The Capture-the-Flag Differential Game: Attack, Transition and Retreat

This paper analyzes the classic game of capture-the-flag, modeled as a conflict between an Attacker and a Defender. The game unfolds in distinct phases with changing objectives: first, the Attacker tries to capture a flag while the Defender attempts to intercept; second, if successful, the Attacker tries to reach a safe zone while the Defender again seeks interception. We mathematically derive the optimal state-feedback strategies for both players and the associated Value function for each phase, rigorously proving their correctness. A key contribution is introducing the transition phase, where we analyze the Defender’s optimal repositioning strategy when flag capture becomes inevitable, preparing it for the game’s second phase. This novel transition connects the game’s stages, critically enabling us to solve the overall Game of Kind – determining the winner from any starting condition – and define the precise circumstances under which the Attacker can both capture the flag and successfully escape to the safe zone.