Proposing a new security game with reward and penalty

We propose a new security game that takes into account both rewards and penalties. Security games have been widely studied in the field of multi-agent systems for many years. We abstract the security game that targets a situation where the defender must leave one vault open for a transaction as a number guessing game with reward and penalty. We investigate this security game in both simultaneous and sequential forms. We analyze the attacker's Max-Min strategy and the defender's Min-Max strategy in these games. Furthermore, we clarify the sufficient conditions for the penalty in which the defender chooses all remaining numbers. If the penalty increment is large, i.e., if the defender also incurs sufficient cost to defend the later safes, both the attacker and the defender must consider the possibility of using all the safes, but the defender can minimize the expected value of the loss by a simple strategy. On the other hand, if the penalty increment is small, the number of safes to be used is small, and thus the amount of loss may be improved, but the defender's strategy will not be as simple. The defender's strategy is formulated as a linear programming problem in this paper. We show what strategies the defender chooses as the penalty increases in the proposed security game for both simultaneous and sequential forms through computational experiments.