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

Sina Sanjari, Subhonmesh Bose, Tamer Başar

TL;DR

This work studies stochastic incentive design in Stackelberg games with a single leader and many followers under dynamic, decentralized information. The authors develop an indirect, constructive approach that builds incentive strategies from near-optimal leader plans, establish symmetry and continuity properties, and analyze mean-field limits. Key results include existence of symmetric incentive equilibria for finite follower populations, ill-posedness of full mean-field limits for large populations, and a major–minor follower framework that yields well-defined mean-field incentives and finite-energy strategies, supported by quadratic Gaussian examples. The paper also proves existence of approximate randomized mean-field incentive equilibria and shows how mean-field solutions provide practical finite-N approximations. These findings advance scalable, smooth incentive design in large decentralized systems and have implications for policy design under uncertainty and dynamic information sharing.

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.

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

TL;DR

This work studies stochastic incentive design in Stackelberg games with a single leader and many followers under dynamic, decentralized information. The authors develop an indirect, constructive approach that builds incentive strategies from near-optimal leader plans, establish symmetry and continuity properties, and analyze mean-field limits. Key results include existence of symmetric incentive equilibria for finite follower populations, ill-posedness of full mean-field limits for large populations, and a major–minor follower framework that yields well-defined mean-field incentives and finite-energy strategies, supported by quadratic Gaussian examples. The paper also proves existence of approximate randomized mean-field incentive equilibria and shows how mean-field solutions provide practical finite-N approximations. These findings advance scalable, smooth incentive design in large decentralized systems and have implications for policy design under uncertainty and dynamic information sharing.

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.
Paper Structure (14 sections, 6 theorems, 59 equations, 1 figure)

This paper contains 14 sections, 6 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Contributions.
  3. A Symmetric Stochastic Stackelberg Game with One leader and A Finite Number of Followers $\mathcal{P}_N$.
  4. Characterization of Incentive Strategies for Stochastic Symmetric Stackelberg Games.
  5. Ill-posedness of Incentive Strategies in the Large Population Limit.
  6. A Hierarchical Incentive Stackelberg Game with a Major Follower.
  7. A QG Game Example of ${\mathcal{P}_N^{\textrm{maj}}}$.
  8. The Mean-Field Limit for ${\mathcal{P}_N^{\textrm{maj}}}$.
  9. Existence of an Approximate Randomized Mean-Field Incentive Equilibrium.
  10. Conclusions and Future Research Directions
  11. Proof of Theorem \ref{['the:symmetric']}.
  12. Proof of Theorem \ref{['the:Ma-Mi']}.
  13. Proof of Proposition \ref{['lem:exi-1']}.
  14. Proof of Theorem \ref{['the:Existence']}.

Key Result

Theorem 1

Consider $\mathcal{P}_{N}$ under the leader's information structure $I_{{{{\sf OCS}}}}^{0}$ or $I_{{{{\sf OPCS}}}}^{0}$.

Figures (1)

  • Figure 1: The leader's performance under leader-major-optimal and leader-optimal solutions.

Theorems & Definitions (16)

  • Definition 1: $\epsilon$-Stackelberg Equilibrium
  • Remark
  • Definition 2: $\epsilon_{0}$-Leader-Optimality
  • Definition 3: $\epsilon$-Incentive Strategy
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Definition 4: $\epsilon$-Stackelberg Equilibrium
  • Definition 5: $\epsilon$-leader-major optimality
  • Definition 6: $\epsilon$-Incentive Strategy
  • ...and 6 more