# Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

arXiv: https://arxiv.org/abs/2106.03278 · HTML: https://mlsanity.com/paper/2106.03278/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2021.

## Built on by (most cited first, 10 of 10)

- [Oracles & Followers: Stackelberg Equilibria in Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2210.11942/) (2022)
- [Learning to Mitigate AI Collusion on Economic Platforms](https://mlsanity.com/paper/2202.07106/) (2022)
- [First-Order Methods for Linearly Constrained Bilevel Optimization](https://mlsanity.com/paper/2406.12771/) (2024)
- [Multi-Sender Persuasion: A Computational Perspective](https://mlsanity.com/paper/2402.04971/) (2024)
- [Social Environment Design](https://mlsanity.com/paper/2402.14090/) (2024)
- [Learning to Steer Markovian Agents under Model Uncertainty](https://mlsanity.com/paper/2407.10207/) (2024)
- [Deep Incentive Design with Differentiable Equilibrium Blocks](https://mlsanity.com/paper/2603.07705/) (2026)
- [Differentiable Arbitrating in Zero-sum Markov Games](https://mlsanity.com/paper/2302.10058/) (2023)
- [Steering No-Regret Agents in MFGs under Model Uncertainty](https://mlsanity.com/paper/2503.09309/) (2025)
- [The Complexity of Optimizing Atomic Congestion](https://mlsanity.com/paper/2312.10219/) (2023)

## Abstract (arXiv)

A growing body of work in game theory extends the traditional Stackelberg game to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria in these games reformulate the followers' best response as constraints in the leader's optimization problem. These reformulation approaches can sometimes be effective, but often get trapped in low-quality solutions when followers' objectives are non-linear or non-quadratic. Moreover, these approaches assume a unique equilibrium or a specific equilibrium concept, e.g., optimistic or pessimistic, which is a limiting assumption in many situations. To overcome these limitations, we propose a stochastic gradient descent--based approach, where the leader's strategy is updated by differentiating through the followers' best responses. We frame the leader's optimization as a learning problem against followers' equilibrium, which allows us to decouple the followers' equilibrium constraints from the leader's problem. This approach also addresses cases with multiple equilibria and arbitrary equilibrium selection procedures by back-propagating through a sampled Nash equilibrium. To this end, this paper introduces a novel concept called equilibrium flow to formally characterize the set of equilibrium selection processes where the gradient with respect to a sampled equilibrium is an unbiased estimate of the true gradient. We evaluate our approach experimentally against existing baselines in three Stackelberg problems with multiple followers and find that in each case, our approach is able to achieve higher utility for the leader.