Abstract
This paper studies the Nash solution to non-convex bargaining problems. Given the multiplicity of the Nash solution in this context, we refine the Nash solution by incorporating an equity consideration. The proposed refinement is defined as the composition of the Nash solution and a variant of the Kalai–Smorodinsky solution. We then present an axiomatic characterization of the new solution.
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Notes
Note that Conley and Wilkie (1996) propose an “equitable” Nash extension solution to nonconvex problems, which is taken as a hybrid of the Nash and the KS solutions. However, their solution is very different from ours proposed in this paper as their solution does not satisfy the efficiency criterion and is not a refinement of the Nash solution.
The KS solution can be defined as usual: for all \(A\in \Sigma \), \( F^{KS}(A)=\{x\in A\mid x_{1}/m_{1}(A)=\cdots =x_{n}/m_{n}(A),\) and there exists no \(y\in A\) such that \(y\gg x\}\). It may be noted that the solution given by \(F(A)=\{a\in A\mid \min (a_{1}/m_{1}(A),\cdots ,a_{n}/m_{n}(A))\ge \min (x_{1}/m_{1}(A),\cdots ,x_{n}/m_{n}(A)),\forall x\in A\}\) for all \(A\in \Sigma \) is a natural extension of the KS solution to nonconvex problems, as proposed by Nagahisa and Tanaka (2002). The reason that we use this variant of the KS solution (instead of the KS solution itself) is that \(F^{N}(A)\cap F^{KS}(A)=\varnothing \) for some \(A\in \Sigma \).
We are grateful to the Associate Editor for pointing this out and for suggesting the example below.
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An earlier version of the paper was presented at the SEA meetings in Atlanta, Georgia, November 2010 and at the CEPET meeting in Udine, Italy, June 2011. We are grateful to M.A. Ballester, Youngsub Chun, Marco Mariotti, Hans Peters, Koichi Tadenuma, and William Thomson for helpful and encouraging comments. We are also grateful to the Associate Editor and the referee for very helpful comments on an earlier version of the paper.
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Xu, Y., Yoshihara, N. An equitable Nash solution to nonconvex bargaining problems. Int J Game Theory 48, 769–779 (2019). https://doi.org/10.1007/s00182-019-00658-4
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DOI: https://doi.org/10.1007/s00182-019-00658-4
Keywords
- Non-convex bargaining problem
- Nash solution
- Equitable Nash solution
- Equity principle
- Binary weak axiom of revealed preference