Abstract
We provide two alternative characterizations of the Nash bargaining solution. We introduce new simple axioms, strong undominatedness by the disagreement point, and egalitarian Pareto optimality. First, we prove that the Nash solution is characterized by symmetry, scale invariance, independence of irrelevant alternatives, and strong undominatedness by the disagreement point. Second, we replace the independence of irrelevant alternatives axiom with the sandwich axiom (Rachmilevitch in Theory Decis 80:427–442, 2016) and egalitarian Pareto optimality. We then demonstrate that the Nash solution is characterized by symmetry, scale invariance, strong undominatedness by the disagreement point, the sandwich axiom, and egalitarian Pareto optimality.
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Notes
See, for example Mariotti (2000).
Although we deal with the two-person case throughout, all arguments in this paper work in the more-than-two-person case.
For \(s,r\in \mathbb {R}^{2}, s \ge r \) if \(s_{i }\ge r_{i}\) for each i , \(s > r\) if \(s_{i }\ge r_{i}\) for each i and \(s \ne r\), and \(s\gg r\) if \(s_{i }>r_{i}\) for each i.
We say that x is weakly dominated by y if \(x< y\) for \(x,y\in \mathbb {R}^{2}\).
I appreciate an anonymous reviewer’s comment.
Lemma 1 is analogous to Lemma 2 in Vartiainen (2007), which has been provided to characterize the extended Nash solution determining a solution and a reference point simultaneously.
If there was a point \(r \in \mathbb {R}^{2}\), such that \(r_{1}+r_{2 }>2\), then the convex combination of r and N(S, 0) contains a point \(u \in S\), such that \(u_{1}u_{2 }>\)1. That is a contradiction. See Nash (1950).
See Kalai and Smorodinsky (1975).
Rachmilevitch (2016) has provided a characterization of the Nash solution using the sandwich axiom (Theorem 1).
The interpretation is based on Rachmilevitch (2016).
References
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Acknowledgements
I am deeply grateful for the helpful comments of two anonymous reviewers and the editor. In addition, I would like to thank Mr. Jerre Bush and Enago for the English language review. I acknowledge support from Jobu University.
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Mori, O. Two simple characterizations of the Nash bargaining solution. Theory Decis 85, 225–232 (2018). https://doi.org/10.1007/s11238-017-9624-x
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DOI: https://doi.org/10.1007/s11238-017-9624-x