Attackability in games of pursuit and evasion with antagonizing players

For the games of pursuit evasion with antagonizing players, the following three stages have been proposed: detection, attack and engagement, in which the roles of the two players are symmetric and each one is meant to search and attack its opponent. In the general framework established in our previous work, while the fundamental concepts, such as detectability, for the first stage have been laid before, this paper is dedicated to the second stage and to develop the associated fundamental concepts including the attackability, which describes whether one player (say $P-1$) could attack its opponent ($P-2$) before $P-2$ could see $P-1$, under the assumptions that (i) each player has a limited range vision zone and a limited range attack zone, and (ii) $P-2$ would follow a predefined trajectory and $P-1$ could choose its trajectory so as to attack $P-2$ since $P-1$ could see $P-2$ first. To demonstrate the concepts of attackability by detailed analysis, % as a preliminary work, a simple yet typical planar PEAP game is discussed in this contribution, where two players are moving along two straight lines with constant speeds and each player has circular vision zone and attack zone. Sufficient and necessary conditions for all possible cases of attackability are given under several natural assumptions, which yields a complete analysis for the new concepts of attackability.