Smooth Information Gathering in Two-Player Noncooperative Games

We present a mathematical framework for modeling two-player noncooperative games in which one player (the defender) is uncertain of the costs of the game and the second player's (the attacker's) intention but can preemptively allocate information-gathering resources to reduce this uncertainty. We obtain the defender's decisions by solving a two-stage problem. In Stage 1, the defender allocates information-gathering resources, and in Stage 2, the information-gathering resources output a signal that informs the defender about the costs of the game and the attacker's intent, and then both players play a noncooperative game. We provide a gradient-based algorithm to solve the two-stage game and apply this framework to a tower-defense game which can be interpreted as a variant of a Colonel Blotto game with smooth payoff functions and uncertainty over battlefield valuations. Finally, we analyze how optimal decisions shift with changes in information-gathering allocations and perturbations in the cost functions.