Games for the optimal deployment of security forces

Abstract

CHAPTER 1 is already known. Therefore, the game value for Stackelberg equilibria can be different from the game values for Nash equilibria. However, in this thesis, we mainly consider zero-sum games (in which the gain for one player equals the loss for the other player) and for these games, the game value and agent's strategy coincide [133]. In the following example, we give a basic security game to explain the different game elements. Example 1.1 (Basic security game). Consider a patrolling game on a part of the North Sea which can be divided into two areas A and B. Since this is a protected area, it is not allowed to fish in A and B. However, in both areas, there is a lot of fish available, so these are popular places for fishermen (intruder) to fish illegally. To prevent illegal fishing in both areas, the coast guard (agent) has one patrolling ship available. This ship is able to patrol and protect one area from illegal fishing each day. At the beginning of a day, the intruder chooses one area to fish. When the intruder fishes successfully, i.e., without being caught by the agent, he obtains a gain. Assume that fishing in B is better than in A: the gain of successfully fishing in area A equals 3 and for area B, the fisherman's gain equals 5. The gain for the intruder equals the loss for the agent if he is patrolling the area where the intruder is not fishing. However, when the agent is patrolling the area where the intruder is fishing, the intruder is caught and the gain for the agent equals 1 (which is also the loss for the intruder). The game above can be described in a matrix displaying the actions and payoffs for all players. In this game, the matrix is given by: