Nash equilibria, variational inequalities, and dynamical systems (Q1862192)

In the paper, the authors investigate some relationships between the Nash equilibrium problem and the corresponding variational inequality problem, which can serve as an optimality condition. A gap function of a game, which is based on the known normal bifunction, is suggested. Under certain additional uniform monotonicity assumptions, a convergence result for the continuous variant of the Rosen type projection method is also established. However, there are several comments on this paper. Firstly, Theorem 3.3 is wrong since the usual (Stampacchia) variational inequality problem does not coincide with the dual (Minty) problem under the assumptions of this theorem. Moreover, the assertion at p.497, line 16 below also contradicts to this theorem. Secondly, it should be mentioned that the mapping \(\nabla J\) was introduced by \textit{J. B. Rosen} [Econometrica 33, 520-534 (1965; Zbl 0142.17603)] and that the normal bifunction was introduced by \textit{H.~Nikaido} and \textit{K.~Isoda} [Pac. J. Math. 5, 807-815 (1955; Zbl 0171.40903)].