A General Lotto game with asymmetric budget uncertainty

We consider General Lotto games of asymmetric information where one player's resource endowment is randomly assigned one of two possible values, and the assignment is not revealed to the opponent. We completely characterize the Bayes-Nash equilibria for two such formulations -- namely, one in which the opponent's endowment is fixed and common knowledge, and another where the opponent has a per-unit cost to utilize resources. We then highlight the impact these characterizations have on resource allocation problems involving a central commander that decides how to assign available resources to two sub-colonels competing in separate Lotto games against respective opponents. We find that randomized assignments, which induce the Bayesian game interactions, do not offer strategic advantages over deterministic ones when the opponents have fixed resource endowments. However, this is not the case when the opponents have per-unit costs to utilize resources. We find the optimal randomized assignment strategy can actually improve the commander's payoff two-fold when compared to optimal deterministic assignments, and four-fold in settings where the commander also pays a per-unit cost for resources.