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Social Choice and Welfare

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09-12-2015 | Original Paper

Bargaining, conditional consistency, and weighted lexicographic Kalai-Smorodinsky Solutions

Author: Bram Driesen

Published in: Social Choice and Welfare | Issue 4/2016

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Abstract

We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (Econ Lett 73:131–136, 2001), and using methods of Imai (Econometrica 51:389–401, 1983), extend their characterization to the domain of multilateral bargaining problems. Aside from standard axioms in the literature, this result involves a new property that weakens the axiom Bilateral Consistency (Lensberg, J Econ Theory 45:330–341, 1988), by making the notion of consistency dependent on how ideal values in a reduced problem change relative to the original problem.

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Lensberg calls this axiom Bilateral Stability.

This claim is more carefully motivated in Sect. 2.3.1.

These papers provide non-cooperative support for the asymmetric Nash solution (Nash 1950; Harsanyi and Selten, 1972; Kalai 1977a).

Vector inequalities: \(\geqq \), \(\ge \), \(>\).

Inclusion is denoted \(\subseteq \), and strict inclusion \(\subset \).

For \(N\in \mathcal {N}\) and \(S\in \Sigma ^{N}\), the Nash solution (Nash 1950) is defined as the unique maximizer of \(\prod _{i\in N}x_{i}\) on \(S\cap \mathbb {R}^{N}_{+}\). A proportional solution is defined as \(\beta ^{*}w\) where w is some vector in \(\mathbb {R}^{N}_{++}\), and \(\beta ^{*}:=\max \{\beta \mid \beta w\in S\}\). The Kalai-Smorodinsky solution (Kalai and Smorodinsky 1975) is defined as \(K(S):=\beta ^{*}u(S)\), where \(\beta ^{*}:=\max \{\beta \mid \beta u(S)\in S\}\). The Raiffa solution is defined as the (possibly infinite) sum \(\frac{1}{|N|}u(S)+\frac{1}{|N|}u(S-\frac{1}{|N|}u(S))+\frac{1}{|N|}u(S-\frac{1}{|N|}u(S)-\frac{1}{|N|}u(S-\frac{1}{|N|}u(S)))+\ldots \).

See also Luce and Raiffa (1957, p. 133).

Peters et al. refer to reduced problems as reduced games.

Given \(N\in \mathcal {N}\) and \(V\subset \mathbb {R}^{N}\), \(cch\, V\) denotes the convex comprehensive hull of (the points in) V. It is defined as the intersection of all convex and comprehensive sets in \(\mathbb {R}^{N}\) that contain (the points in) V.

See the solution D, defined in Sect. 4.1.

This proof is included in the Appendix.

Parts (i) and (ii) of Lemma 3.5 respectively correspond with Lemmas 3 and 8 of Imai (1983, pp. 395, 397).

For all \(Q,N\in \mathcal {N}\) with \(Q\subset N\) and \(|Q|=2\), and for all \(S\in \Sigma ^{N}\).

It is not known in general whether Imai’s (1983) lexicographic Kalai-Smorodinsky solution – and by extension, the weighted generalizations considered in this paper—can be characterized on this smaller domain.

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Title: Bargaining, conditional consistency, and weighted lexicographic Kalai-Smorodinsky Solutions
Author: Bram Driesen
Publication date: 09-12-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2016
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0936-x

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