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Indifference and the uniform rule
*
Lars Ehlers
Maastricht University, Department of Quantitative Economics, P.O. Box 616, 6200 MD Maastricht, The Netherlands
Received 29 March 1999; accepted 3 November 1999
Abstract
We consider private good economies with single-plateaued preferences. We show that the uniform rule is the only allocation rule satisfying indifference (in terms of welfare), strategy-proofness and no-envy. Indifference (in terms of welfare) means that if one allocation is recommended, then another allocation is recommended if and only if all agents are indifferent between these allocations. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Uniform rule; Single-plateaued preferences; Indifference; Strategy-proofness; No-envy
JEL classification: D63; D71
1. Introduction
We consider the problem of allocating some amount of a perfectly divisible good among a finite set of agents. A rule associates an allocation with each profile of preferences. A certain rule has been defined for economies with single-peaked preferences, the uniform rule, and in a number of studies this rule has been shown to play the essential role. Sprumont (1991) was the first to provide an axiomatic characterization of it. He showed that it is the only rule satisfying strategy-proofness (telling the truth is a weakly dominant strategy), Pareto-optimality (only efficient allocations are chosen) and no-envy (no agent wants to exchange his share with the share of another agent). Ching (1994) showed that in this characterization no-envy can be weakened to equal treatment of equals (when two agents have the same preferences, then they should be indifferent between their shares according to their common preference). Ching and Serizawa (1998) showed that the maximal domain
1
ensuring the compatibility of these properties is the domain of single-plateaued preferences : instead of a single best consumption (the peak), a whole segment of most preferred consumptions is allowed.
*Tel.:131-43-388-3932; fax: 131-43-325-8535. E-mail address: [email protected] (L. Ehlers) 1
Moulin (1984) introduced single-plateaued preferences in the context of public good provision.
Ching (1992) considered choice correspondences on the domain of single-plateaued preferences and extended the uniform rule. He showed that the extended uniform rule is the only rule satisfying
2
strategy-proofness, Pareto-optimality, no-envy, and indifference (in terms of welfare) , defined as follows. Under indifference (in terms of welfare), if a rule recommends an allocation, then another allocation is recommended if and only if all agents are indifferent between them. Thus, in terms of welfare there is no discrimination among allocations and agents. Using the same arguments as in Ching (1994), it can be shown that if no-envy is replaced by equal treatment of equals, then the same uniqueness result holds. However, Ching (1992), Footnote 5, raised the question whether his characterization is tight. In this note we show that Pareto-optimality is superfluous. The uniform rule is the only rule satisfying indifference (in terms of welfare), strategy-proofness, and no-envy. The proof involves arguments different from former characterizations of the uniform rule. However, it is not yet clear whether the replacement of no-envy by equal treatment of equals would also yield uniqueness.
2. The model and the result
A collective endowment of a perfectly divisible commodity has to be allocated among a finite set of agents. Let N5h1, . . . ,nj denote the set of agents and E the collective endowment. A feasible
N
allocation is a vector z[R1 such thatoi[Nzi5E, i.e., free disposal is not allowed. Denote by Z the set of all feasible allocations. Each agent i[N is equipped with a preference relation R over [0, E].i
Denote by P the strict relation associated with R , and by I the indifference relation. The preferencei i i
]
relation R is single-plateaued if there is an interval [p(R ), p(R )]i ] i i #[0, E] such that for all x , yi i[[0,
] ]
E], if yi,xi#p(R ) or p(R )i i #xi,y , then x P y , and if x , yi i i i i i[[p(R ), p(R )], then x I y . Ai i i i i
] ] ]
preference relation R is single-peaked if it is single-plateaued and p(R )5p(R ). Denote by 5 the set
i ] i i
N
of all single-plateaued preferences. A preference profile (R , . . . ,R )[5 is denoted by R. For
1 n
S#N, denote by R the restriction of the profile R to the coalition S, i.e., RS S5(R )i i[S. Since the
N
endowment is fixed, an economy is simply denoted by R[5 .
N
An allocation rule is a non-empty correspondencew:5 →Z, i.e., an allocation rule recommends
for each economy a non-empty set of allocations. We are interested in identifying the rules satisfying the following axioms. The first one says that if a certain allocation is recommended, then another allocation is recommended if and only if all agents are indifferent between them. Thus, in terms of welfare there is no discrimination among allocations and agents.
N
Indifference (in terms of welfare): For all R[5 , all z[w(R), and all z9[Z: z9[w(R) if and
9
only if for all i[N, z I z .i i i
In general, different sets of allocations are recommended for different preference profiles. Strategy-proofness prevents agents from gaining by mispresenting their preferences.
N
9
9
In Ching (1992), rules are essentially single-valued. His fourth property is originally called Pareto-indifference. Here, we merge these two properties and call the result indifference(in terms of welfare).
3
Note that we only consider rules satisfying indifference. Therefore, we could equivalently use weaker versions of
9 9
The last axiom says that every agent weakly prefers his consumption to any other agent’s consumption.
N
No-envy: For all R[5 , all z[w(R), and all i, j[N, z R z .
i i j
Ching (1992) extended the uniform rule to the single-plateaued domain as follows.
N
Note that in cases (i) and (iii) U(R) is a singleton. Whenever U(R) is a singleton, we also write U (R) to indicate the amount of agent i at the unique uniform allocation. Now we are able to state thei result. Section 3 provides a proof of Theorem 2.1
Theorem 2.1. The uniform rule is the only rule satisfying indifference, strategy-proofness, and no-envy.
4
Note that the axioms of Theorem 2.1 imply Pareto-optimality. The axioms of Theorem 2.1 are independent. The following three examples establish independence.
N e e
satisfies indifference and no-envy, but not strategy-proofness.
N ¯ ¯
Example 2.4. For all R[5 , if E,o p(R ), then w (R)5min(p(R ),E ), w (R)5min(p(R ),E2
i[N] i 1 ] 1 2 ] 2
¯ ¯ ¯
w1(R)), and so on. Otherwise,w(R)5U(R). Then, w satisfies indifference and strategy-proofness, but not no-envy.
3. Proof of Theorem 2.1
The following property turns out to be useful for the proof of Theorem 2.1. A rule satisfies unanimity if it recommends the allocations where each agent receives one of his most preferred amounts, whenever it is possible. It is a very weak form of Pareto-optimality.
N ]
Lemma 3.1. Let w be a rule satisfying indifference and strategy-proofness. Then w satisfies
Next, we prove that a rule satisfying the axioms of Theorem 2.1 coincides with the uniform rule for all profiles at which there is excess demand.
t11
Subcase 2.1: j[h1, . . . ,tj. Similar to Case 1, we obtain a contradiction to strategy-proofness.
t11 ¯ t11 ¯
By using the same arguments as in Lemma 3.2, it is easy to prove that a rule satisfying the axioms of Theorem 2.1 coincides with the uniform rule for all profiles at which there is excess supply.
Lemma 3.3. Let w be a rule satisfying indifference, strategy-proofness, and no-envy. Then for all
N ]
R[5 such that o p(R ),E, w(R)5U(R).
j[N j
Now, Lemma 3.1, 3.2 and 3.3 imply Theorem 2.1.
Acknowledgements
Useful comments of Hans Peters, Ton Storcken, Stephen Ching, Yves Sprumont, William Thomson, and an anonymous referee are acknowledged.
References
Ching, S., 1992. A simple characterization of the uniform rule. Economics Letters 40, 57–60.
Ching, S., 1994. An alternative characterization of the uniform rule. Social Choice and Welfare 11, 131–136.
Moulin, H., 1984. Generalized Condorcet-winners for single-peaked preferences and single-plateaued preferences. Social Choice and Welfare 1, 127–147.
Sprumont, Y., 1991. The division problem with single-peaked preferences: A characterization of the uniform allocation rule. Econometrica 59, 509–519.
