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.

翻译：本文提出一种数学框架，用于建模两玩家非合作博弈场景，其中一方玩家（防御者）对博弈成本及另一方玩家（攻击者）的意图存在不确定性，但可通过预先分配信息获取资源来降低这种不确定性。通过求解两阶段问题获得防御者的决策：第一阶段，防御者分配信息获取资源；第二阶段，信息获取资源输出信号，向防御者揭示博弈成本与攻击者意图，随后双方进行非合作博弈。我们提出一种基于梯度的算法求解该两阶段博弈，并将该框架应用于塔防游戏（可视为具有平滑收益函数与战场估值不确定性的Colonel Blotto博弈变体）。最后，我们分析最优决策如何随信息获取分配变化及成本函数扰动发生转移。