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Social Choice and Welfare

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07-07-2017 | Original Paper

Bribe-proofness for single-peaked preferences: characterizations and maximality-of-domains results

Author: Takuma Wakayama

Published in: Social Choice and Welfare | Issue 2/2017

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Abstract

This paper considers the problem of allocating an amount of a perfectly divisible resource among agents. We are interested in rules eliminating the possibility that an agent can compensate another to misrepresent her preferences, making both agents strictly better off. Such rules are said to be bribe-proof (Schummer in J Econ Theory 91:180–198, 2000). We first provide necessary and sufficient conditions for any rule defined on the single-peaked domain to be bribe-proof. By invoking this and Ching’s (Soc Choice Welf 11:131–136, 1994) result, we obtain the uniform rule as the unique bribe-proof and symmetric rule on the single-peaked domain. Furthermore, we examine how large a domain can be to allow for the existence of bribe-proof and symmetric rules and show that the convex domain is such a unique maximal domain.

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The model has been given several interpretations. See Sprumont (1991) and Barberà et al. (1997).

“Single-peakedness” denotes that each agent has a most preferred share and that above or below this share, welfare decreases.

A number of studies characterize the uniform rule, such as Thomson (1994a, b, 1995, 1997), Schummer and Thomson (1997), Chun (2000, 2003), Kesten (2006), Serizawa (2006), Mizobuchi and Serizawa (2006), and Sakai and Wakayama (2012). For a survey of these characterizations of the uniform rule, see Thomson (2014). See also Thomson (2011) for a survey of fair allocation theory including a discussion of the uniform rule.

Massò and Neme (2003, 2007) call this axiom “bribe-proofness.”

Bonifacio (2015) applies coalitional bribe-proofness to the problem of reallocating a perfectly divisible good among a group of agents with individual endowments and single-peaked preferences. He also identifies the class of coalitionally bribe-proof reallocation rules.

The author thanks William Thomson for suggesting this term and the interpretation of this axiom. In addition, a similar axiom exists in the framework of bankruptcy problems. See, for example, Subsection 3.4 in Thomson (2015).

Throughout this paper, we assume that preferences are continuous. Therefore, strictly speaking, we look for maximal domains including the single-peaked domain and included in the domain of continuous preferences.

“Single-plateaued preferences” are variants of single-peaked preferences, for which the sets of most preferred shares are intervals.

A preference is “convex” if its upper contour set of any share is convex.

Let \(a, b \in \mathbb {R}\) be such that \(a \le b\). Then, we denote by \([a, b ]\) and \(]a, b [\) the closed interval from a to b and the open interval from a to b, respectively. We also denote by \([a, b[\) and ]a, b] the half-open intervals from a to b.

See Schummer (2000) for a more detailed discussion.

It is easy to confirm this fact by using Example 1 in Massò and Neme (2003).

To be more precise, Massò and Neme (2007) use a stronger version of coalitional bribe-proofness; it requires that bribers (i.e., agents in \(S {\setminus } T\)) weakly gain after reallocating their shares, while bribed agents (i.e., agents in T) strictly gain. Therefore, by using this idea, we can define a stronger version of bribe-proofness. See Sect. 4.1 for a more detailed discussion.

This axiom is a weaker version of “replacement monotonicity” introduced by Barberà et al. (1997). Weak replacement monotonicity is first studied under the name “one-sided welfare domination under preference replacement” in Thomson (1997). Although the conclusion as regards the latter is written in welfare terms, both axioms are equivalent in the presence of efficiency.

This rule \(f^{ \text{ MN }}(\ \cdot \ , \Omega )\) is based on the rule introduced by Massò and Neme (2007, Section 5).

This axiom is a weaker version of non-bossiness introduced by Satterthwaite and Sonnenschein (1981). The notion of non-bossiness is formally defined as follows: for each \(R \in \mathscr {P}^{N}_{\Omega }\), each \(i \in N\), and each \(R'_i \in \mathscr {P}_{\Omega }\), if \(f_i(R, \Omega ) = f_i(R'_i, R_{-i}, \Omega )\), then \(f(R, \Omega ) = f(R'_i, R_{-i}, \Omega )\).

The notion of no-envy is formally defined as follows: for each \(R \in \mathscr {D}^N\) and each \(i,j \in N\), \(f_i(R, \Omega ) \mathrel {R_i} f_j(R, \Omega )\).

This notion is first introduced by Schmeidler and Vind (1972) in pure exchange economies. Klaus et al. (1997) use this notion to characterize the uniform reallocation rule.

Bonifacio (2015) considers a notion of fairness, called “equal-treatment,” that is somewhat different from the notion of symmetry but is in the spirit of symmetry. It says that if the differences between the individual endowment and the “peak” (i.e., the unique element in the top set) of any two agents are the same, then each of them should be indifferent between her share change and the other agent’s share change. Bonifacio (2015, Theorem 5) shows that the uniform reallocation rule is the only reallocation rule that satisfies coalitional bribe-proofness, “equal-treatment,” and an auxiliary axiom called “reversibility.”

This axiom is suggested by an anonymous referee.

Massò and Neme (2001) obtain a maximal domain result for rules satisfying efficiency, strategy-proofness, and a stronger version of symmetry in the fixed-resource framework.

The condition of “tops-only” states that the amount allocated to agents depends only on their top set.

To be more precise, Serizawa (2006) shows that the uniform rule on \(\mathscr {P}^{N}_{\Omega }\) is non-bossy. As we stated in Footnote 16, weak non-bossiness is weaker than non-bossiness.

Suppose, on the contrary, that \(\Omega \le \sum _{j \in N} T(R_j)\). Then, by \(x_i \ne T(R_i)\) and the definition of the uniform rule, \(T(R_i) > x_i = \min \{T(R_i), \lambda \} = \lambda \). If \(\Omega \le \sum _{j \in N {{\setminus }} \{i\}} T(R_j) + T(R'_i)\), then \(\lambda = x_i < x'_i = T(R'_i) = \min \{T(R'_i), \lambda '\} \le \lambda '\). It then follows that for each \(j \in N {{\setminus }} \{ i\}\), \(x'_j \ge x_j\). Hence \(\Omega = \sum _{j \in N} x'_j > \sum _{j \in N} x_j = \Omega \), a contradiction. If \(\sum _{j \in N {{\setminus }} \{i\}} T(R_j) + T(R'_i) < \Omega \), then \(x'_j = \max \{T(R_j), \lambda ' \} \ge \min \{T(R_j), \lambda \} = x_j\) for each \(j \in N {\setminus } \{ i\}\). This implies that \(\Omega = \sum _{j \in N} x'_j > \sum _{j \in N} x_j = \Omega \), a contradiction.

Whether the extended uniform rule is the only rule satisfying bribe-proofness and symmetry on the convex domain is open.

Barberà S, Jackson MO, Neme A (1997) Strategy-proof allotment rules. Games Econ Behav 18:1–21CrossRef

Benassy J-P (1982) The economics of market disequilibrium. Academic, New York

Bonifacio AG (2015) Bribe-proof reallocation with single-peaked preferences. Soc Choice Welf 55:617–638CrossRef

Ching S, Serizawa S (1998) A maximal domain for the existence of strategy-proof rules. J Econ Theory 78:157–166CrossRef

Ching S (1994) An alternative characterization of the uniform rule. Soc Choice Welf 11:131–136CrossRef

Chun Y (2000) Distributional properties of the uniform rule in economies with single-peaked preferences. Econ Lett 67:23–27CrossRef

Chun Y (2003) One-sided population-monotonicity, separability, and the uniform rule. Econ Lett 78:343–349CrossRef

Foley D (1967) Resource allocation and the public sector. Yale Econ Essays 7:45–98

Kesten O (2006) More on the uniform rule: characterizations without Pareto-optimality. Math Soc Sci 51:192–200CrossRef

Klaus B, Peters H, Storken T (1997) Reallocation of an infinitely divisible good. Econ Theory 10:305–333CrossRef

Massò J, Neme A (2001) Maximal domain of preferences in the division problem. Games Econ Behav 37:367–387CrossRef

Massò J, Neme A (2003) Bribe-proof rules in the division problem. UFAE and IAE Working papers 571.03

Massò J, Neme A (2007) Bribe-proof rules in the division problem. Games Econ Behav 61:331–343CrossRef

Mizobuchi H, Serizawa S (2006) Maximal domain for strategy-proof rules in allotment economies. Soc Choice Welf 27:195–210CrossRef

Mizukami H (2003) On the constancy of bribe-proof solutions. Econ Theory 22:211–217CrossRef

Sakai T, Wakayama T (2012) Strategy-proofness, tops-only, and the uniform rule. Theory Decis 72:287–301CrossRef

Satterthwaite MA, Sonnenschein H (1981) Strategy-proof allocation mechanisms at differentiable points. Rev Econ Stud 48:587–597CrossRef

Schmeidler D, Vind K (1972) Fair net trade. Econometrica 60:637–642CrossRef

Schummer J (2000) Manipulation thorough bribes. J Econ Theory 91:180–198CrossRef

Schummer J, Thomson W (1997) Two derivations of the uniform rule. Econ Lett 55:333–337CrossRef

Serizawa S (2006) Pairwise strategy-proofness and self-enforcing manipulation. Soc Choice Welf 26:305–331CrossRef

Sprumont Y (1991) The division problem with single-peaked preferences: a characterization of the uniform allocation rule. Econometrica 59:509–519CrossRef

Thomson W (1994a) Resource-monotonic solutions to the problem of fair division when preferences are single-peaked. Soc Choice Welf 11:205–223

Thomson W (1994b) Consistent solutions to the problem of fair division when preferences are single-peaked. J Econ Theory 63:219–245CrossRef

Thomson W (1995) Population-monotonic solutions to the problem of fair division when preferences are single-peaked. Econ Theory 5:229–246CrossRef

Thomson W (1997) The replacement principle in economies with single-peaked preferences. J Econ Theory 76:145–168CrossRef

Thomson W (2011) Fair allocation rules. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of Social Choice and Welfare, vol 2. Elsevier, Amsterdam, pp 393–506

Thomson W (2014) Fully allocating a commodity among agents with single-peaked preferences. Working paper, University of Rochester

Thomson W (2015) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update. Math Soc Sci 74:41–59CrossRef

Title: Bribe-proofness for single-peaked preferences: characterizations and maximality-of-domains results
Author: Takuma Wakayama
Publication date: 07-07-2017
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2017
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-017-1068-2

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