Incentive Designs for Stackelberg Games with a Large Number of Followers and their Mean-Field Limits

We study incentive designs for a class of stochastic Stackelberg games with one leader and a large number of (finite as well as infinite population of) followers. We investigate whether the leader can craft a strategy under a dynamic information structure that induces a desired behavior among the followers. For the finite population setting, under sufficient conditions, we show that there exist symmetric incentive strategies for the leader that attain approximately optimal performance from the leader's viewpoint and lead to an approximate symmetric (pure) Nash best response among the followers. Driving the follower population to infinity, we arrive at the interesting result that in this infinite-population regime the leader cannot design a smooth "finite-energy" incentive strategy, namely, a mean-field limit for such games is not well-defined. As a way around this, we introduce a class of stochastic Stackelberg games with a leader, a major follower, and a finite or infinite population of minor followers, where the leader provides an incentive only for the major follower, who in turn influences the rest of the followers through her strategy. For this class of problems, we are able to establish the existence of an incentive strategy with finitely many minor followers. We also show that if the leader's strategy with finitely many minor followers converges as their population size grows, then the limit defines an incentive strategy for the corresponding mean-field Stackelberg game. Examples of quadratic Gaussian games are provided to illustrate both positive and negative results. In addition, as a byproduct of our analysis, we establish existence of a randomized incentive strategy for the class mean-field Stackelberg games, which in turn provides an approximation for an incentive strategy of the corresponding finite population Stackelberg game.