On a finite state space pursuit evasion game with dynamic information

The game to be discussed belongs to the class of two-person zero-sum dynamic games. The players move in a plane and are unaware of each other´s positions unless a player flashes a search light which illuminates an area of known shape around this player. By flashing his search light a player discloses his position to the other player, wherever he is. Termination occurs when this other player finds himself trapped within the area illuminated by the search light. Both players want to catch the other player before they are caught themselves. This problem has been investigated for a finite state space (the players move on a network with a finite number of nodes) and for the discrete-time case. Both a normal and an extensive form description of the game can be given; they are equivalent and the former one is more amenable for obtaining numerical results. In essence, finding the saddle-point strategies boils down to the solution of a two-level linear programming problem. Explicit results have been obtained when the players are confined to the circumference of a circle and during each time step each player can either move to one of the neighbouring nodes or stay where he is. Specific results depend on the number of nodes and the shape of the illluminated area. If the total number of flashes (available to both players) does not exceed two, then the class of mixed and behavioral strategies coincide.