Dominant strategy implementation of bargaining solutions

Journal Pre-proof Dominant strategy implementation of bargaining solutions Hideki Mizukami, Takuma Wakayama

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S0165-4896(20)30016-0 https://doi.org/10.1016/j.mathsocsci.2020.01.008 MATSOC 2099

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Mathematical Social Sciences

Received date : 30 January 2019 Revised date : 27 January 2020 Accepted date : 27 January 2020 Please cite this article as: H. Mizukami and T. Wakayama, Dominant strategy implementation of bargaining solutions. Mathematical Social Sciences (2020), doi: https://doi.org/10.1016/j.mathsocsci.2020.01.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Elsevier B.V. All rights reserved.

*Highlights (for review)

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H IGHLIGHTS • We study dominant strategy implementation in bargaining environments. • We characterize the class of dominant strategy implementable (DSI) rules.

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• We identify the class of (bargaining) solutions induced by desirable DSI rules. • Only dictatorial solutions are induced by desirable DSI rules.

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• Some desirable DSI rules are non-dictatorial but they all cannot induce any solution.

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DOMINANT STRATEGY IMPLEMENTATION OF BARGAINING SOLUTIONS∗ HIDEKI MIZUKAMI† AND TAKUMA WAKAYAMA‡

A BSTRACT. We consider the problem where agents bargain over their shares of a perfectly divisible commodity. The aim of this paper is to identify the class of bargaining solutions induced by dominant strategy implementable allocation rules. To this end, we characterize the class of dominant strategy implementable allocation rules and impose the property of welfarism, which makes it possible for any allocation rule to induce a bargaining solution. Our main result is that an allocation rule is dominant strategy implementable and satisfies welfarism and some mild requirements if and only if it induces a dictatorial solution. Keywords: bargaining problem; bargaining solution; dominant strategy implementation; welfarism; dictatorial solution

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JEL codes: C78; D78; C71; C72; D71

Date: January 27, 2020. ∗ The paper was initially titled “Implementability and the axioms of bargaining theory.” We are grateful to the anonymous referee for constructive comments and suggestions. We would like to thank Masaki Aoyagi, Atsushi Kajii, Tatsuyoshi Saijo, Tadashi Sekiguchi, Ken-ichi Shimomura, Koji Takamiya, Naoki Yoshihara for their helpful comments and suggestions. We also benefited from discussions with Chiaki Hara, Haruo Imai, Ryo-ichi Nagahisa, Akira Okada, Shigehiro Serizawa, and the participants in the 2003 Annual Meeting of the Japanese Economic Association, the 8th International Meeting of the Society for Social Choice and Welfare, and the seminars at Kyoto University and Osaka University. This work was supported by JSPS KAKENHI Grant Number JP22730165, JP22330061, JP15H03328, JP16K03567, and JP17K03628. † College of Economics, Aoyama Gakuin University, 4-4-25 Shibuya, Shibuya-ku, Tokyo 150-8366, Japan; [email protected]. ‡ Corresponding author. Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimiku, Kyoto 612-8577, Japan; [email protected]. 1

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HIDEKI MIZUKAMI AND TAKUMA WAKAYAMA

1. I NTRODUCTION

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We consider problems where agents bargain over their shares of a perfectly divisible commodity. Examples of such problems include bargaining over the share of profit between an employer and a labor union or bargaining over the distribution of property. In such situations, only when all agents agree on a feasible outcome, can they arrive at it; otherwise, they arrive at the predetermined outcome or nothing. To avoid disagreements, they may decide to follow a recommendation made by an impartial arbitrator (e.g., a central arbitration committee like that in the UK presiding over labor disputes, a judge presiding over civil trials). Axiomatic bargaining theory, initiated by Nash (1950), deals with these situations.1 This theory provides numerous “bargaining solutions” that associate a profile of utility levels with each utility possibility set. As pointed out by Raiffa (1953), a bargaining solution can be interpreted as the recommendation of the arbitrator. The shape of the utility possibility set is based on agents’ preferences. Since preferences are usually unknown to the arbitrator, selfish agents may have an incentive to gain by manipulating the bargaining solution through misrepresentation of their preferences. Thus, the arbitrator faces the problem of constructing a mechanism whose equilibrium outcomes are always outcomes the arbitrator wishes to recommend. If the arbitrator can construct such a mechanism, then the bargaining solution is said to be “implementable.” Unfortunately, however, we cannot directly apply the standard notion of implementability to bargaining solutions. This is because the notion of implementability is usually formalized by means of “allocation rules” that are defined on the set of underlying physical outcomes, whereas bargaining solutions are defined on the utility space. Thus, we explicitly consider the underlying physical outcomes and allocation rules.2 This paper focuses on the class of allocation rules satisfying essential single-valuedness (each agent is indifferent among all the selected allocations) and Pareto-indifference (if an allocation is chosen, then any other allocation that all agents find indifferent to it should also be chosen). The aim of this paper is to identify bargaining solutions induced by implementable allocation rules.3 To verify this, we consider welfarism (Roemer, 1986, 1988); it requires that if two preference profiles correspond to the same utility possibility set, then the allocation rule should assign to each of the preference profiles outcomes that are indistinguishable in terms of utility across the preference profiles. This condition is necessary and sufficient for any allocation rule to induce a bargaining solution. For a survey on axiomatic bargaining theory, see Thomson (2009). Since the early work of Moulin (1984), the literature on the implementation of bargaining solutions has taken a similar approach. Roemer (1986, 1988) and Chun and Thomson (1988) are other early studies that focus on allocation rules in the context of axiomatic bargaining theory. For relatively recent studies, see Chen and Maskin (1999) and Yoshihara (2003). They do not, however, consider the incentive issue in their frameworks. 3 One might think that there are some similarities between implementation theory and the Nash program, according to which a good solution may not only be supported by axiomatic analysis but also have a noncooperative foundation. Trockel (2002), in fact, shows that any support result in the Nash program can be embedded into implementation theory. See, also, Dagan and Serrano (1998) and Bergin and Duggan (1999).

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There have been many studies on Nash or subgame-perfect implementation in the context of the bargaining problem.4 Unlike previous studies, we examine dominant strategy implementation. One motivation for dominant strategy implementation is that the use of dominant strategies does not require an agent to believe that other agents will behave rationally. That is, under dominant strategy implementation, we can construct a mechanism with weak rationality assumptions.5 We first characterize the class of dominant strategy implementable allocation rules in our setting (Proposition 1). This result tells us that any dominant strategy implementable allocation rule must be single-valued. When we turn our attention to single-valued allocation rules, the well-known revelation principle (Proposition 2) states that dominant strategy implementability implies strategy-proofness (truth-telling is a dominant strategy for everyone). Moreover, given that an allocation rule is single-valued, we establish that dictatorial rules are the only allocation rules satisfying strategy-proofness, welfarism, and the two auxiliary conditions called non-disagreement and non-bossiness (Proposition 3). By invoking these results, it turns out that only dictatorial solutions are induced by dominant strategy implementable allocation rules satisfying non-disagreement, non-bossiness, and welfarism (Theorem 1). Our dictatorial results are similar to the well-known Gibbard-Satterthwaite theorem (Gibbard, 1973; Satterthwaite, 1975). Since the Gibbard-Satterthwaite theorem cannot be applied to restricted preference environments such as ours, our dictatorial results are not a by-product of this theorem. In fact, in our setting, there is a non-dictatorial rule satisfying dominant strategy implementability, non-disagreement, and non-bossiness. However, such non-dictatorial rules cannot induce any bargaining solution; that is, they all violate welfarism. This suggests that welfarism is the source of our dictatorial results. The remainder of this paper is organized as follows: Section 2 provides basic notation and definitions. Section 3 offers a characterization of dominant strategy implementable allocation rules and identifies the class of bargaining solutions induced by dominant strategy implementable allocation rules. Section 4 concludes with some remarks. Appendices A–C contain proofs that are omitted from the main text. 2. P RELIMINARIES

2.1. M ODEL

Let N ≡ {1, 2, . . . , n} be the finite set of agents, with generic element i. There is a perfectly divisible commodity that has to be shared among a group of agents. Without loss of 4

In abstract settings, Moulin (1984) and Howard (1992) construct mechanisms that subgame-perfect implement allocation rules inducing the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) and the Nash solution (Nash, 1950), respectively. Miyagawa (2002) provides a simple mechanism that subgame-perfect implements a wide class of allocation rules inducing bargaining solutions such as the Nash solution and the Kalai-Smorodinsky solution in a general economic framework, which includes our setting. Samejima (2005) extends Miyagawa’s results to the three-or-more-agent case. Vartiainen (2007) demonstrates that no efficient and symmetric bargaining solution is induced by any Nash implementable allocation rule in our settings. 5 Mizukami and Wakayama (2007) provide a necessary and sufficient condition for dominant strategy implementation in economic environments. Their results are applicable to our model.

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generality, we assume that the amount of the commodity is equal to 1. The set of feasible allocations is ( ) X n X ≡ x = ( xi ) i∈ N ∈ R + : xi ≤ 1 , where xi represents agent i’s share. Let ( X≡

x∈

i∈ N

R n+ :

X i∈ N

xi = 1

)

be the set of balanced allocations. We denote by △ the set of lotteries (probability distributions on X) with finite support. A typical lottery is denoted by ℓ. Given ℓ ∈ △ and x ∈ X, let ℓ( x) be the probability that ℓ assigns to x. We abuse notation and write x both for the allocation x and the degenerate lottery that assigns probability one to x. Therefore, the set X also stands for the set of degenerate lotteries. Given ℓ ∈ △ and i ∈ N, let ℓi be the marginal distribution of ℓ onto agent i’s share. For each i ∈ N, let Θi be the set of agent i’s types, each of which prescribes von Neumann-Morgenstern (henceforth vNM) preferences over △. We assume that each agent i ∈ N is equipped with a utility function ui : [0, 1] × Θi → R satisfying the following: for each θi ∈ Θi , ui ( · ; θi ) : [0, 1] → R is a continuous, strictly increasing, and strictly concave function.6 To simplify the exposition, we also assume that for each i ∈ N and each θi ∈ Θi , ui (0; θi ) = 0.7 Given ℓ ∈ △, the expected utility of agent i ∈ N with her type θi ∈ Θi is defined by X Ui (ℓ; θi ) = ℓ( x)ui ( xi ; θi ),

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x ∈supp (ℓ)

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where supp(ℓ) ≡ { x ∈ X : ℓ( x) > 0}. Note that there is a pair {θi , θi′ } ⊂ Θi such that θi 6= θi′ , but θi and θi′ yield the same vNM preferences. A type profile is a list θ ≡ (θi )i∈ N . Let Θ ≡ ∏ i∈ N Θi be the domain of type profiles. Given ℓ ∈ △ and θ ∈ Θ, let U (ℓ; θ ) ≡ (Ui (ℓ; θi ))i∈ N . We often denote N \ {i} by “−i.” With this notation, (θi′ , θ−i ) is the type profile where agent i’s type is θi′ and the type of agent j 6= i is θ j . 2.2. A LLOCATION

RULES AND DOMINANT STRATEGY IMPLEMENTATION

An allocation rule, or briefly, a rule is a (possibly multi-valued) mapping f : Θ ։ △ that assigns a nonempty subset of △ to each θ ∈ Θ.8 Throughout this paper, we make two assumptions on rules: A1 (Essential single-valuedness). For each θ ∈ Θ and each pair {ℓ, ℓ′ } ⊆ f (θ ), U (ℓ; θ ) = U (ℓ′ ; θ ). 6We discuss what happens if we weaken the strict concavity of utility functions to concavity in Remark 7. 7This normalization does not affect the essence of the analysis in this paper. Vartiainen (2007), which studies

Nash implementation of bargaining solutions, imposes the same assumption to simplify the exposition. 8The arrow symbol ։ is used for a multi-valued mapping.

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A2 (Pareto-indifference). For each θ ∈ Θ and each pair {ℓ, ℓ′ } ⊂ △, if ℓ ∈ f (θ ) and U (ℓ′ ; θ ) = U (ℓ; θ ), then ℓ′ ∈ f (θ ).

2.3. B ARGAINING

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A1 says that the lotteries that are chosen by a rule are welfare equivalent. A2 says that a rule chooses a set of all lotteries that all agents are indifferent to. For each i ∈ N, let Mi be agent i’s strategy set. A typical element of Mi is denoted by mi . A mechanism is a pair Γ = ( M, g), where M ≡ ∏ i∈ N Mi and g : M → △ is an outcome function. A strategy profile is denoted by m ≡ (mi )i∈ N . Given θ ∈ Θ, we say that m∗ ∈ M is a dominant strategy equilibrium of Γ at θ if for each i ∈ N, each m′i ∈ Mi , and each m−i ∈ M−i ≡ ∏ j6=i M j , Ui ( g(m∗i , m−i ); θi ) ≥ Ui ( g(m′i , m−i ); θi ). Given θ ∈ Θ, let DSEΓ (θ ) ⊆ M and g(DSEΓ (θ )) be the sets of dominant strategy equilibria of Γ at θ and dominant strategy equilibrium outcomes of Γ at θ, respectively. Given a rule f , we say that Γ = ( M, g ) dominant strategy implements f if for each θ ∈ Θ, g(DSEΓ (θ )) = f (θ ). If such a mechanism does exist, we say that the rule is dominant strategy implementable.9 PROBLEM AND BARGAINING SOLUTIONS

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Let θ ∈ Θ. We define the utility possibility set at θ by S(θ ) ≡ {U (ℓ; θ ) ∈ R n : ℓ ∈ △}. Let ∂S(θ ) ≡ {s ∈ S(θ ) : there is no s′ ∈ S(θ ) such that s′ > s} be the utility possibility frontier at θ.10 The disagreement point at θ is given by d(θ ) = (di (θ ))i∈ N ≡ U (0; θ ), where 0 ≡ (0, . . . , 0). A bargaining problem for θ is a pair (S(θ ), d(θ )). By our assumption that ui (0; θi ) = 0 for each i ∈ N, d(θ ) = 0. Therefore, for simplicity, we write S(θ ) for (S(θ ), d(θ )). Note that in our setting, S(θ ) is compact, strictly convex, and strictly S comprehensive11. Let Σ ≡ θ ′ ∈Θ S(θ ′ ) be the set of bargaining problems. A bargaining solution, or briefly, a solution is a single-valued mapping F : Σ → R n , which assigns a list of utility levels in S ⊂ R n to each S ∈ Σ. Given a solution F, we denote the utility level assigned to agent i by Fi (S). For each pair {θ, θ ′ } ⊂ Θ, if S(θ ) = S(θ ′ ), then F (S(θ )) = F (S(θ ′ )). In what follows, we often write F (S) for F (S(θ )). 3. R ESULTS

OF DOMINANT STRATEGY IMPLEMENTABLE RULES

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3.1. C HARACTERIZATION

We now offer a characterization of dominant strategy implementable rules in our setting. Let i ∈ N. We say that a mapping λi : R → R is a monotone transformation if λi is continuous and increasing. Let Λi be the set of all monotone transformations. Given λi ∈ Λi and θ ∈ Θ, let λi (S(θ )) ≡ {s′ ∈ R n : s′ = (λi (si ), s−i ) for some s ∈ S(θ )}. Remark 1. Since any distinct agent i’s type induces the same ordinal ordering on [0, 1], we obtain the following fact: Let θ ∈ Θ, i ∈ N, and θi′ ∈ Θi . Then, there is λi ∈ Λi

9We consider full implementation of rules; that is, all equilibrium outcomes of a mechanism are optimal

rather than just some (i.e., partial implementation).

10Vector inequalities are denoted as follows: given x, x ′ ∈ R n , x ≧ x ′ means x ≥ x ′ for each i ∈ N; x ≥ x ′ i i

means x ≧ x ′ and x 6= x ′ ; and x > x ′ means xi > xi′ for each i ∈ N. 11A set S ⊂ R n is strictly comprehensive if for each x, y ∈ S, if x ≥ y, then there is z ∈ S with z > y.

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such that for each xi ∈ [0, 1], ui ( xi ; θi′ ) = λi (ui ( xi ; θi )). Moreover, for each λi ∈ Λi , if ♦ S(θi′ , θ−i ) = λi (S(θ )), then for each xi ∈ [0, 1], ui ( xi ; θi′ ) = λi (ui ( xi ; θi )).

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Proposition 1 below characterizes the class of dominant strategy implementable rules in terms of the following three conditions.

• Single-valuedness: For each θ ∈ Θ, f (θ ) is a singleton.12

• Ordinality: For each pair {θ, θ ′ } ⊂ Θ, if f (θ ) 6= f (θ ′ ), then there are i ∈ N and {ℓ, ℓ′ } ⊂ △ such that Ui (ℓ; θi ) ≥ Ui (ℓ′ ; θi ) and Ui (ℓ′ ; θi′ ) > Ui (ℓ; θi′ ).

• Invariance to individual monotone transformations: For each θ ∈ Θ, each i ∈ N, each θi′ ∈ Θi , and each λi ∈ Λi , if S(θi′ , θ−i ) = λi (S(θ )), then for each ℓ′ ∈ f (θi′ , θ−i ) and each ℓ ∈ f (θ ), Ui (ℓ′ ; θi′ ) = λi (Ui (ℓ; θi )).

Proposition 1. A rule is dominant strategy implementable if and only if it is single-valued, ordinal, and invariant to individual monotone transformations. Proof. See Appendix A.



Remark 2. Given θ ∈ Θ, a lottery ℓ ∈ △ is efficient at θ if there is no ℓ′ ∈ △ such that for each i ∈ N, Ui (ℓ′ ; θi ) > Ui (ℓ; θi ). Note that in our model, irrespective of θ ∈ Θ, the set of efficient lotteries is equivalent to the set X. It is easy to confirm that if a rule f is singlevalued, then A2 implies that for each θ ∈ Θ, f (θ ) ∈ X ∪ {0}; that is, any single-valued rule chooses either an efficient lottery or the disagreement outcome at each type profile. This, together with Proposition 1, implies that any dominant strategy implementable rule chooses either an efficient lottery or the disagreement outcome at each type profile. ♦ INDUCED BY DOMINANT STRATEGY IMPLEMENTABLE RULES

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3.2. S OLUTIONS

This paper aims to identify the class of “implementable” solutions. However, we cannot directly apply the standard notion of implementability to solutions because the notion of implementability is formalized by means of rules. Hence, we identify the class of solutions induced by dominant strategy implementable rules. In doing so, we impose the following condition on rules.

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• Welfarism: For each pair {θ, θ ′ } ⊂ Θ, if S(θ ) = S(θ ′ ), then for each ℓ ∈ f (θ ) and each ℓ′ ∈ f (θ ′ ), U (ℓ; θ ) = U (ℓ′ ; θ ′ ).

This condition, introduced by Roemer (1986, 1988), states that if two type profiles give rise to the same bargaining problem, then the rule should assign to each of the type profiles lotteries that are indistinguishable in terms of utility across the type profiles. This condition is necessary and sufficient for any rule f to induce a solution F; that is, for each θ ∈ Θ and each ℓ ∈ f (θ ), F (S(θ )) = U (ℓ; θ ).13 We further introduce two auxiliary conditions on rules. The first condition is very mild.

• Non-disagreement: There is no θ ∈ Θ such that 0 ∈ f (θ ).

12If f (θ ) is a singleton, then we abuse notation and denote the single element by f (θ ).

13The proof of this fact is available from the authors upon request.

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This condition states that a rule should not choose the disagreement outcome. The second condition is a version of “non-bossiness” (Satterthwaite and Sonnenschein, 1981). Roughly speaking, it states that every agent cannot change the other agents’ share by announcing different types without affecting her own share. Although several studies consider this condition for single-valued rules, Bogomolnaia, Deb, and Ehlers (2005) extend it to multi-valued rules in the following manner:14

• Non-bossiness: For each θ ∈ Θ, each i ∈ N, and each θi′ ∈ Θi , if (i) ℓ ∈ f (θ ) and (ii) for each ℓ′ ∈ f (θi′ , θ−i ), ℓi = ℓ′i , then ℓ ∈ f (θi′ , θ−i ).

Our notion of non-bossiness requires that whenever the deviating agent has the same marginal distributions, the initially chosen lottery should be chosen at the new type profile. Note that under A1, non-bossiness implies that if (i) ℓ ∈ f (θ ) and (ii) for each ℓ′ ∈ f (θi′ , θ−i ), ℓi = ℓ′i , then U (ℓ; θ ) = U (ℓ′ ; θ ). The next fact states that whenever there are two agents, any dominant strategy implementable rule satisfying non-disagreement must be non-bossy. Fact 1. Suppose n = 2. If a rule f is dominant strategy implementable and satisfies nondisagreement, then it is non-bossy. Proposition 1 tells us that any dominant strategy implementable rule must be single-valued. That is, we can focus on single-valued rules whenever we insist on dominant strategy implementability. The next condition is a well-known incentive compatibility condition for single-valued rules; it states that truth-telling is a dominant strategy for everyone.

• Strategy-proofness: For each θ ∈ Θ, each i ∈ N, and each θi′ ∈ Θi , Ui ( f (θ ); θi ) ≥ Ui ( f (θi′ , θ−i ); θi ).

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Given that a rule is single-valued, the so-called revelation principle states that dominant strategy implementability implies strategy-proofness. Proposition 2 (Revelation principle). If a single-valued rule is dominant strategy implementable, then it is strategy-proof.

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We now provide a closed characterization result for single-valued rules satisfying strategyproofness.15 We say that a rule f is dictatorial if there is i ∈ N such that for each θ ∈ Θ, f (θ ) = xi , where xii = 1 and for each j 6= i, xij = 0. Proposition 3. A single-valued rule satisfies strategy-proofness, non-disagreement, non-bossiness, and welfarism if and only if it is dictatorial. Proof. See Appendix B.



It is easy to confirm that any dictatorial rule induces the following solution: a solution F is dictatorial if there is i ∈ N such that for each S ∈ Σ, Fi (S) = ai (S) and each j 6= i, Fj (S) = 0, where ai (S) ≡ max {si : s ∈ S}. Next is our main result. It states that only dictatorial solutions are induced by dominant strategy implementable rules satisfying nondisagreement, non-bossiness, and welfarism. 14See Thomson (2016) for a survey on non-bossiness.

15We are grateful to the anonymous referee for suggesting this statement.

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Theorem 1. A rule is dominant strategy implementable and satisfies non-disagreement, nonbossiness, and welfarism if and only if it induces a dictatorial solution. Proof. We first show the “only if” part. Let f be a dominant strategy implementable rule satisfying non-disagreement, non-bossiness, and welfarism. By Propositions 1 and 2, f is single-valued and strategy-proof. Thus, Proposition 3 implies that f is a dictatorial rule, which induces a dictatorial solution. Next, we show the “if” part. Let f be a rule inducing a dictatorial solution, that is, it is dictatorial. Then, it is easy to see that f is single-valued, ordinal, and invariant to individual monotone transformations. Thus, it follows from Proposition 1 that f is dominant strategy implementable. Moreover, by Proposition 3, f satisfies non-disagreement, non-bossiness, and welfarism.  By using Fact 1, as a corollary to Theorem 1, we can obtain the following:

Corollary 1. Suppose n = 2. A rule is dominant strategy implementable and satisfies nondisagreement and welfarism if and only if it induces a dictatorial solution.

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Remark 3. It should be noted that Theorem 1 is not a by-product of the well-known Gibbard-Satterthwaite theorem (Gibbard, 1972; Satterthwaite, 1975), because we cannot apply the Gibbard-Satterthwaite theorem to restricted (preference) domain environments such as ours. In fact, in our setting, some non-dictatorial rules (e.g., the equal sharing rule) are dominant strategy implementable and satisfy both non-disagreement and non-bossiness. Therefore, one might wonder if only dictatorial solutions remain. This is because non-dictatorial rules that are dominant strategy implementable and satisfy both non-disagreement and non-bossiness cannot induce any solution; that is, they all violate welfarism. This means that while welfarism is indispensable to establish the connection between rules and solutions, it is quite demanding in the sense that it drastically reduces the class of “implementable” solutions. ♦

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Remark 4. When the set of types is fairly restricted, Theorem 1 might not hold. For example, consider the set of types satisfying the following: for each i ∈ N, (i) Θi = (0, 1), and (ii) for each θi ∈ Θi , agent i has a utility function ui ( · ; θi ) : [0, 1] → R defined by ui ( xi ; θi ) = xiθi . In this setting, welfarism is fairly weak because there is a one-to-one mapping from this domain to the set of bargaining problems. Thus, any rule vacuously satisfies welfarism. This implies that every constant rule, including the equal sharing rule, can induce a solution. Hence, there are many non-dictatorial solutions induced by dominant strategy implementable rules satisfying non-disagreement, non-bossiness, and welfarism. ♦ Remark 5. Given Corollary 1, one might wonder if Theorem 1 holds without non-bossiness. Whenever there are three or more agents, there is a solution induced by a “bossy” rule satisfying dominant strategy implementability, non-disagreement, and welfarism ; however, the solution is not dictatorial. To see this, let N = {1, 2, 3}. Consider the following rule:

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for each θ ∈ Θ,

(0, 0, 1)

if u1 (1; θ1 ) ≥ 2

otherwise.

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f ∗ (θ ) =

 (0, 1, 0)

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Note that rule f ∗ is dominant strategy implementable and satisfies non-disagreement and welfarism.16 This rule induces the following solution F ∗ : for each S ∈ Σ,  (0, a (S), 0) if a (S) ≥ 2 2 1 F ∗ (S) = (0, 0, a3 (S)) otherwise.

Obviously, solution F ∗ is not dictatorial.

♦

Roemer (1996, Theorem 2.7) shows that dictatorial solutions are the only ones satisfying the three axioms that are standard in the axiomatic bargaining theory literature. The first axiom requires that there should be no feasible point at which all the agents are better off.

• Efficiency: For each S ∈ Σ, F (S) ∈ ∂S.

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The second axiom entails that the solution should yield the same list of utility levels if two bargaining problems are generated by the type profiles that prescribe the same vNM preference profile. For the formal definition of this axiom, some notation is required. Let i ∈ N. We say that a mapping τi : R → R is a positive linear transformation if τi : t 7→ ai t for each t ∈ R, where ai > 0. Let Ti be the set of such positive linear transformations. Let T ≡ ∏ j∈ N Tj . Given s ∈ R n , let τ (s) ≡ (τi (s j )) j∈ N = ( a j s j ) j∈ N . Given S ∈ Σ, let τ (S) ≡ {s′ ∈ R n : s′ = τ (s) for some s ∈ S}.

• Scale invariance: For each pair {S, S′ } ⊂ Σ and each τ ∈ T, if S′ = τ (S), then F (S′ ) = τ ( F (S)).

The last axiom, first introduced by Kalai (1977), is related to an agent’s attitude toward risk.

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• Strong monotonicity: For each pair {S, S′ } ⊂ Σ, if S ⊆ S′ , then F (S) ≦ F (S′ ). Proposition 4 (Roemer, 1996). A solution is efficient, scale invariant, and strongly monotonic if and only if it is dictatorial. Proof. For the original proof, see Roemer (1996, Theorem 2.7). We provide an alternative proof in Appendix C.  When we combine Proposition 4 with Theorem 1 or Corollary 1, we obtain the following results, which show the relationship between the axioms for bargaining solutions and the dominant strategy implementability of rules. 16It is easy to check that f ∗ satisfies single-valuedness, ordinality, and invariance to individual monotone transfor-

mations. Therefore, by Proposition 1, it is dominant strategy implementable.

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Corollary 2. A rule is dominant strategy implementable and satisfies non-disagreement, nonbossiness, and welfarism if and only if it induces a bargaining solution satisfying efficiency, scale invariance, and strong monotonicity. Corollary 3. Suppose n = 2. A rule is dominant strategy implementable and satisfies nondisagreement and welfarism if and only if it induces a bargaining solution satisfying efficiency, scale invariance, and strong monotonicity. Remark 6. Note that non-bossiness is indispensable for Corollary 2 to hold. In fact, whenever there are three or more agents, there is a solution induced by a “bossy” rule satisfying dominant strategy implementability, non-disagreement, and welfarism, but the solution violates strong monotonicity. Solution F ∗ introduced in Remark 5 is an example of such a solution. ♦

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Remark 7. If we enlarge the domain of utility functions by weakening the strict concavity of utility functions to concavity, all our results (Theorem 1, Propositions 1–4, and Corollaries 1–3) remain true.17 All proofs of our results, except for Proposition 3, remain valid for this larger domain. It is interesting to note that on this larger domain, dictatorial rules are the only rules that satisfy strategy-proofness, non-disagreement, and non-bossiness; that is, we obtain dictatorial results without welfarism.18 Proposition 3 immediately follows from this result. Moreover, this implies that only dictatorial rules are only rules satisfying dominant strategy implementability, non-disagreement, and non-bossiness. On the other hand, recall that some non-dictatorial rules satisfy these three properties if all utility functions are strictly concave utility functions (Remark 3). Therefore, if we enlarge the domain of utility functions, welfarism is no longer the source of our dictatorial results. ♦

4. C ONCLUDING

REMARKS

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We studied dominant strategy implementation in bargaining environments. This paper considered the problem where agents bargain over their shares of a perfectly divisible commodity. We identified the class of bargaining solutions induced by dominant strategy implementable rules satisfying non-disagreement, non-bossiness, and welfarism: only dictatorial solutions are induced by dominant strategy implementable rules satisfying the three conditions. By invoking the characterization result of Roemer (1996), this result clarified the relationship between axioms for solutions and dominant strategy implementability of rules. We found that efficiency, scale invariance, and strong monotonicity are necessary and sufficient for a solution to be induced by a dominant strategy implementable rule satisfying non-disagreement, non-bossiness, and welfarism.

17We thank an anonymous referee for suggesting this discussion. 18A proof of this claim is available from the authors upon request.

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Finally, we mention some open questions for consideration in future research. First, we focused on the class of rules that satisfy essential single-valuedness (A1) and Paretoindifference (A2). If essential single-valuedness is dropped, then we have to consider multivalued solutions.19 If Pareto-indifference is dropped, then many rules would appear.20 It would be interesting to identify necessary and sufficient conditions for any solution to be induced by dominant strategy implementable rules that are neither essentially single-valued nor Pareto-indifferent. Second, we considered the problem of dividing a commodity (e.g., a pie) among agents. In this problem, there is a common worst outcome at every type profile (i.e., the outcome 0). This structure helps in deriving the results in this paper. This can be attributed to the fact that the structure allows us to apply the result of Mizukami and Wakayama (2007) to our setting. It is an open question whether the results in this paper would hold for different economic environments. Third, we assumed that the set of type profiles takes a product form. Our method of the proof of Proposition 1 depends on this assumption. Thus, it is open whether the results in this paper would hold without assuming the product form of the set of type profiles. These issues will be considered in future research. A PPENDIX A. P ROOF

OF

P ROPOSITION 1

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We begin with proving five lemmas that are useful in the proof of Proposition 1. To do this, we now introduce additional notions. Given i ∈ N, θi ∈ Θi , and x ∈ X, let UC (θi , x) ≡ {ℓ ∈ △ : Ui (ℓ; θi ) ≥ Ui ( x; θi )} be the upper contour set of θi at x. Given i ∈ N and {θi , θi′ } ⊂ Θi , we say that θi is strictly more risk averse than θi′ if for each x ∈ X with xi ∈ (0, 1), UC (θi , x) ⊂ UC (θi′ , x). Lemma 1. Let θ ∈ Θ, i ∈ N, and θi′ ∈ Θi be such that f (θ ) is not a singleton and θi is strictly more risk averse than θi′ . Then, for each x ∈ f (θ ) ∩ X, if xi 6= 0, (i) for each ℓ ∈ f (θ ), Ui (ℓ; θi′ ) ≥ Ui ( x; θi′ ); (ii) there is ℓ′ ∈ f (θ ) such that Ui (ℓ′ ; θi′ ) > Ui ( x; θi′ ).

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Proof. Let θ ∈ Θ, i ∈ N, and θi′ ∈ Θi be such that f (θ ) is not a singleton and θi is strictly more risk averse than θi′ . Since f (θ ) is not a singleton, by A2, f (θ ) ∩ X 6= ∅. Let x ∈ f (θ ) ∩ X. Then, xi 6= 1; otherwise, f (θ ) = { x} by A1 and A2, that is, f (θ ) is a singleton. Since we assume xi 6= 0, xi ∈ (0, 1).

(1)

We first prove (i). Since θi is strictly more risk averse than θi′ , by (1), UC (θi′ , x) ⊃ UC (θi , x).

(2)

19By considering multi-valued solutions, Vartiainen (2007) derives possibility results for Nash implementa-

tion. 20See Barber`a, Bogomolnaia, and van der Stel (1998) for details.

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Note that by A1 and A2, f (θ ) = {ℓ ∈ △ : U (ℓ; θ ) = U ( x; θ )}. Then, by (1),

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UC (θi , x) = {ℓ ∈ △ : Ui (ℓ; θi ) ≥ Ui ( x; θi )} ⊃ {ℓ ∈ △ : U (ℓ; θ ) = U ( x; θ )} = f (θ ).

(3)

Thus, (2) and (3) together imply UC (θi′ , x) ⊃ f (θ ). This establishes that for each ℓ ∈ f (θ ), Ui (ℓ; θi′ ) ≥ Ui ( x; θi′ ). We next prove (ii). Suppose, by contradiction, that for each ℓ ∈ f (θ ), Ui (ℓ; θi′ ) ≤ Ui ( x; θi′ ). Then, by (i), for each ℓ ∈ f (θ ), Ui (ℓ; θi′ ) = Ui ( x; θi′ ).

(4)

Since (1) and θi is strictly more risk averse than θi′ , by Roth (1979, Theorem 4), there is an increasing and strictly concave function k : R → R such that for each xi′ ∈ [0, 1], ui ( xi′ ; θi ) = k(ui ( xi′ ; θi′ )). Let ℓˆ ∈ f (θ ) \ { x}. Since k is strictly concave, by Jensen’s inequality and (4),     X
ˆ xˆ )ui ( xˆ i ; θ ′ ) ℓ( Ui ( x; θi ) = k Ui ( x; θi′ ) = k Ui (ℓˆ ; θi′ ) = k  i

>

X

ˆ xˆ ∈supp ( ℓ)

contradicting A1.




ˆ xˆ ∈supp ( ℓ)

ˆ xˆ )k ui ( xˆi ; θ ′ ) = Ui (ℓˆ ; θi ), ℓ( i



Lemma 2. If a rule is dominant strategy implementable, then it is single-valued.

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Proof. Suppose, by contradiction, that a rule f is not single-valued. Then, there is θ ∈ Θ such that f (θ ) is not a singleton. Then, by A2, f (θ ) ∩ X 6= ∅. Let x ∈ f (θ ) ∩ X. Then, there is i ∈ N with xi ∈ (0, 1).21 Let i ∈ N and θ i ∈ Θi be such that xi ∈ (0, 1) and θi is strictly more risk averse than θ i . By A2, f (θ i , θ−i ) ∩ X 6= ∅. Let x ∈ f (θ i , θ−i ) ∩ X. Since f is dominant strategy implementable, then there is a mechanism Γ = ( M, g) such that g(DSEΓ (θ )) = f (θ );

(5)

g(DSEΓ (θ i , θ−i )) = f (θ i , θ−i ).

(6)

Let m∗ ∈ DSEΓ (θ ) and m ∈ DSEΓ (θ i , θ−i ). Note that (m∗i , m−i ) ∈ DSEΓ (θ ). Then, for each m′i ∈ Mi and each m−i ∈ M−i , Ui ( g(m∗i , m−i ); θi ) ≥ Ui ( g(m′i , m−i ); θi ). This implies that Ui ( g(m∗i , m−i ); θi ) ≥ Ui ( g(m i , m−i ); θi ).

(7)

Ui ( g(m∗i , m∗−i ); θi ) = Ui ( g(m∗i , m−i ); θi ).

(8)

Since (m∗i , m−i ) ∈ DSEΓ (θ ), (5) implies { g(m∗ ), g(m∗i , m−i )} ⊆ f (θ ). By A1,

21If there exists no i ∈ N with x ∈ (0, 1), then either (i) for each i ∈ N, x = 0 or (ii) for some i ∈ N, x = 1 i i i

and for each j 6= i, x j = 0. In both cases, f (θ ) = { x } by A1 and A2; that is, f (θ ) is a singleton.

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It thus follows from (7) and (8) that Ui ( g(m∗ ); θi ) ≥ Ui ( g(m ); θi ). Combined with (5) and (6), this implies that for each ℓ ∈ f (θ ) and each ℓ ∈ f (θ i , θ−i ),

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Ui (ℓ; θi ) ≥ Ui (ℓ; θi ).

(9)

Similarly, for each ℓ ∈ f (θ i , θ−i ) and each ℓ ∈ f (θ ),

Ui (ℓ; θ i ) ≥ Ui (ℓ; θ i ).

(10)

Then, (9) and A1 together imply that Ui ( x; θi ) ≥ Ui ( x; θi ), which implies that, by the strict increasingness of ui ( · ; θi ), xi ≥ x i .

(11)

Recall that xi 6= 0, f (θ ) is not a singleton, and θi is strictly more risk averse than θ i . It thus follows from Lemma 1 that there is ℓ˜ ∈ f (θ ) such that Ui (ℓ˜ ; θ i ) > Ui ( x; θ i ).

(12)

Since ℓ˜ ∈ f (θ ), by (10), for each ℓ ∈ f (θ i , θ−i ),

Ui (ℓ; θ i ) ≥ Ui (ℓ˜ ; θ i ).

(13)

Then (12) and (13) together imply that for each ℓ ∈ f (θ i , θ−i ), Ui (ℓ; θ i ) > Ui ( x; θ i ). Since x ∈ f (θ i , θ−i ), Ui ( x; θ i ) > Ui ( x; θ i ). Hence, the strict increasingness of ui ( · ; θ i ) implies  that xi > xi , which contradicts (11). Lemma 3. If a rule is dominant strategy implementable, then it is ordinal.

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Proof. Suppose, by contradiction, that a rule f is not ordinal. Then there is {θ, θ ′ } ⊂ Θ such that f (θ ) 6= f (θ ′ ), but there are no i ∈ N and {ℓ, ℓ′ } ⊂ △ such that Ui (ℓ; θi ) ≥ Ui (ℓ′ ; θi ) and Ui (ℓ′ ; θi′ ) > Ui (ℓ; θi′ ). This means that θ and θ ′ yield the same vNM preference profile. Since f is dominant strategy implementable, then there is a mechanism Γ = ( M, g) such that g(DSEΓ (θ )) = f (θ );

(14)

g(DSEΓ (θ ′ )) = f (θ ′ ).

(15)

Since θ and θ ′ yield the same vNM preference profile, DSEΓ (θ ) = DSEΓ (θ ′ ). On the other hand, by f (θ ) 6= f (θ ′ ) and (14)–(15), we have g(DSEΓ (θ )) = f (θ ) 6= f (θ ′ ) = g(DSEΓ (θ ′ )), which implies that DSEΓ (θ ) 6= DSEΓ (θ ′ ). This is a contradiction.  Lemma 4. If a rule is dominant strategy implementable, then it is invariant to individual monotone transformations. Proof. Suppose, by contradiction, that a rule f is not invariant to individual monotone transformations. Then, there are θ ∈ Θ, i ∈ N, θi′ ∈ Θi , ℓ ∈ f (θ ), ℓ′ ∈ f (θi′ , θ−i ), and λi ∈ Λi such that S(θi′ , θ−i ) = λi (S(θ )) and Ui (ℓ′ ; θi′ ) 6= λi (Ui (ℓ; θi )). By single-valuedness (Lemma 2), A2 implies that { f (θ ), f (θi′ , θ−i )} ⊂ X ∪ {0}. It then follows that Ui ( f (θi′ , θ−i ); θi′ ) 6=

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λi (Ui ( f (θ ); θi )). This implies Ui ( f (θi′ , θ−i ); θi′ ) 6= Ui ( f (θ ); θi′ ). Since both ui ( · ; θi ) and ui ( · ; θi′ ) are strict increasing, either

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(i) Ui ( f (θ ); θi ) > Ui ( f (θi′ , θ−i ); θi ) and Ui ( f (θ ); θi′ ) > Ui ( f (θi′ , θ−i ); θi′ ), or (ii) Ui ( f (θ ); θi ) < Ui ( f (θi′ , θ−i ); θi ) and Ui ( f (θ ); θi′ ) < Ui ( f (θi′ , θ−i ); θi′ ).

On the other hand, since f is dominant strategy implementable and single-valued, the revelation principle (Proposition 2) implies that f is strategy-proof. This implies that Ui ( f (θ ); θi ) ≥ Ui ( f (θi′ , θ−i ); θi ) and Ui ( f (θi′ , θ−i ); θi′ ) ≥ Ui ( f (θ ); θi′ ), contradicting (i) and (ii).



Lemma 5. If a single-valued rule is invariant to individual monotone transformations, then it is strategy-proof. Proof. We prove this by contraposition. Suppose that a single-valued rule f is not strategyproof. Then, there are θ ∈ Θ, i ∈ N, and θi′ ∈ Θi such that Ui ( f (θ ); θi ) < Ui ( f (θi′ , θ−i ); θi ). By single-valuedness, A2 implies that { f (θ ), f (θi′ , θ−i )} ⊂ X ∪ {0}. Let λi ∈ Λi be such that S(θi′ , θ−i ) = λi (S(θ )). Note that by Remark 1, for each xi ∈ [0, 1], ui ( xi ; θi′ ) = λi (ui ( xi ; θi )). Then, we have Ui ( f (θi′ , θ−i ); θi′ ) = λi (Ui ( f (θi′ , θ−i ); θi )) > λi (Ui ( f (θ ); θi )). Hence, f is not invariant to individual monotone transformations. 

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Proof of Proposition 1. The “only if” part immediately follows from Lemmas 2–4. Thus, we now show the “if” part. Let f be a rule satisfying single-valuedness, ordinality, and invariance to individual monotone transformations. Then, it follows from Lemma 5 that f is strategy-proof. Let   ′ ∈ Θ , there are ℓ, ℓ′ ∈ △ such that   for each θ i   i ⊂ Θi . Θ∗i ≡ θi ∈ Θi : Ui (ℓ; θi ) > Ui (ℓ′ ; θi ), Ui (ℓ′ ; θi′ ) > Ui (ℓ; θi′ ),     ′ ′ ′ Ui (ℓ; θi ) > Ui (0; θi ), and Ui (ℓ ; θi ) > Ui (0; θi )

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When we consider Θ∗ ≡ ∏i∈ N Θ∗i , we are in a so-called “weak separable environment.”22 Thus, we can use the Mizukami and Wakayama’s (2007) mechanism Γ = ( M, g) for dominant strategy implementation. By single-valuedness and strategy-proofness, Mizukami and Wakayama’s (2007) result implies that for each θ ′ ∈ Θ∗ , f (θ ′ ) = g(DSEΓ (θ ′ )).

(16)

Let θˆ ∈ Θ \ Θ∗ . Then, there is θ ′ ∈ Θ∗ such that θˆ and θ ′ yield the same vNM preference profile.23 Thus, DSEΓ (θˆ) = DSEΓ (θ ′ ). By ordinality and (16), f (θˆ ) = f (θ ′ ) = g(DSEΓ (θ ′ )) = g(DSEΓ (θˆ )). Hence, we can conclude that f is dominant strategy implementable.  22Roughly speaking, an environments is weak separable one if (i) there is a common worst alternative; (ii)

we can always punish one agent without changing the outcome received by the other agent; and (iii) strict value distinction with respect to the common worst alternative is satisfied. See Mizukami and Wakayama (2007) for the formal definition. 23The proof of this claim is available from the authors upon request.

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Before proving Proposition 3, we provide the following useful lemma. We say that a rule is constant if for each {θ, θ ′ } ⊂ Θ, f (θ ) = f (θ ′ ). Lemma 6. A single-valued rule is strategy-proof and non-bossy if and only if it is constant. Proof. Since the “if” part is obvious, it suffices to show the “only if” part. Let f be a single-valued rule satisfying strategy-proofness and non-bossiness. Let {θ, θ ′ } ⊂ Θ. We now show that f (θ ) = f (θ ′ ). Since f is single-valued, A2 implies that { f (θ ), f (θ ′ )} ⊂ X ∪ {0}. For simplicity, we abuse notation and let f i (θˆ) denote the share assigned to agent i at θˆ ∈ Θ. Since both u1 ( · ; θ1 ) and u1 ( · ; θ1′ ) are strictly increasing, strategy-proofness implies f1 (θ ) = f1 (θ1′ , θ−1 ). Thus, by non-bossiness, we have f (θ ) = f (θ1′ , θ−1 ). Since u2 ( · ; θ2 ) and u2 ( · ; θ2′ ) are strictly increasing, strategy-proofness implies f2 (θ1′ , θ−1 ) = f2 (θ1′ , θ2′ , θ−{1,2} ). Thus, by non-bossiness and f (θ ) = f (θ1′ , θ−1 ), we have f (θ ) = f (θ1′ , θ2′ , θ−{1,2} ). Repeating this argument for i = 3, . . . , n, we have f (θ ) = f (θ ′ ). 

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Proof of Proposition 3. We first show the “if” part. Let f be a dictatorial rule. Then, there is i ∈ N such that for each θˆ ∈ Θ, f (θˆ ) = xi . Without loss of generality, assume i = 1. It is easy to confirm that f satisfies strategy-proofness, non-bossiness, and non-disagreement. We now show that f satisfies welfarism. Let {θ, θ ′ } ⊂ Θ be such that S(θ ) = S(θ ′ ). Since x11 = 1 and u1 (1; θ1 ) = u1 (1; θ1′ ), U1 ( f (θ ); θ1 ) = U1 ( f (θ ′ ); θ1′ ). Moreover, for each j 6= 1, since x1j = 0 and u j (0; θ j ) = u j (0; θ j′ ), Uj ( f (θ ); θ j ) = Uj ( f (θ ′ ); θ j′ ). Hence, U ( f (θ ); θ ) = U ( f (θ ′ ); θ ′ ), which implies that f satisfies welfarism. We next show the “only if” part. Let f be a single-valued rule satisfying the four conditions. By Lemma 6, f is constant. Then, by non-disagreement and single-valuedness, there ˆ Suppose, by contradiction, that f is not is xˆ ∈ X such that for each θˆ ∈ Θ, f (θˆ) = x. dictatorial. Then, there is { j, k} ⊆ N such that 0 < xˆ j < 1 and

0 < xˆ k < 1.

(17)

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Without loss of generality, assume j = 1. Let {θ, θ ′ } ⊂ Θ be such that for each i ∈ N, ui (1; θi ) = ui (1; θi′ ) and each xi ∈ (0, 1), ui ( xi ; θi′ ) > ui ( xi ; θi ). Then, S(θ ) ⊂ S(θ ′ ). Thus, f (θ ) = f (θ ′ ) = xˆ and (17) together imply that U1 ( f (θ ′ ); θ1′ ) > U1 ( f (θ ); θ1 ).

(18)

Let θ ′′ ∈ Θ be such that S(θ ′ ) ⊂ S(θ ′′ ), θ1′′ = θ1 , and for each i 6= 1 and each xi ∈ (0, 1], ui ( xi ; θi′′ ) > ui ( xi ; θi ). Since f (θ ′′ ) = f (θ ) and θ1′′ = θ1 , U1 ( f (θ ′′ ); θ1′′ ) = U1 ( f (θ ); θ1 ).

(19)

Let θ ∗ ∈ Θ be such that S(θ ∗ ) = S(θ ′′ ), θ1∗ = θ1′ , and for each i 6= 1 and each xi ∈ (0, 1], ui ( xi ; θi∗ ) > ui ( xi ; θi′ ). Since f (θ ∗ ) = f (θ ′ ) and θ1∗ = θ1′ , U1 ( f (θ ∗ ); θ1∗ ) = U1 ( f (θ ′ ); θ1′ ).

(20)

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By (18)–(20), U1 ( f (θ ∗ ); θ1∗ ) > U1 ( f (θ ′′ ); θ1′′ ). This implies U ( f (θ ∗ ); θ ∗ ) 6= U ( f (θ ′′ ); θ ′′ ). On the other hand, since S(θ ∗ ) = S(θ ′′ ), welfarism implies U ( f (θ ∗ ); θ ∗ ) = U ( f (θ ′′ ); θ ′′ ). This is a contradiction.  A PPENDIX C. A LTERNATIVE

PROOF OF

P ROPOSITION 4

Since the “if” part is straightforward, we prove the “only if” part. Let F be a solution satisfying efficiency, scale invariance, and strong monotonicity. Given S ∈ Σ and i ∈ N, let ai (S) = ( a1i (S), . . . , ain (S)) be such that aii (S) ≡ ai (S) and for each j 6= i, aij (S) = 0. We proceed in two steps.  S Step 1: For each S ∈ Σ, F (S) ∈ i∈ N ai (S) . Suppose, by contradiction, that there is  S S ∈ Σ such that F (S) ∈ / i∈ N ai (S) . Then, efficiency implies that o [ n F (S) ∈ ∂S ai ( S ) . (21) i∈ N

Let S′ ∈ Σ be such that S′ ⊃ S and for each i ∈ N, ai (S′ ) = ai (S). Then, strong monotonicity implies F (S′ ) ≧ F (S). Next, let j ∈ N. Consider S j ∈ Σ and τ j ∈ T such that S j ⊃ S′ and S j = τ j (S), where n o j τ j (S) = s j ∈ R n : s j = τ j (s) = (s j , τ− j (s− j )) for some s ∈ S . Then, scale invariance implies F (S j ) = τ j ( F (S)). Since S j ⊃ S′ , strong monotonicity implies F (S j ) ≧ F (S′ ). Thus, we obtain F (S j ) = τ j ( F (S)) ≧ F (S′ ) ≧ F (S).

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By Fj (S j ) = Fj (S), this implies Fj (S′ ) = Fj (S). Since j is arbitrary, we obtain that for each i ∈ N, Fi (S′ ) = Fi (S), that is, F (S′ ) = F (S). Then, (21) and S′ ⊃ S together imply F (S′ ) ∈ / ∂S′ , contradicting efficiency.  S Step 2: F is dictatorial. Let S ∈ Σ. Then, by Step 1, we obtain F (S) ∈ i∈ N ai (S) . Without loss of generality, we assume that F (S) = a1 (S). Let S′ ∈ Σ. We now show F (S′ ) = a1 (S′ ). Let S ∈ Σ be such that (i) S ⊂ S′ ; (ii) a1 (S) = a1 (S′ ); and (iii) S = τ (S) for some τ ∈ T. Then, scale invariance implies F (S) = τ ( F (S)). Thus F (S) = a1 (S). By (i), strong monotonicity implies that F (S) ≦ F (S′ ). Combined with (ii), these imply a1 (S) = a1 (S′ ) ≦ F (S′ ), which establishes F (S′ ) = a1 (S′ ). Hence F is dictatorial.  R EFERENCES

B ARBER A` , S., A. B OGOMOLNAIA , AND H. VAN DER S TEL (1998): “Strategy-proof probabilistic rules for expected utility,” Mathematical Social Sciences, 35, 89–103. B ERGIN , J. AND J. D UGGAN (1999): “Implementation theoretic approach to non-cooperative foundations,” Journal of Economic Theory, 86, 50–76. B OGOMOLNAIA , A., R. D EB , AND L. E HLERS (2005): “Strategy-proof assignment on the full preference domain,” Journal of Economic Theory, 123, 161–186. C HEN , M. AND E. M ASKIN (1999): “Bargaining, production, and monotonicity in economic environments,” Journal of Economic Theory, 89, 140–147.

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