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Erschienen in: Social Choice and Welfare 2/2017

07.07.2017 | Original Paper

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

verfasst von: Takuma Wakayama

Erschienen in: Social Choice and Welfare | Ausgabe 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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Fußnoten
1
The model has been given several interpretations. See Sprumont (1991) and Barberà et al. (1997).
 
2
“Single-peakedness” denotes that each agent has a most preferred share and that above or below this share, welfare decreases.
 
3
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.
 
4
Massò and Neme (2003, 2007) call this axiom “bribe-proofness.”
 
5
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.
 
6
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).
 
7
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.
 
8
“Single-plateaued preferences” are variants of single-peaked preferences, for which the sets of most preferred shares are intervals.
 
9
A preference is “convex” if its upper contour set of any share is convex.
 
10
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 ]ab] the half-open intervals from a to b.
 
11
See Schummer (2000) for a more detailed discussion.
 
12
It is easy to confirm this fact by using Example 1 in Massò and Neme (2003).
 
13
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.
 
14
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.
 
15
This rule \(f^{ \text{ MN }}(\ \cdot \ , \Omega )\) is based on the rule introduced by Massò and Neme (2007, Section 5).
 
16
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 \(f(R, \Omega ) = f(R'_i, R_{-i}, \Omega )\).
 
17
The notion of no-envy is formally defined as follows: for each \(R \in \mathscr {D}^N\) and each \(i,j \in N\), \(f_i(R, \Omega ) \mathrel {R_i} f_j(R, \Omega )\).
 
18
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.
 
19
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.”
 
20
This axiom is suggested by an anonymous referee.
 
21
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.
 
22
The condition of “tops-only” states that the amount allocated to agents depends only on their top set.
 
23
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.
 
24
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.
 
25
Whether the extended uniform rule is the only rule satisfying bribe-proofness and symmetry on the convex domain is open.
 
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Metadaten
Titel
Bribe-proofness for single-peaked preferences: characterizations and maximality-of-domains results
verfasst von
Takuma Wakayama
Publikationsdatum
07.07.2017
Verlag
Springer Berlin Heidelberg
Erschienen in
Social Choice and Welfare / Ausgabe 2/2017
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-017-1068-2

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