We present a mathematical framework for modeling two-player noncooperative games in which one player is uncertain of the other player's costs but can preemptively allocate information-gathering resources to reduce this uncertainty. We refer to the players as the uncertain player (UP) and the certain player (CP), respectively. We obtain UP's decisions by solving a two-stage problem where, in Stage 1, UP allocates information-gathering resources that smoothly transform the information structure in the second stage. Then, in Stage 2, a signal (that is, a function of the Stage 1 allocation) informs UP about CP's costs, and both players execute strategies which depend upon the signal's value. This framework allows for a smooth resource allocation, in contrast to existing literature on the topic. We also identify conditions under which the gradient of UP's overall cost with respect to the information-gathering resources is well-defined. Then we provide a gradient-based algorithm to solve the two-stage game. Finally, we apply our 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. We include an analysis of how optimal decisions shift with changes in information-gathering allocations and perturbations in the cost functions.

翻译：本文提出一个数学框架，用于建模双人非合作博弈，其中一名参与者对另一名参与者的成本不确定，但可以预先分配信息收集资源以降低这种不确定性。我们分别将两名参与者称为不确定参与者（UP）与确定参与者（CP）。我们通过求解一个两阶段问题来获得UP的决策：在第一阶段，UP分配信息收集资源，这些资源平滑地改变了第二阶段的信息结构；随后在第二阶段，一个信号（即第一阶段分配的函数）向UP告知CP的成本，双方参与者根据该信号值执行策略。与现有文献相比，此框架允许平滑的资源分配。我们还确定了UP总成本相对于信息收集资源的梯度有定义的条件，并给出一种基于梯度的算法来求解该两阶段博弈。最后，我们将框架应用于一个塔防游戏，该游戏可解释为具有平滑支付函数且战场估值不确定的Colonel Blotto博弈变体。我们分析了最优决策如何随信息收集分配的变化以及成本函数的扰动而改变。