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：暂无翻译