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01.11.2015

The Nash solution is more utilitarian than egalitarian

verfasst von: Shiran Rachmilevitch

Erschienen in: Theory and Decision | Ausgabe 3/2015

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Abstract

I state and prove formal versions of the claim that the Nash (Econometrica 18: 155–162, 1950) bargaining solution creates a compromise between egalitarianism and utilitarianism, but that this compromise is “biased”: the Nash solution puts more emphasis on utilitarianism than it puts on egalitarianism. I also extend the bargaining model by assuming that utility can be transferred between the players at some cost (the transferable and non-transferable utility models are polar cases of this more general one, corresponding to the cases where the transfer cost is zero and infinity, respectively); I use the extended model to better understand the connections between egalitarianism and utilitarianism.

Vorheriger Artikel Cooperative games with homogeneous groups of participants

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It is assumed that \(\mathbf 0 \equiv (0,0)\in S\) for every bargaining problem \(S\), so zero payoffs are always feasible; also, it is assumed that there is an \(x\in S\) with \(x>\mathbf 0 \), so cooperation is worthwhile (\(u R v\) means that \(u_i R v_i\) for both \(i\), for each \(R\in \{>,\ge \}\); \(u\gneqq v\) means that \(u\ge v\) and \(u\ne v\)).

Comprehensiveness means that \(\{y\in {\mathbb {R}}_+^2: y\le x\}\subset S\) for all \(x\in S\).

\(E\) was axiomatized for the first time by Kalai (1977). \(U\), in general, is multi-valued; in this paper, I will only consider problems for which it is single-valued.

For problems \(S\) whose Pareto frontier is strictly concave, \(U(S)\) is the unique maximizer of the utility sum over \(S\) and \(E(S)\) is the unique maximizer of \(\text {min}\{x_1,x_2\}\) over \(x\in S\) (in the following Section, I formally introduce an important class of such problems—smooth bargaining problems). Compromising on precision just a tiny bit, I will sometimes refer to \(\sum _i x_i\) and \(\text {min}\{x_1,x_2\}\) as the utilitarian and egalitarian objectives, respectively.

Suppes (1966), Sen (1970).

Related results have been obtained by Anbarci and Sun (2011). See also Mariotti (2000).

Later in this paper, I will prove two generalizations of Proposition 1—Propositions 6 and 7 below. Hence, for brevity, a proof of Proposition 1 is not provided.

A bargaining problem is normalized if for each player the minimum and maximum utilities are \(0\) and \(1\).

Cao refers to the relative utilitarian solution as the modified Thomson solution and to the Kalai-Smorodinsky as the Raiffa solution. I will introduce these solutions formally in Sect. 3.

Both results are related to the fact that the Nash solution is the only solution that jointly satisfies the egalitarian and utilitarian objectives for some rescaling of the individual utilities (Harsanyi 1959, Shapley 1969).

CES solutions have been studied by Sobel (2001), Bertsimas et al. (2012), and Haake and Qin (2013).

Locally, the value of the Nash product does not decrease, as one gets closer to the Nash solution point.

Alvarez-Cuadrado and van Long (2009) consider a maximization of a convex combination of utilitarian and egalitarian objectives in the context of intergenerational equity (their objectives are defined on infinite utility streams).

This is due to midpoint domination (Sobel 1981).

Like \(U\), the solution \(RU\) is also, in principle, multi-valued. For simplicity, I assume that the problems under consideration in this paper are such that it is single-valued (this is the case, for example, on the domain \({\mathcal {B}}^*\)).

A solution, \(\mu \), is scale invariant if \(\mu (l\circ S)=l\circ \mu (S)\) for every \(S\) and every pair of positive linear transformations \(l=(l_1,l_2)\). A positive linear transformation is also called a rescaling.

This assumption is wlog, since each \(\mu \in \{E,U,\mu ^\rho \}\) is an anonymous solution; a solution \(\mu \) is anonymous if for each \(S\) it is true that \(\pi \circ \mu (S)=\mu (\pi \circ S)\), where \(\pi (a,b)\equiv (b,a)\).

A solution \(\mu \) is continuous if \(\mu (S_n)\rightarrow \mu (S)\), provided that \(\{S_n\}\) converges to \(S\) in the Hausdorff topology.

The last equality here is due to the fact that we just proved that \(N^{h(p)}\) is \(p\)-EU robust.

This means that the solution point is to the right of \(E(S)\).

See Fleurbaey et al. (2008) for illuminating discussions on the subject.

I am grateful to a thorough referee for offering this interpretation.

\(\alpha =\frac{1}{2}\) corresponds to the solution \(NA\).

For the sake of brevity, I omit the proof. It is available upon request.

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Titel: The Nash solution is more utilitarian than egalitarian
verfasst von: Shiran Rachmilevitch
Publikationsdatum: 01.11.2015
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 3/2015
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-014-9477-5

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