  Entities in multi-agent systems may seek conflicting subobjectives, and this
leads to competition between them. To address performance degradation due to
competition, we consider a bi-level lottery where a social planner at the high
level selects a reward first and, sequentially, a set of players at the low
level jointly determine a Nash equilibrium given the reward. The social planner
is faced with efficiency losses where a Nash equilibrium of the lottery game
may not coincide with the social optimum. We propose an optimal bi-level
lottery design problem as finding the least reward and perturbations such that
the induced Nash equilibrium produces the socially optimal payoff. We formally
characterize the price of anarchy and the behavior of public goods and Nash
equilibrium with respect to the reward and perturbations. We relax the optimal
bi-level lottery design problem via a convex approximation and identify mild
sufficient conditions under which the approximation is exact.
