A Geometric Tool for Two-Phase Multiplayer Reach-Avoid Games: Ellipses

This paper focuses on the utilization of geometric tools in two-phase multiplayer reach-avoid games. In these games, a thief aims to enter a target region and subsequently reach a safe region while evading capture by guarders. We first obtain the solution to the game of kind in the game scenario where the safe region is a single point by using ellipses. Then we show that ellipses are also useful to solve the game of kind when two guarders cooperatively play against the thief. Furthermore, we study the dominance region for two-phase games with the help of ellipses. The construction method of the boundary of dominance regions is provided and illustrated with a numerical example.