Optimum strategies for two large systems in a multistep game (Q1086178)

The contents and presentation (which may have been hampered by the translation into English somewhat) are poor. The only message of the paper is an algorithm which calculates the saddlepoint of a continuously differentiable, bounded and strictly convex (in u)-concave (in v) function w; here \(u\in {\mathbb{R}}^ n\), \(v\in {\mathbb{R}}^ m\). The algorithm calculates a sequence \((u^{(i)},v^{(i)})\), \(i=1,2,...\). Given \(v^{(i)}\) one determines \(u^{(i+1)}\) by minimizing \(w(u,v^{(i)})\) with respect to u and subsequently one determines \(v^{(i+1)}\) by maximizing \(w(u^{(i+1)},v)\) with respect to v. A proof of the convergence to the saddlepoint is given. Instead of u (v) the authors use the notation K(x) (K(y)), but the role of x respectively y is not clear to the reviewer; they are no state vectors, since state vectors for the two interacting systems are given by X and Y. Constraints on the controls are present; they are introduced as the ''effort distribution is finite'', i.e. bounded by one. The title of the paper and the last sentence of the introduction (''the aim of this work is the determination of an optimal control for operation of a large stochastic system in a multistep game problem of large dimension'') promise a lot. It is disappointing that the multistep character and the stochastic behaviour do not play any role in the paper. Also the largeness of the dimension is mentioned in a few places but not discussed at all.