Multi-defender capture-the-flag game with convex target regions

This study addresses the capture-the-flag game with general convex target regions and multiple defenders, a special type of reach-avoid games with two target regions wherein an attacker aims to first reach the flag region and next the return region without being captured by a team of defenders. The game process can be divided into two phases: before the attacker reaches the flag region and after. We handle the two game phases in a recursive manner to characterize the value of the game through optimization problems. We establish that the value of the second game phase equals the optimal value of a convex optimization problem, and the value of the overall capture-the-flag game equals the optimal value of a quasi-convex optimization problem. We demonstrate that the quasi-convex optimization problem has a unique minimum point when the value of the capture-the-flag game is larger than that of the second game phase. Furthermore, we prove that the maximum number of defenders ensuring the game value is two, which is a useful result for assigning multiple defenders to capture multiple attackers. Finally, we propose an iterative algorithm to solve the quasi-convex optimization problem.