## A two-person game in mixed strategies as a model of training.(Russian, English)Zbl 1088.91002

Zh. Vychisl. Mat. Mat. Fiz. 45, No. 9, 1566-1574 (2005); translation in Comput. Math. Math. Phys. 45, No. 9, 1511-1519 (2005).
An education process is modeled by static and dynamic components of a bimatrix two-person game with mixed strategies $x* \in \text{Argmax}{ \langle S^Tx, y*\rangle + (1/2)\langle Bx, x\rangle | \langle e, x\rangle = 1, x \geq 0},$
$y* \in \text{Argmax}{\langle x*, Py\rangle + (1/2)\langle Dy, y\rangle | \langle e, y\rangle = 1, y \geq 0},$ where $$S, P, B, D$$ are matrices, $$B \leq 0, D \leq 0$$. The target functions present a sum of the bi-linear and quadratic functions. A solution is found by an extra-approximate method, which is based upon incompressibility of the game operator due to regularization and splitting of the proximal step. A splitting procedure affords to reduce the original problem to another semi-definite negative one. The convergence of the solution to the Nash equilibrium is proved.

### MSC:

 91A05 2-person games 97A90 Fiction and games (MSC2000)

### Keywords:

Nash equilibrium; convergence; education process