A Maximal Domain for the Existence of Strategy-Proof Rules

Journal of Economic Theory  ET2337 journal of economic theory 78, 157166 (1998) article no. ET972337

NOTES, COMMENTS, AND LETTERS TO THE EDITOR

A Maximal Domain for the Existence of Strategy-Proof Rules* Stephen Ching Department of Economics and Finance, Faculty of Business, City University of Hong Kong, Kowloon Tong, Hong Kong

and Shigehiro Serizawa  Department of Economics, Shiga University, 1-1-1 Banba, Hikone, Shiga 522, Japan Received February 28, 1995; revised June 13, 1997

In a recent paper, Sprumont (1991, Econometrica 59, 509519) showed that the uniform rule (Benassy, 1982, ``The Economics of Market Disequilibrium,'' Academic Press, 1982) is the only rule satisfying strategy-proofness, anonymity, and efficiency on the single-peaked domain (Black, 1948, J. Polit. Econ. 56, 2334). This result motivates us to investigate whether there is a larger domain on which there exists a nontrivial strategy-proof rule. We want such a domain to be as large as possible. We show that the single-plateaued domain (Moulin, 1984, Soc. Choice Welfare 1, 127147) is the unique maximal domain for strategy-proofness, symmetry, and efficiency. Thus, we conclude that the assumption of single-peakedness essentially cannot be weakened if one insists on strategy-proofness, together with the other two basic requirements. Journal of Economic Literature Classification Numbers: C72, D78.  1998 Academic Press

* We acknowledge useful comments from the participants of the Theory Workshop at the University of Rochester and the 1994 International Conference on Game Theory at SUNY Stony Brook. We thank Salvador Barbera and Jordi Masso for motivating this research, and Ehud Kalai, Herve Moulin, and Alejandro Neme for their useful communications. We also thank the associate editor and the anonymous referee for their detailed comments. E-mail: efstfccityu.edu.hk. Stephen Ching is grateful to William Thomson and the participants of his seminar, particularly Tar@k Kara and Tayfun Sonmez, for their helpful comments. Ching acknowledges the support of Competitive Earmarked Research Grant 9040250.  Shigehiro Serizawa did his share of the work while he was at the Institute of Social and Economic Research at Osaka University. He is thankful for its good research atmosphere.

157 0022-053198 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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1. INTRODUCTION We investigate under what conditions we can induce agents to reveal their true preferences in the context of allocation problems. Given an allocation problem, the agents in that economy will interact with each other. A rule describes which allocation is the outcome for each economy. Since a change in the preferences of an agent usually changes the allocation chosen by the rule, he may benefit from misrepresenting his preferences. Then the resulting allocation may not be the one intended for the ``true economy.'' A desirable rule should therefore provide incentives for the agents to reveal their true preferences. The strongest incentive compatibility requirement is strategy-proofness: no agent can ever benefit from misrepresenting his preferences, regardless of whether the other agents misrepresent or not. It is well-known that strategy-proof rules essentially do not exist when preferences are unrestricted. However, strategy-proof rules do exist when preferences are restricted in some natural way. When preferences are restricted, announced preferences can also be restricted in the same way. Strategy-proofness becomes a less demanding property. 1 In this paper, we are interested in identifying domains of preferences on which strategy-proof rules exist. In a recent paper, Sprumont [14] considered the problem of allocating a perfectly divisible commodity when agents have ``single-peaked'' preferences (Black [6]). Preferences are single-peaked if more is strictly preferred to less up to a point, and less to more beyond that point. 2 In addition to strategy-proofness, he also considered the distributional requirement of anonymity (the names of the agents do not matter) and the optimality requirement of efficiency (an allocation is chosen only if there is no other allocation that is preferred by all agents and strictly preferred by at least one agent). He showed that a rule, known as the ``uniform rule'' 3 (Benassy [5]), is the only rule on the single-peaked domain that satisfies strategy-proofness, anonymity, and efficiency. Ching [9] strengthened this characterization by showing that anonymity can be weakened to symmetry (any two agents with the same preferences receive indifferent consumption). These results leave open the question whether there is a domain larger than the single-peaked domain on which strategy-proof, efficient, and 1

Blin and Satterthwaite [7] studied strategy-proofness of majority rule with Borda completion. They showed that if the true preferences are ``single-peaked,'' but announced preferences are unrestricted, then strategy-proof rules essentially do not exist. Since strategy-proofness remains quite demanding, their impossibility result can be understood intuitively. 2 See Sprumont for models in which single-peakedness of preferences is a natural restriction. 3 The uniform rule can be described as follows: agents are allowed to choose their preferred consumption subject to a common upper bound, or a common lower bound, the bound being chosen to obtain feasibility.

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symmetric rules exist. We want such a domain to be as large as possible. A domain is a maximal domain for a list of properties if there is a rule defined on this domain satisfying the properties, and there is no rule defined on any strictly larger domain satisfying the same properties. Note that a maximal domain for a list of properties need not be unique. We show that a maximal domain including single-peaked preferences for strategy-proofness, efficiency, and symmetry is unique and slightly larger than the singlepeaked domain. The maximal domain we identify is known as the singleplateaued domain (Moulin [12]). Preferences are single-plateaued if the set of preferred points is a closed interval; more is strictly preferred to less up to the lower bound of the interval, and less to more beyond the upper bound. We establish that the single-plateaued domain is the unique maximal domain including single-peaked preferences for strategy-proofness, symmetry, and efficiency. Hence, we conclude that the assumption of single-peakedness essentially cannot be weakened if one insists on strategy-proofness, together with the optimality requirement of efficiency and the distributional requirement of symmetry. Barbera et al. [4] established the first maximal domain result in a public commodity model. They considered the domain which they called separable preference. They showed that a rule is strategy-proof on that domain if and only if it is a member of the class of rules called voting by committees, and that if a voting by committees rule defined on a sufficient large domain is strategy-proof and the power structure of the rule is nonextreme, 4 then that domain is a subdomain of the domain of separable preferences. Subsequently, Barbera et al. [2] and Serizawa [13] established similar results in a generalized model with multiple levels of public commodities, and Barbera et al. [3] obtain a similar result in a model with feasibility constraints. As the model was generalized, the preference domain was also generalized to the domain of multidimensional single-peaked preferences. This notion coincides with single-peaked preference when there is only one good and with separable preferences when the level of goods is restricted to zero or one. In this sense, multidimensional single-peakedness is a generalization of separability and single-peakedness. The main difference of our paper from theirs is that our maximal domain result is established by only imposing properties on rules. That is, no pre-specified structure of the rules is used in our result while their maximal domain results are established for some special class of rules such as voting by committees. A similar approach is employed by Alcalde and Barbera [1] in a matching model, and by Le Breton and Sen [8] in a general public commodity 4 This is the requirement that in the decision on each public good, every agent has some influence, but no agent has veto or decisive power. This condition excludes particular rules such as dictatorship. See Barbera et al. [4] or Serizawa [13] for the formal definition.

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model. Closely related but different questions were posed by Kalai and Muller [10] and Kalai and Ritz [11]. They studied the general conditions for domains which admit the existence of nondictatorial Arrow-type social welfare functions. The paper is organized as follows: Section 2 describes the model and the results. Section 3 presents the proofs. Section 4 concludes.

2. THE MODEL AND THE RESULTS Let M # R ++ be an amount of a perfectly divisible commodity. Let N=[1, ..., n] be a group of agents with n2. Each agent is equipped with a preference relation R 0 defined over R + _ []. Let P 0 be the strict relation associated with R 0 , and I 0 the indifference relation. Given a preference relation R 0 , let G(R 0 )=[(x 0 , y 0 ) # R 2+ | x 0 R 0 y 0 ] be the graph of R 0 . A preference relation is continuous if its graph is closed. Let R be the domain of continuous preference relations. Given a relation R 0 # R, let p(R 0 )= [x 0 # R + _ [] | \y 0 # R + , x 0 R 0 y 0 ] be the set of preferred consumption according to R 0 . When p(R 0 ) is a singleton, we slightly abuse notation and use p(R 0 ) to denote its single element. Let p(R 0 )=inf p(R 0 ) and p(R 0 )=  1 , ..., R n ) # R n. An economy is sup p(R 0 ). A preference profile is a list R=(R n 5 a pair (R, M) # R _R ++ . Given an economy (R, M) # R n_R ++ , a feasible allocation, or simply an allocation, is a vector z # R n+ such that  z i =M. Let Z(M) be the set of all allocations for the economy (R, M). A rule is a systematic way to associate an allocation with each economy. A rule is a function . : R n_R ++  R n+ such that for all (R, M) # R n_R ++ , .(R, M) # Z(M). Our main property is the strongest incentive compatibility requirement that no agent can ever benefit from misrepresenting his preferences in the direct revelation game associated with the rule. Strategy-Proofness. For all (R, M) # R n _R ++ , all i # N, and all R$i # R, . i (R, M)R i . i (R$i , R &i , M). We also impose the distributional requirement called symmetry that any two agents with the same preferences receive indifferent consumption. A stronger requirement is strong symmetry: any agents with the same preference receive the same consumption. 5 Our results rely on the assumption that the amount of the commodity is allowed to vary. Therefore, we deal with a large class of allocation problems with variable preferences and commodity level. If the amount is fixed, we have examples to show that the results do not hold.

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Symmetry. For all (R, M) # R n_R ++ and all i, j # N such that R i =R j , .i (R, M) I i . j (R, M). Strong Symmetry. For all (R, M) # R n_R ++ and all i, j # N such that Ri =R j , . i (R, M)=. j (R, M). Our optimality requirement is that an allocation be chosen only if there is no other allocation that is preferred by all agents and strictly preferred by at least one agent. Efficiency. For all (R, M) # R n_R ++ , there is no z # Z(M) such that for all i # N, z i R i . i (R, M), and for some i # N, z i P i . i (R, M). Sprumont [14] first analyzed this model when agents have single-peaked preferences (Black [5]). Definition. A preference relation R 0 # R is single-peaked if p(R 0 ) is a singleton; and for all x 0 , y 0 # R + with [ y 0
min[ p(R i ), *(R, M)]

{max[ p(R ), *(R, M)] i

if M p(R j ) otherwise,

(1)

where *(R, M) solves  U j (R, M)=M. Another example of a rule satisfying strategy-proofness and symmetry is the equal division rule E: for all (R, M) # Rn_R ++ and all i # N, E i (R, M)= Mn. Since the equal division rule satisfies strategy-proofness and symmetry on any domain, our objective is to examine whether there are domains larger than the single-peaked domain on which there is a rule satisfying strategy-proofness, symmetry, and the optimality requirement of efficiency. We want the domains to be as large as possible. Definition. A domain R nm is a maximal domain for a list of properties if (i) R nm R n; (ii) there is a rule on R nm satisfying the properties; and (iii) there is no rule satisfying the same properties for all R na such that R nm /R na R n. 6

The property that the names of the agents do not matter.

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Note that a maximal domain for a list of properties need not be unique. We show that a maximal domain including single-peaked preferences for strategy-proofness, efficiency, and symmetry is unique and slightly larger than the single-peaked domain. The additional admissible preferences are known as single-plateaued preferences. Definition. A preference relation R 0 # R is single-plateaued if p(R 0 ) is an interval [p(R 0 ), p(R 0 )]; and for all x 0 , y 0 # R + with [ y 0
3. PROOF OF THE THEOREM We need two preliminary results to prove the theorem. The first one is a slightly stronger result than the following statement: if a rule on the singlepeaked domain satisfies efficiency, then symmetry and strong symmetry are equivalent. Lemma 1. Let R ns R na R n. If there is a rule . on R na satisfying symmetry and efficiency, then for all (R, M) # R na _R ++ and all i, j # N such that Ri =R j # Rs , . i (R, M)=. j (R, M). Proof. Let (R, M) # R na _R ++ and i, j # N be such that R i =R j # Rs . First, efficiency implies that either . i (R, M)p(R i ) and . j (R, M)p(R j ); or p(Ri ). i (R, M) and p(R j ). j (R, M). Then, symmetry and R i =R j # Rs together imply that . i (R, M)=. j (R, M). K The second one is a corollary of the following characterization: the uniform rule on the single-peaked domain is the only rule satisfying strategy-proofness, symmetry, and efficiency (Ching [9]).

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Corollary 1. Let R ns R na R n. If there is a rule . on R na satisfying strategy-proofness, symmetry, and efficiency, then .(R, M)=U(R, M) for all (R, M) # R ns _R ++ . The proof of the theorem below consists of two steps (Step 1 and 2). In Step 1, we establish that the domain for the three properties must be included in the domain of ``convex'' preferences. Definition. A preference relation R 0 # R is convex if p(R 0 ) is an interval [p(R 0 ), p(R 0 )]; and for all x 0 , y 0 # R + with [ y 0
Step 1. We first show that R na R nc . Suppose, by contradiction, that there exists a relation R 0 # Ra "Rc . Then there are three points x 0 < y0
{ max[x$ # [x , y ] | x$ I z ] min[z$ # [ y , z ] | z$ I x ] z*= { min[z$ # [y , z ] | z$ I z ]

x* 0=

0

0

0

0 0 0

0

0

0

0 0

0

0

0

0 0 0

0

0

if z 0 R 0 x 0 otherwise; if z 0 R 0 x 0 otherwise.

(2) (3)

Since R 0 is continuous, x* 0 and z* 0 are well-defined. Note that x 0 x* 0< z , x* I z* , and for all x$ # (x* , z* ), z* P x$ . Let M=nz* y 0
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.1(R 1 , R$&1 , M){. 1(R$, M)(=z* 0 ). To establish Step 1, we derive a contradiction in each case. Case 1. Assume that . 1(R 1 , R$&1 , M)=z* 0 . By Lemma 1, . &1(R 1 , , ..., z* ). Note that (x* , (M&x* )(n&1), ..., (M&x* R$&1 , M)=(z* 0 0 0 0 0 ) (n&1)) I z* . For all i{1, since R$ # R and MP$ z* , it follows that # Z(M) and x* 0 1 0 i s i 0 ) (n&1) P$ z* . These two statements together contradict efficiency. (M&x* 0 i 0 Case 2. Assume that . 1(R 1 , R$&1 , M){z*0 . There are three subcases: (i) If z* 0 <. 1(R 1 , R$&1 , M), then . 1(R 1 , R$&1 , M) P$1 . 1(R$, M). (ii) If <. (R then . 1(R$, M) P 1 . 1(R 1 , R$&1 , M). (iii) If x* 0 1 1 , R$&1 , M)0 be such that x 0 &= # ( p(R 0 ), x 0 ) and x 0 += # (x 0 , y 0 ). The feasible allocation (x 0 &=, x 0 +=, x 0 , ..., x 0 ) Pareto dominates (x 0 , ..., x 0 ), contradicting efficiency. Step 2 is completed. The proof is completed by constructing a rule on the single-plateaued domain that satisfies strategy-proofness, symmetry, and efficiency. We obtain such a rule by extending the uniform rule as follows: for all (R, M) # R np _R ++ and all i # N, min[p(R i ), *(R, M)]  U ei(R, M)= min[ p(R i ), p(R i )+*(R, M)]

{

max[ p(R i ), *(R, M)]

M p(R j )  if  p(R j )
where *(R, M) solves  U ej(R, M)=M. Remark 2. It is interesting to note that if there exists a rule defined on a domain satisfying strong symmetry and efficiency, then that domain is a subdomain of the single-plateaued domain. Strategy-proofness can be dropped altogether! The proof of this result can be found in the Appendix.

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4. CONCLUSIONS We establish that the maximal domain including single-peaked preferences for strategy-proofness, efficiency, and symmetry is unique and slightly larger than the single-peaked domain. This result shows that the assumption of single-peakedness essentially cannot be weakened if one insists on strategy-proofness, together with the distributional requirement of symmetry and the optimality requirement of efficiency. Alternatively, this result can be understood as a general impossibility theorem. For instance, the theorem can be stated as follow: There is no rule satisfying strategy-proofness, efficiency, and symmetry that is defined on any domain strictly larger than the singleplateaued domain, say the convex domain. A. APPENDIX In Remark 2, we mention the following strong domain restriction implied by strong symmetry and efficiency. Proposition A. Let R na R n. If there is a rule on R na satisfying strong symmetry and efficiency, then R na R np . Proof. Let R na R n. Suppose that there is a rule . on R na satisfying strong symmetry and efficiency. Suppose, by contradiction, that there exists R0 # Ra "Rp . Let R 0 # Ra "Rp . There are two cases: Case 1. There exist x 0 < y 0
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Finally, note that the extended uniform rule satisfies strong symmetry and efficiency on the single-plateaued domain. K Remark A. The following example shows that Proposition A is no longer true if we use symmetry instead of strong symmetry. Let R i # R be the preference relation that can be represented by the sine function on R + , i.e., u~i (x i )=sin(x i ) for all x i # R + . Let n=2 and M>0. Let .(R, M)= (M&2?m)2, 2?m+((M&2?m2)), where m is the integer such that M # [2?m, 2?(m+1)]. The rule . is symmetric and efficient on [R i ] 2, but Ri  Rp . Remark B. In classical exchange economies, we can prove a result similar to Proposition A. If there is a rule defined on a domain satisfying strong symmetry and efficiency, then that domain is a subdomain of the weakly convex domain.

REFERENCES 1. J. Alcalde and S. Barbera, Top dominance and the possibility of strategy-proof stable solutions to matching problems, Econ. Theory 4 (1994), 417435. 2. S. Barbera, F. Gul, and E. Stacchetti, Generalized median voter schemes and committees, J. Econ. Theory 61 (1994), 262289. 3. S. Barbera, J. Masso, and A. Neme, Voting under constraints, mimeo, Universitat Autonoma de Barcelona, 1995. 4. S. Barbera, H. Sonnenschein, and L. Zhou, Voting by committees, Econometrica 59 (1991), 595609. 5. J-P. Benassy, ``The Economics of Market Disequilibrium,'' Academic Press, New York, 1982. 6. D. Black, On the rationale of group decision-making, J. Polit. Econ. 56 (1948), 2334. 7. J-M. Blin and M. Satterthwaite, Strategy-proofness and single-peakedness, Public Choice 30 (1975), 5158. 8. M. Le Breton and A. Sen, Strategyproof social choice functions over product domains with unconditional preferences, mimeo, 1992. 9. S. Ching, An alternative characterization of the uniform rule, Soc. Choice Welfare 11 (1994), 131136. 10. E. Kalai and E. Muller, Characterization of domains admitting nondictorial social welfare functions and nonmanipulable voting procedures, J. Econ. Theory 16 (1977), 457469. 11. E. Kalai and Z. Ritz, Characterization of the private alternatives domains admitting arrow social welfare functions, J. Econ. Theory 22 (1980), 2336. 12. H. Moulin, Generalized Condorcet-winners for single peaked and single-plateau preferences, Soc. Choice Welfare 1 (1984), 127147. 13. S. Serizawa, Power of voters and domain of preferences where voting by committees is strategy-proof, J. Econ. Theory 67 (1995), 599608. 14. Y. Sprumont, The division problem with single-peaked preferences: A characterization of the uniform rule, Econometrica 59 (1991), 509519.

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