A Numerical Construction Algorithm of Nash and Stackelberg Solutions for Two-person Non-zero Sum Linear Positional Differential Games

Abstract

The report evolves a method, which uses the formalization and results of positional antagonistic differential games theory, developed by N. N. Krasovskii and his scientific school, for constructing solutions of a class of non-antagonistic differential games. The method transforms 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. Numerical Nash solution construction algorithm based upon auxiliary bimatrix games sequence is presented. Used computational geometry algorithms include convex hull construction, union and intersection of polygons and a Minkowski sum for polygons. Results of numerical experiment for a material point motion in plane are presented. The point is moved by force formed by two players. Every 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.