Nash Smoothing on the Test Bench: $H_{\alpha}$ -Essential Equilibria

We extend the analysis of van Damme (1987, Section 7.5) of the famous smoothing demand in Nash (1953) as an argument for the singular stability of the symmetric Nash bargaining solution among all Pareto efficient equilibria of the Nash demand game. Van Damme's analysis provides a clean mathematical framework where he substantiates Nash's conjecture by two fundamental theorems in which he proves that the Nash solution is among all Nash equilibria of the Nash demand game the only one that is *H*-essential. We show by generalizing this analysis that for any asymmetric Nash bargaining solution a similar stability property can be established that we call $H_{\alpha}$-essentiality. A special case of our result for α = 1/2 is $H_{1/2}$-essentiality that coincides with van Damme's *H*-essentiality. Our analysis deprives the symmetric Nash solution equilibrium of Nash's demand game of its exposed position and fortifies our conviction that, in contrast to the predominant view in the related literature, the only structural diffeerence between the asymmetric Nash solutions and the symmetric one is that the latter one is symmetric. While our proofs are mathematically straightforward given the analysis of van Damme (1987), our results change drastically the prevalent interpretation of Nash's smoothing of his demand game and dilute its conceptual importance.