Detectability in games of pursuit evasion with antagonizing players

In reality, the games of pursuit and evasion actually involve antagonizing players, rather than the commonly studied pursuit and evasion games with pre-fixed pursuers or evaders. In these practical scenarios, the game process can normally be divided into three stages: detection, attack and engagement stages. In such games, the roles of players are symmetric and each one is to search and attack the other one. This key point distinguishes the games of pursuit and evasion with antagonizing players from most existing pursuit-evasion games. In this paper, a basic framwork for such games is established first by introducing several essential concepts including vision zone, range of view, field of view, etc. Then, a fundamental and important concept, detectability, is presented under the assumptions that (i) each player has a limited range vision zone and (ii) the players would follow their own predefined trajectories until one player could detect its opponent. At the detection stage, to demonstrate concepts of detectability and related concepts, a simple yet typical case is investigated in this contribution, where two players are moving along two straight lines with constant speeds and each player has a circular vision zone. Sufficient and necessary conditions for all possible cases of detectability are given under several natural assumptions, which consequently yields a complete analysis.