在沒有被捕獲懲罰的情況下，我們發現最佳策略涉及根據所使用的攻擊者模型，最小化預期的到達時間或返回時間。此外，我們發現，在高懲罰情況下增加巡邏日程的隨機性顯著降低了攻擊者的預期收益。為了應對一般情況下呈現的挑戰，我們定義了一個雙標準優化問題，並比較了四種算法，旨在平衡最大化預期獎勵和增加巡邏日程隨機性之間的權衡。;We explore an extended version of the zero-sum patrolling security game where the attacker has the flexibility to decide the timing, location, and duration of their attack, examined under three distinct attacker models. In this game, the attacker′s payoff is determined by the utilities gained from the attack minus any penalties incurred if caught by the patrolling defender. Our primary objective is to minimize the attacker′s overall payoff. To achieve this, we transform the game into a combinatorial minimax problem with a clearly defined objective function.
In cases where there is no penalty for getting caught, we establish that the optimal strategy involves minimizing either the expected hitting time or return time, contingent on the attacker model employed. Furthermore, we find that enhancing the randomness of the patrol schedule significantly reduces the attacker′s expected payoff in scenarios involving high penalties. To address the challenges presented in general scenarios, we developed a bi-criteria optimization problem and compare four algorithms designed to balance the trade-off between maximizing expected rewards and increasing randomness in patrol scheduling.
