Discrete games of infiltration

Abstract

Gal has suggested the following scenario for a two-person game. An Infiltrator starts at the first of an ordered set of p points. At discrete intervals of time t = 1,2,... he chooses to move to one of the adjacent points or to stay where he is. A Guard starts from any point and at each of the same intervals of time moves to a point up to u points away. He then searches for the Infiltrator, detecting him with probability μ if the players are at the same point, and with probability zero otherwise. Neither player is aware of his opponent's moves unless detection occurs. At the last of the p points the Infiltrator is safe from detection. We look at a number of zero-sum games which are based on this scenario. These include both infinite move games and games in which the number of moves is restricted by a time limit of n. The objective of the Infiltrator is either to reach the last point undetected, or just to evade the Guard. We show that the infinite move games have mixed strategy solutions which can be constructed from solutions to finite move games. In addition, we study a further set of games in which the Infiltrator is also safe from detection at the first point. His objective then is to reach the last point undetected. In this context we extend the work of Lalley to a more general set of points. Some examples of optimal strategies are presented. Finally we discuss some possible generalisations to other discrete infiltration games.

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