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.
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Notes
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.
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.
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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Acknowledgments
Insightful and informative reports from several anonymous referees are gratefully acknowledged. The comments of the participants in the T.S. Kim Memorial Seminar at Seoul National University have also contributed significantly to the paper; I am grateful to the seminar participants, and, in particular, to Youngsub Chun and Biung-Ghi Ju.
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Rachmilevitch, S. The Nash solution is more utilitarian than egalitarian. Theory Decis 79, 463–478 (2015). https://doi.org/10.1007/s11238-014-9477-5
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DOI: https://doi.org/10.1007/s11238-014-9477-5