  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 convexity of the leader's
cost and other sufficient conditions, we show that there exist symmetric
\emph{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. Leveraging functional analytic
tools, we further show that there exists a symmetric incentive strategy, which
is affine in the dynamic part of the leader's information, comprising partial
information on the actions taken by 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. For this class of problems, we establish the
existence of an incentive strategy and 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 the 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.
