Polymatrix games and optimization problems

Consideration was given to the properties of the polymatrix game, a finite noncooperative game of $N$ players ($N\geqslant3$). A theorem of reduction of the search for Nash equilibria to an optimization problem was proved. This clears the way to the numerical search of equilibria. Additionally, a simple proof of the Nash theorem of existence of equilibrium in the polymatrix game was given using the theorem of existence of solution in the optimization problem.