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.