Ship differential game approach for multiple players: Stackelberg security games

Casting the problem as a pursuit-evasion behavior in continuous time, this paper addresses a class of multiplayer ship differential Stackelberg security game. We are concerned with the conditions under which the defenders can capture the attackers. We represent the Stackelberg game as a Nash game for relaxing the interpretation of the noncooperative solution and the equilibrium selection problem. The weights of the players for the Nash solution are determined by their role in the Stackelberg game. The defenders try to minimize the capture condition. The attackers, knowing that they are being pursued by defenders, try to maximize the capture condition and minimize the distance to a certain target. For computing the equilibrium of the game, we employ a saddle-point method approach. The method consists of two half-steps iterated procedure where the functional of the game decrease and finally converges to an equilibrium point. We present the analysis of the convergence. Finally, we give a numerical example to illustrate the effectiveness and usefulness of our approach.