# title
MINIMAX METHODS FOR OPEN-LOOP EQUILIBRIA IN N-PERSON DIFFERENTIAL-GAMES .1. LINEAR QUADRATIC GAMES AND CONSTRAINED GAMES

# abstract
<jats:title>Synopsis</jats:title><jats:p>In [13], Nikaido and Isoda generalised von Neumann's symmetrisation method for matrix games. They showed that <jats:italic>N</jats:italic>-person noncooperative games can be treated by a minimax method.</jats:p><jats:p>We apply this method to <jats:italic>N</jats:italic>-person differential games. Lukes and Russell [11] first studied <jats:italic>N</jats:italic>-person nonzero sum linear quadratic games in 1971. Here we have reproduced and strengthened their results. The existence and uniqueness of equilibria are completely determined by the <jats:italic>invertibility</jats:italic> of the decision operator, and the nonuniqueness of equilibrium strategies is only up to a finite dimensional subspace of the space of all admissible strategies.</jats:p><jats:p>In the constrained case, we have established an existence result for games with a much weaker convexity assumption subject to compact convex constraints. We have also derived certain results for games with noncompact constraints. Several examples of quadratic and non-quadratic games are given to illustrate the theorem.</jats:p><jats:p>Numerical computations are also possible and are given in the sequel [3].</jats:p>

# publicationVenue.label
Proceedings of the Royal Society of Edinburgh Section A Mathematics

# publicationOutlet
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS

# keywords[]
4903 Numerical And Computational Mathematics