  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.
