Indifference and the uniform rule

Economics Letters 67 (2000) 303–308 www.elsevier.com / locate / econbase

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 ensuring the compatibility of these properties is the domain of single-plateaued preferences 1 : 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. 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 99 )00274-8

 2000 Elsevier Science S.A. All rights reserved.

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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 strategy-proofness, Pareto-optimality, no-envy, and indifference ( in terms of welfare) 2 , 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 N 5 h1, . . . ,nj denote the set of agents and E the collective endowment. A feasible allocation is a vector z [ R N1 such that o i [ N z i 5 E, 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 i over [0, E]. Denote by Pi the strict relation associated with R i , and by Ii the indifference relation. The preference relation R i is single-plateaued if there is an interval [p(R i ), ]p(R i )] # [0, E] such that for all x i , y i [ [0, ] E], if y i , x i #p(R i ) or ]p(R i ) # x i , y i , then x i Pi y i , and if x i , y i [ [p(R i ), ]p(R i )], then x i Ii y i . A ] ]] preference relation R i is single-peaked if it is single-plateaued and p(R i ) 5p(R i ). Denote by 5 the set ] of all single-plateaued preferences. A preference profile (R 1 , . . . ,R n ) [ 5 N is denoted by R. For S # N, denote by R S the restriction of the profile R to the coalition S, i.e., R S 5 (R i ) i [S . Since the endowment is fixed, an economy is simply denoted by R [ 5 N . An allocation rule is a non-empty correspondence w :5 N → 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. Indifference (in terms of welfare): For all R [ 5 N , all z [ w (R), and all z9 [ Z: z9 [ w (R) if and only if for all i [ N, z i Ii z 9i . In general, different sets of allocations are recommended for different preference profiles. Strategyproofness prevents agents from gaining by mispresenting their preferences. Strategy-proofness: For all R [ 5 N , all i [ N, all R 9i [ 5, all z [ w (R), and all z9 [ w (R 9i , R N \hi j ), z i R i z 9i .3 2

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 strategy-proofness. For example, ‘for all z9 [ w (R i9 , R N \hi j ) there exists z [ w (R) such that z i R i z i9 ’ is sufficient.

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The last axiom says that every agent weakly prefers his consumption to any other agent’s consumption. No-envy: For all R [ 5 N , all z [ w (R), and all i, j [ N, z i R i z j . Ching (1992) extended the uniform rule to the single-plateaued domain as follows. Uniform rule, U : For all R [ 5 N ,

1. when E # o j [N p(R j ), there exists l such that U(R) 5 hz [ Z u for all i [ N, z i 5 min(p(R i ), l)j, ] ] ] )j, and 2. when o j [N p(R j ) # E # o j [N ]p(R j ), U(R) 5 hz [ Z u for all i [ N, p(R i ) # z i #p(R i ] ] ] ), l)j. 3. when E $ o j [N ]p(R j ), there exists l such that U(R) 5 hz [ Z u for all i [ N, z i 5 max(p(R i Note that in cases (i) and (iii) U(R) is a singleton. Whenever U(R) is a singleton, we also write Ui (R) to indicate the amount of agent i at the unique uniform allocation. Now we are able to state the 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. Note that the axioms of Theorem 2.1 imply Pareto-optimality.4 The axioms of Theorem 2.1 are independent. The following three examples establish independence. Example 2.2. For all R [ 5 N , let w e (R) 5 h(E /n, . . . ,E /n)j. Then w e satisfies strategy-proofness and no-envy, but not indifference. N Example 2.3. For all R [ 5 , if E , o i [N p(R i ), for some j [ N, ]p(R j ) 5 E, and for all i [ N\h jj, ] 0Pi (E 2p(R j )), then w˜ j (R) 5 E and for all i [ N\h jj, w˜ i (R) 5 0. Otherwise, w˜ (R) 5 U(R). Then, w˜ ] satisfies indifference and no-envy, but not strategy-proofness. N Example 2.4. For all R [ 5 , if E , o i [N p(R i ), then w¯ 1 (R) 5 min(p(R 1 ),E), w¯ 2 (R) 5 min(p(R 2 ),E 2 ] ] ] w¯ 1 (R)), and so on. Otherwise, w¯ (R) 5 U(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 Unanimity: For all R [ 5 , if o j [N p(R j ) # E # o j [N ]p(R j ), then z [ w (R) if and only if for all ] ] ). i [ N, p(R i ) # z i #p(R i ] 4

Consider the rule which recommends at every preference profile the allocations which are Pareto-optimal and envy-free. In our model this rule satisfies Pareto-indifference. Note that this is not the case for standard exchange economies with weakly convex preferences (see Thomson, 1999).

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Lemma 3.1. Let w be a rule satisfying indifference and strategy-proofness. Then w satisfies unanimity. Proof. Let R [ 5 N be such that

O p(R ) # E # O ]p(R ) ] j

j [N

j

(1)

j [N

and z [ w (R). By indifference, to show unanimity it suffices to prove that for all i [ N, p(R i ) # z i # ] ]p(R ). Suppose that w does not satisfy unanimity, i.e., for some j [ N, z [ ] i j ⁄ [p(R j ), p(R j )]. Say j 5 1. ] Let R¯ 1 [ 5 be such that [p(R¯ 1 ), ]p(R¯ 1 )] 5 [0, E]. By strategy-proofness, for all z9 [ w (R¯ 1 , R N \h1j ), ] ] ] )]. Say z 91 [ ⁄ [p(R 1 ), p(R 1 )]. By indifference and (1), there exists j [ N\h1j such that z 9j [ ⁄ [p(R j ), p(R j ] ] ] )]. By indifference and j 5 2. By strategy-proofness, for all z0 [ w (R¯ 1 , R¯ 1 , R N \h1,2j ), z 992 [ ⁄ [p(R 2 ), p(R 2 ] (1), there exists j [ N\h1,2j such that z 99j [ ⁄ [p(R j ), ]p(R j )]. Since N is finite, by induction we have that ] ] ⁄ [p(R n ), p(R n )]. By indifference, w ((R¯ 1 ) i [N ) 5 Z, which contradicts for all z¯ [ w ((R¯ 1 ) i [N \hn j , R n ), z¯ n [ ] strategy-proofness. Therefore, w satisfies unanimity. h 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. Lemma 3.2. Let w be a rule satisfying indifference, strategy-proofness, and no-envy. Then for all R [ 5 N such that E , o j [N p(R j ), w (R) 5 U(R). ] Proof. By Lemma 3.1, w satisfies unanimity. Let R [ 5 N be such that E , o j [N p(R j ) and let l [ R 1 ] N be such that for all i [ N, Ui (R) 5 min(p(R i ), l). Let R¯ [ 5 be such that for all i [ N, ] ] ¯ )] 5 [U (R),E]. By unanimity, [p(R¯ i ),p(R i i ] w (R¯ ) 5 U(R¯ ). (2) t Let Sl 5 hi [ N u p(R i ) $ lj and R 5 (R h1, . . . ,tj ,R¯ N \h1, . . . ,tj ). For all S # N, let us consider the profile ] (R S ,R¯ N \S ). By induction on uSu we will show that for all S # N,

w (R S ,R¯ N \S ) 5 U(R¯ ).

(3)

If uSu 5 0, then S 5 5. Thus, (R 5 ,R¯ N \5 ) 5 R¯ and the induction basis follows from (2). Induction basis: w (R 5 ,R¯ N \5 ) 5 U(R¯ ). By induction, we assume that (3) holds for all S # N such that uSu # t. Induction step: (3) holds for all S # N such that uSu 5 t 1 1. Proof: Without loss of generality, we may assume that S 5 h1, . . . ,t 1 1j. By induction,

w (R t ) 5 U(R¯ ).

(4)

t 11 t 11 We have to prove that w (R ) 5 U(R¯ ). Let z [ w (R ). We consider two cases. Case 1: t 1 1 [ Sl . Strategy-proofness and (4) imply z t11 $ l. By no-envy, for all j [ ht 1 2, . . . ,nj, t11 z j $p(R j ). Therefore, if z ± U(R¯ ), then there exists j [ h1, . . . ,tj such that ] (5) z j , Uj (R¯ ) 5 min(p(R j ), l). ]

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t11 Consider the profile (R h1, . . . ,t 11j\h j j ,R¯ N \(h1, . . . ,t 11j\h j j) ) 5 R˜ . Note that uh1, . . . ,t 1 1j\h jju 5 t. Thus, t11 by induction and (3), wj (R˜ ) 5 Uj (R¯ ) 5 min(p(R j ), l). Because of (5), the last statement contradicts ] strategy-proofness. Case 2: t 1 1 [ ⁄ Sl . Suppose that z ± U(R¯ ). Strategy-proofness and (4) imply z t 11 $ Ut11 (R¯ ). Thus, there exists j [ N\ht 1 1j such that z j , Uj (R¯ ). We consider two subcases. Subcase 2.1: j [ h1, . . . ,tj. Similar to Case 1, we obtain a contradiction to strategy-proofness. ⁄ h1, . . . ,t 1 1j. Hence, R tj 11 5 R¯ j . First, suppose that for all k [ Sl , R tk11 5 R¯ k . Subcase 2.2: j [ t 11 Hence, for all i [ h1, . . . ,t 1 1j, p(R i ) 5p(R¯ i ), and ] ]

O p(R ]

i [N

t11 i

)5

O ]p(R¯ ) 5 E. i

i [N

t11 t11 Unanimity implies w (R ) 5 U(R¯ ). Thus, there exists k [ Sl such that R k 5 R k ± R¯ k . Since t11 R j 5 R¯ j and z j , Uj (R¯ ) # l, no-envy implies

z k # z j , l #p(R k ). ] Thus, k [ h1, . . . ,tj and z k , Uk (R¯ ). By Subcase 2.1, the last inequality cannot occur. 앳 We proved (3). Since U(R) 5 U(R¯ ), applying (3) for S 5 N yields w (R) 5 U(R), the desired conclusion. h 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 j [N ]p(R j ) , E, w (R) 5 U(R). 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. Ching, S., Serizawa, S., 1998. A maximal domain for the existence of strategy-proof rules. Journal of Economic Theory 78, 157–166.

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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. Thomson, W., 1999. Monotonic extensions on economic domains. Review of Economic Design 4, 13–33.