Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Follower Agnostic Methods for Stackelberg Games

Chinmay Maheshwari, James Cheng, S. Shankar Sasty, Lillian Ratliff, Eric Mazumdar

TL;DR

An efficient algorithm to solve online Stackelberg games, featuring multiple followers, in a follower-agnostic manner, using a unique gradient estimator, leveraging specially designed strategies to probe followers.

Abstract

In this paper, we present an efficient algorithm to solve online Stackelberg games, featuring multiple followers, in a follower-agnostic manner. Unlike previous works, our approach works even when leader has no knowledge about the followers' utility functions or strategy space. Our algorithm introduces a unique gradient estimator, leveraging specially designed strategies to probe followers. In a departure from traditional assumptions of optimal play, we model followers' responses using a convergent adaptation rule, allowing for realistic and dynamic interactions. The leader constructs the gradient estimator solely based on observations of followers' actions. We provide both non-asymptotic convergence rates to stationary points of the leader's objective and demonstrate asymptotic convergence to a \emph{local Stackelberg equilibrium}. To validate the effectiveness of our algorithm, we use this algorithm to solve the problem of incentive design on a large-scale transportation network, showcasing its robustness even when the leader lacks access to followers' demand.

Follower Agnostic Methods for Stackelberg Games

TL;DR

An efficient algorithm to solve online Stackelberg games, featuring multiple followers, in a follower-agnostic manner, using a unique gradient estimator, leveraging specially designed strategies to probe followers.

Abstract

In this paper, we present an efficient algorithm to solve online Stackelberg games, featuring multiple followers, in a follower-agnostic manner. Unlike previous works, our approach works even when leader has no knowledge about the followers' utility functions or strategy space. Our algorithm introduces a unique gradient estimator, leveraging specially designed strategies to probe followers. In a departure from traditional assumptions of optimal play, we model followers' responses using a convergent adaptation rule, allowing for realistic and dynamic interactions. The leader constructs the gradient estimator solely based on observations of followers' actions. We provide both non-asymptotic convergence rates to stationary points of the leader's objective and demonstrate asymptotic convergence to a \emph{local Stackelberg equilibrium}. To validate the effectiveness of our algorithm, we use this algorithm to solve the problem of incentive design on a large-scale transportation network, showcasing its robustness even when the leader lacks access to followers' demand.
Paper Structure (20 sections, 13 theorems, 78 equations, 2 figures, 1 algorithm)

This paper contains 20 sections, 13 theorems, 78 equations, 2 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Related works
  3. PROBLEM FORMULATION
  4. MOTIVATING EXAMPLE: INCENTIVE DESIGN IN ROUTING GAMES
  5. ALGORITHM AND ANALYSIS
  6. Algorithmic structure
  7. Leader's strategy update
  8. Gradient estimator
  9. Algorithm
  10. Convergence to stationary points
  11. Proof of Theorem \ref{['thm: ConvergenceStationary']}
  12. Non-convergence to saddle points
  13. Proof of Theorem \ref{['thm: NonconvergenceSaddle']}
  14. NUMERICAL EXPERIMENTS
  15. CONCLUSIONS
  16. ...and 5 more sections

Key Result

Theorem 1

Let Assumption assm: BasicAssumptionSetup-assm: FollowerUpdatesConvergence hold. If we choose $\eta_t = \bar{\eta} (t+1)^{-1/2}d^{-1}, \delta_t = \bar{\delta} (t+1)^{-1/4}d^{-1/2}$ such that $\bar{\eta}\leq d/2\tilde{\ell}$. Then the updates $(x_t)_{t\in[T]}$ in Algorithm alg: ZerothOrderTwoPointAlg where $\alpha = CK^{-\lambda}$ if Assumption assm: FollowerUpdatesConvergence(1a) hold, or $\alpha

Figures (2)

  • Figure 1: Schematic depiction of Sioux Falls transportation network network. The numbers on the edges and nodes are identifiers.
  • Figure 2: The evolution of planners objective function with iterates of the algorithm. The shaded blue region denotes the confidence interval calculated over 12 runs.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 4
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • ...and 7 more