A numerical construction algorithm of nash and stackelberg solution for two-person non-zero sum linear positional differential games

Abstract

The report proposes a numerical method of Stackelberg and Nash solutions construction in a class of differential games. It is based upon results of the positional antagonistic differential games theory developed by N. N. Krasovskii and his scientific school. The method transforms a non-antagonistic game into so-called non-standard optimal control problem. Numerical solutions for Stackelberg games are constructed by an algorithm developed by S. Osipov. For Nash solution construction we build auxiliary bimatrix games sequence. Both algorithms make use of known antagonistic game value computation procedures and are ultimately based upon computational geometry algorithms including convex hull construction, union, intersection, and Min-kowski sum of flat polygons. Results of numerical experiment for a material point motion in plane are presented. The point is moved by force formed by two players. Each player has his personal target point. Among the obtained results, there is a Nash solution such, that along the corresponding trajectory the position of the game is non-antagonistic at first, and then becomes globally antagonistic starting from some moment of time. © Springer Science+Business Media B.V. 2010.