A simple characterization of the uniform rule

Economics Letters North-Holland

57

40 (1992) 57-60

A simple characterization of the uniform rule Stephen Ching * University of Rochester, Rochester NY, USA Received Accepted

6 July 1992 17 August 1992

We prove, by a simple and direct argument, that the uniform rule is the only allocation rule satisfying Pareto efficiency, Pareto indifference, strategy-proofness, and no-envy on the domain of economies with single-plateaued preferences and a collective endowment. A similar result can be obtained with single-plateaued preferences replaced by single-peaked preferences [Sprumont (199111.

1. Introduction

In a recent paper, Sprumont (1991) considers the problem of allocating a collective endowment of a perfectly divisible commodity in economies where all agents have single-peaked preferences. He shows that a certain allocation rule, known as the uniform rule, is the only rule satisfying Pareto efficiency, strategy-proofiess, and no-envy. His proof relies on another characterization in which no-envy is replaced by a significantly weaker axiom, anonymity. In this note, we show how to make use of the full strength of no-envy and prove the characterization by a simple and direct argument. ’ We also consider economies where all agents have single-plateaued preferences. The difference between single-peaked and single-plateaued preferences is slight: there is a single preferred consumption in the former versus a segment of preferred consumptions in the latter. Then we show that a similar result can be obtained on the single-plateaued domain. The note is divided into three sections. Section 2 describes the model. Section 3 gives formal statements of the results. Since the proofs of the results are very similar, only the proof of the result pertaining to the single-plateaued domain is given. 2. The model

The problem is to allocate a collective endowment of a perfectly divisible commodity among a group N of n agents, indexed by i. The endowment is denoted by R. A feasible allocation, or Correspondence to: Stephen Ching, Department of Economics, University of Rochester, * I am indebted to Professor William Thomson for his inspiration and patient guidance. Ryo-ichi Nagahisa for use’ful comments. All errors are mine. ’ The argument used here is similar to Thomson (n.d.). 0165-1765/92/$05.00

0 1992 - El sevier Science

Publishers

B.V. All rights reserved

Rochester, NY 14627, USA. I am grateful to Hideo Konishi

and

S. Ching /A

58

simple characterization of the uniform rule

simply an allocation, is a vector z = (z,, . . . , zn) E rW: such that Cz, = 0. (Note that free disposability is not assumed.) Let 2 be the set of all allocations. Each agent i EN is equipped with a continuous preference relation defined over R,, 2 denoted by Ri. Let Pi be the strict relation associated with R,, and li be the indifference relation. The preference relation Ri is single-plateaued if there is an interval [p(R,), p(Ri)] c [w, such that for all xi, yi E R,, if yi
An allocation rule, or simple a rule, is a non-empty correspondence 4 : LP + 2 satisfying the property of essential single-valuedness: for all R EL&?~,for all z, z’ E 4(R), and for all

i EN,

z,Z,zi.

We are interested in rules satisfying the following axioms. The first one says that an allocation is recommended only if there is no other allocation that is preferred by all agents to it, and strictly preferred by at least one agent. For all R EL%?“,and for all z E C+(R), there is no z’ E Z such that for all i EN, Pareto efficiency. ziR,zi, and for some i EN, ziP,z,. The next axiom requires that if two allocations are such that all agents are indifferent them, either both of them are recommended, or none of them. For all R EL??“, and for all z, z’ E Z, if for all i EN, Pareto indifference. and only if z’ E 4(R).

between

ziZizI, then z E 4(R)

if

In general, different sets of allocations are recommended for different preference profiles. As a result, an agent may gain by misrepresenting his preferences. The following axiom prevents profitable misrepresentations. For all R ES”, Strategy-proofness. ziRiz;. 4

for all z E 4(R),

for all i EN,

for all R: ~9,

and for all

z’ E qXR:, R-J,

The last axiom says that no agent prefers any other agent’s consumption to his own. No-entry [Foley (196711. For all R ~9~,

for all z E 4,(R), and for all i, j EN, ziRizj.

Before showing the characterization, we need to extend the uniform rule to the single-plateaued domain. A natural extension is as follows: 2 Similar results can be obtained with preference relations defined over [O, 01. s A preference relation R, is single-peaked if p(Ri) = jj(R,). Let p(Ri) =p(Ri) = jXR,). 4 Note that rules are only essentially single:alued with respect to reported preference profiles. Since there is no unambiguous way to compare different sets of recommended allocations, strategy-proofness can be defined in a number of ways. Here we use the weakest conceivable definition.

S. Ching /A

Uniform rule, U.

59

simple characterization of the uniform rule

Let R ES%“’ be given. For all i EN,

min{p_( 4) yA} {z~~zEZ, max{p(Ri),

andp_(R,)

IZiIp(Ri)}

A}

if

O
if

zg(Ri)

if

0 2 C3(Ri)>

sfi
where A solves CUi(R) = R.

3. The results It is easy to check that Pareto efficiency implies that if there is ‘not enough to divide’, i.e. if 0 5 Cp(Ri), each agent i consumes less than the left edge of his plateau, p_(R,); if there is ‘too much 6 divide’, i.e. if 0 2 Cp(Ri), each agent i consumes more than the right edge of his plateau, p(R,). The following result shows that, in either case, the set of recommended allocations is a singleton. Lemma

1.

Let C$ be a rule satisfying Pareto

R I Cp(R,), or 0 2 CjXR,), -

efficiency.

Then for all R ~9~

such that either

4(R) is a singleton.

Proof.

Fix R ES?‘“. Suppose, without loss of generality, that 0 I Cp(Ri). Suppose, by way of contradiction, that there exist z, z ’ E 4(R) such that z # z’. By Pareto efficiency, zi, zi 2 p(Ri) for all i EN. Then there is i EN such that zi I 2;. Note that either ziPiz!, or zlPizi, contradicting the Q.E.D. assumption that 4 is essentially single-valued. Obviously, if a rule satisfies Pareto efficiency, it is single-valued on the single-peaked subdomain. Consequently, on the single-peaked subdomain, Pareto indifference is vacuously satisfied by a rule satisfying Pareto efficiency. For each agent i, given the preferences of all other agents, we have a counterpart of Sprumont’s result: Pareto efficiency, and strategy-proofness together imply that if there is not enough to divide, the consumption of agent i depends on his preference relation only through the left edge of his plateau; if there is too much to divide, it depends on his preference relation only through the right edge of his plateau. Lemma

2.

Let 4 be a rule satisfying Pareto efficiency, and strategy-proofness. Then for all and for all R,! ~99 such that either a-< Cp(Rj) with p(Rj) = p(Ri), or _ -

R ES”, for all i EN, R 2 E@(R,) with j(Ri)

=~(Ri),

~i(R[I R-i) = +i(R)*

Proof Fix R ~9%‘“. Suppose, without loss of generality, that R I Cp_(Ri). Fix i EN, RI EL%’with p(R[) =p(R,). By Lemma 1, $(R) and $(R(, R-i) are singletons. By Pareto efficiency, $i(R>, qi(Rj, RTi) I p(Ri). Suppose, by way of contradiction, that qbi(R) # 4i(Rj, R-i). Then either 4i(R)Pi’4i(Rj,R_i), or 4i(Ri, R_i)Pi4i(R), contradicting strategy-proofness. Q.E.D.

Sprumont proves that the uniform rule is the only rule satisfying Pareto efficiency, strategy-proofsubdomain. We are now ready to state and prove, by a simple and direct argument, a counterpart of Sprumont’s theorem on the single-plateaued domain.

ness, and no-envy on the single-peaked

60

Theorem

S. Ching / A simple characterization of the uniform rule

1.

The uniform rule is the only rule satisfying Pareto and no-envy. 5

efficiency,

Pareto

indifference,

strategy-proofness,

Proof Let C$ be a rule satisfying no-envy. Fix R ~9~.

Pareto efficiency, Pareto indifference,

strategy-proofness,

and

Case 1: 0 < Cg(R,), or J2 > CjXR,). Suppose, without loss of generality, that $2 < Cp(RJ. By Pareto efficiency, $i(R) 5 p(Rj) for all i EN. By Lemma 1, 4(R) is a singleton. Suppose, by way of contradiction, that 4(R) /U(R). Then there is i EN such that +i(R) < Ui(R) I p(RJ. Let Ri ES%? be such that p(R[) = p(Ri), and p(Rj) 2 0. By Lemma 2, 4JRI, R_i) = +JR>. ByPareto efficiency, and the feasibility constraint, there is j E N such that q(R) < 43,(R:, R-J ip(Rj). Since q.(R)
By Pareto efficiency, and Pareto indifference, $(R) = {z E Z ]p_(Ri)

Sprumont shows that the uniform rule satisfies Pareto efficiency, strategy-proofness, and no-envy on the single-peaked subdomain. A similar argument can be used to show that the uniform rule satisfies the three axioms on this domain. By Lemma 1, Pareto indifference is vacuously satisfied by the uniform rule on the domain of economies R ~9 such that 0 5 Cp(R,), or 52 2 CjXR,). By definition, the uniform rule satisfies Pareto indifference on the domain of economies R ~59~ such Q.E.D. that Cp(Ri) _<0 I Cp(Ri). _ Although Sprumont’s theorem is stated on a smaller domain, his theorem is not implied by Theorem 1. Fortunately, an argument very similar to the proof of Theorem 1 can be used to prove Sprumont’s theorem. To do so, an additional piece of notation is needed. A single-peaked preference relation Ri can be described by a function r, : R + -+ R + U ~0 such that for all 0 with y&zi, and ribi) = m otherwise; for all p(Ri) +t(R).

References Foley, Duncan, 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Sprumont, Yves, 1991, The division problem with single-peaked preferences: A characterization of the uniform rule, Econometrica 59, 509-519. Thomson, William, nd., Consistent solutions to the problem of fair division when preferences are single-peaked, Journal of Economic Theory, forthcoming. 5 Whether Theorem 1 is tight is an open question. The difficulty comes indifference, strategy-proofness, and no-enuy, but not Pareto efficiency.

from constructing

a rule that

satisfies

Pareto