# Inducing Equilibria via Incentives: Simultaneous Design-and-Play Ensures Global Convergence

arXiv: https://arxiv.org/abs/2110.01212 · HTML: https://mlsanity.com/paper/2110.01212/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 53 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Understanding the impact of entropy on policy optimization](https://mlsanity.com/paper/1811.11214/) (2018)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Scalable Gradient-Based Tuning of Continuous Regularization Hyperparameters](https://mlsanity.com/paper/1511.06727/) (2015)
- [Understanding Short-Horizon Bias in Stochastic Meta-Optimization](https://mlsanity.com/paper/1803.02021/) (2018)
- [End-to-End Learning and Intervention in Games](https://mlsanity.com/paper/2010.13834/) (2020)
- [Learning to Defend by Learning to Attack](https://mlsanity.com/paper/1811.01213/) (2018)
- [Coordinating the Crowd: Inducing Desirable Equilibria in Non-Cooperative Systems](https://mlsanity.com/paper/1901.10923/) (2019)
- [Sensitivity analysis for network aggregative games](https://mlsanity.com/paper/1706.08693/) (2017)

## Built on by (most cited first, 8 of 8)

- [Differentiable Bilevel Programming for Stackelberg Congestion Games](https://mlsanity.com/paper/2209.07618/) (2022)
- [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)
- [Learning Optimal Tax Design in Nonatomic Congestion Games](https://mlsanity.com/paper/2402.07437/) (2024)
- [Steering No-Regret Agents in MFGs under Model Uncertainty](https://mlsanity.com/paper/2503.09309/) (2025)
- [A Soft Inducement Framework for Incentive-Aided Steering of No-Regret Players](https://mlsanity.com/paper/2508.21672/) (2025)
- [Follower Agnostic Methods for Stackelberg Games](https://mlsanity.com/paper/2302.01421/) (2023)
- [Barriers to Welfare Maximization with No-Regret Learning](https://mlsanity.com/paper/2411.01720/) (2024)

## Abstract (arXiv)

To regulate a social system comprised of self-interested agents, economic incentives are often required to induce a desirable outcome. This incentive design problem naturally possesses a bilevel structure, in which a designer modifies the rewards of the agents with incentives while anticipating the response of the agents, who play a non-cooperative game that converges to an equilibrium. The existing bilevel optimization algorithms raise a dilemma when applied to this problem: anticipating how incentives affect the agents at equilibrium requires solving the equilibrium problem repeatedly, which is computationally inefficient; bypassing the time-consuming step of equilibrium-finding can reduce the computational cost, but may lead the designer to a sub-optimal solution. To address such a dilemma, we propose a method that tackles the designer's and agents' problems simultaneously in a single loop. Specifically, at each iteration, both the designer and the agents only move one step. Nevertheless, we allow the designer to gradually learn the overall influence of the incentives on the agents, which guarantees optimality after convergence. The convergence rate of the proposed scheme is also established for a broad class of games.