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

TL;DR

This paper investigates incentive strategies in stochastic Stackelberg games with many followers, revealing limitations in mean-field approaches and proposing solutions for games with a major follower and many minor followers.

Contribution

It introduces a new class of Stackelberg games with a major follower, establishing the existence of incentive strategies and mean-field limits, and analyzes the limitations of smooth incentive strategies in infinite populations.

Findings

Symmetric incentive strategies can achieve near-optimal leader performance in finite populations.

Mean-field limits for incentive strategies are not well-defined in infinite-population regimes.

Existence of randomized incentive strategies provides practical approximations for finite games.

Abstract

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.