with preferences R[S _{, that will face a certain problem, and solve}

it by forecasting the amount of labor required to perform the task. Once this is done, suppose that the actual amount of labor required is greater than expected. Then two options are open: either the original distribution is cancelled altogether and the actual problem is solved, or the rule is applied to the incremental amount of labor required, after adjusting the preferences by shifting them by the amounts initially assigned. Agenda-independence requires the recommendation made by the rule to be independent of the chosen option. Formally:

N

Prior to introducing the next property, let us start by defining the usual linearity axiom. An allocation

N _]1 _]1

rule F is linear if for all N[N_{, all R}[S _{, and all T, T}9_{, T}0[R_{1, if T}0 5 _T1 _T9 _{then F(R,}

2 2

T0₎5_{1 / 2F(R, T )}1_{1 / 2F(R,T}9_{). It follows from Lemma 1 below that efficiency,}

agenda-indepen-N

dence,symmetry,and linearity are incompatible. However, if we define, for all N[1 _{and all R}[S _,

A(R)5

O

_{p(R )}2_{nmin p(R ),}

i N i

it can be easily seen that these properties are compatible when we restrict attention to problems (R, T )

*

fulfilling the condition T$_{A(R). Indeed,}E 5h_{(R, T )}[E_:T$_A(R)j _{is the largest domain on which}

2

linearity is compatible with efficiency, symmetry, and agenda-independence. This motivates the following property:

The following rule is the subject of our analysis:

N

Equal-distance rule, D: For all N[1_{, all e}5_{(R, T )}[E _{, and all i}[_N,

1

Note also that the usual meaning of ‘dummy’ in cooperative game theory is different from the one adopted here.

2

D (e)5_maxh_{0, p(R )}1lj_,

i i

where l _{is chosen so that D(e)}[_X(e).

The equal-distance rule rations the agents as equally as possible from their peaks, subject to the

3

condition that all be assigned a nonnegative amount. It can be shown that the rule D selects that feasible point that is closest, according to the Euclidean distance, to the vector of preferred contributions ( p(R )) . Note that if P(R)#_T, l$_{0, and otherwise} l,_0.

i N

3. The characterization

We will need the following lemma (Herrero and Villar, 1998):

Lemma 1. Let F be a rule satisfying efficiency, symmetry, and agenda-independence. Then for all

N

N[1_{, all e}5_{(R, T )}[E _{, and all i}[_{N, if T}$o_{p(R ), then F(e)}5_D(e).

i

Proof. As T$o_{p(R ), we first allocate} o_{p(R ). By efficiency, every agent receives her peak. After}

i i

translating each agent’s preferences by his peak, we obtain a profile of identical preferences. By symmetry, the left over T2o_{p(R ), should be divided equally. The conclusion follows by agenda}

i

independence. h

We now obtain our main result:

Theorem 1. The equal distance rule is the only rule satisfying efficiency, symmetry, dummy, agenda-independence, and constrained linearity.

Proof. The equal-distance rule satisfies all the properties (Thomson, 1994; Herrero and Villar, 1998).

Let us consider the converse implication. In view of Lemma 1 it suffices to consider problems e5_(R,

*

To see that the properties in Theorem 1 are independent, consider the following examples. In each case we mention the property that is not satisfied.

Symmetry. Select a particular agent i in N _{(the set of potential agents). For all N}[N_{, all e}5_(R,

N

Thanks are due to William Thomson and an anonymous referee for helpful comments. Financial ˜

´

support from the Direccion General de Ensenanza Superior, under project PB97-0120, is gratefully acknowledged. The first author is also endebted to the Ministerio de Educacion y Ciencia, PR 97-1361084.

4

Appendix A

Claim 1. Let F be a rule satisfying efficiency, symmetry, and agenda-independence. For all N[1

N

_*

_*

and all R[S _{, let A(R)}5o_{p(R )}2_{nminp(R ), and} E 5h_{(R, T )}[E_:T$_A(R)j_{. Then,} E _{is the}

i i

largest domain on which F can also satisfy linearity.

N

(i) First note that if F is agenda-independent, for all R[S _{and all T}._T9 $_{0, F(R, T )}$_F(R, T9_).

(ii) Let F be a rule onE _{satisfying efficiency, symmetry, and agenda-independence, and linearity}

N

Efficiency, symmetry, and constrained linearity are trivially satisfied. Let us show that it also satisfies agenda-independence. As D is agenda-independent, it is enough to check this property for uNu5_{3 and T}#_A(R).

Let e5_{(R, T )}[E_{, T}9_,T0[R_{1such that T}9 1_T0 5_{T, e}9 5_{(R, T}9_{), R}9_{be obtained by shifting R by} F(e9_{), and e}0 5_(R9_{, T}0_{). We have to show that F(e)}5_F(e9₎1_F(e0_{). There are two cases to be}

considered.

(a) B(R)#_T#_{A(R). Thus, F(e)}5_{(0, b}2_{a, T}2_b1_a).

9

9

(a.1) B(R)#_T9 #_{A(R). In this case F(e}9₎5_{(0, b}2_{a, T}9 2_b1_{a) and thus p(R )}5_{p(R )}5_a,

i j

9

p(R )5_c1_b2_a2_T9_{. Hence, B(R}9₎5_{0, A(R}9₎5_c1_b2_2a2_T9_{. As T}0 #_A(R9_{) by assumption, it}

k

follows that F(e0₎5_{(0, 0, T}0_{). Therefore, F(e)}5_F(e9₎1_F(e0_).

9

9

9

(a.2) T9 #_{B(R). Thus F(e}9₎5_{(0, T}9_{/ 2, T}9_{/ 2) and p(R )}5_{a, p(R )}5_b2_T9_{/ 2, p(R )}5_c2_T9_{/ 2.}

i j k

Hence, B(R9₎5_2(b2_T9_{/ 2}2_{a), A(R}9₎5_c1_b2_2a2_T9_{. As B(R}9₎#_T0 #_A(R9_{), by assumption,} F(e0₎5_{(0, b}2_T9_{/ 2}2_{a, T}0 2_b1_a1_T9_{/ 2), and therefore, F(e)}5_F(e9₎1_F(e0_).

(b) T#_{B(R). Thus F(e)}5_{(0, T / 2, T / 2). Consequently, T}9 #_{B(R) and F(e}9₎5_{(0, T}9_{/ 2, T}9_{/ 2).}

Moreover, T0 #_B(R9₎#_B(R)2_T9_{so that F(e}0₎5_{(0, T}0_{/ 2, T}0_{/ 2) and therefore, F(e)}5_F(e9₎1_F(e0_).

Finally, F fails to satisfy dummy: Let N5h_{i, j, k}j_{be such that p(R )}5_{2, p(R )}5_{3, and p(R )}5_5.

i j k

Let T5_{2 and consider the problem e}5_{(R, T ). Then, F(e)}5_{(0, 1, 1), and thus, i is a dummy in e for} F. If we now remove player i, in the reduced problem e_h , F (e )5₀±_{1, and F (e )}5₂±

j,kj j hj,kj k hj,kj

1. h

References

Ching, S., 1994. An alternative characterization of the uniform rule. Social Choice and Welfare 11, 131–136.

Herrero, C., Villar, A., 1998. Agenda independence in allocation problems with single-peaked preferences, mimeo, Universidad de Alicante.

Herrero, C., Villar, A., 1999. The equal-distance rule in allocation problems with single-peaked preferences. In: Alkan, A., Aliprantis, R., Yannelis, N.C. (Eds.), Current Trends in Economics, Theory and Applications, Springer Verlag, Berlin, pp. 215–223.

Sprumont, Y., 1991. The division problem with single-peaked preferences: a characterization of the uniform allocation rule. Econometrica 59, 509–519.

Thomson, W., 1994. Consistent solutions to the problem of fair division when preferences are single-peaked. Journal of Economic Theory 63, 219–245.