# Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games

arXiv: https://arxiv.org/abs/1909.04694 · HTML: https://mlsanity.com/paper/1909.04694/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2019.

## Built on by (most cited first, 30 of 37)

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- [Approximating Discontinuous Nash Equilibrial Values of Two-Player General-Sum Differential Games](https://mlsanity.com/paper/2207.01773/) (2022)
- [SOCIALGYM 2.0: Simulator for Multi-Agent Social Robot Navigation in Shared Human Spaces](https://mlsanity.com/paper/2303.05584/) (2023)
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- [On a Connection between Differential Games, Optimal Control, and Energy-based Models for Multi-Agent Interactions](https://mlsanity.com/paper/2308.16539/) (2023)
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- [Dynamically Switching Human Prediction Models for Efficient Planning](https://mlsanity.com/paper/2103.07815/) (2021)
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- [Self-Reflective Terrain-Aware Robot Adaptation for Consistent Off-Road Ground Navigation](https://mlsanity.com/paper/2111.06742/) (2021)
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- [How Bad is Selfish Driving? Bounding the Inefficiency of Equilibria in Urban Driving Games](https://mlsanity.com/paper/2210.13064/) (2022)
- [Who Plays First? Optimizing the Order of Play in Stackelberg Games with Many Robots](https://mlsanity.com/paper/2402.09246/) (2024)
- [Constraint Learning in Multi-Agent Dynamic Games from Demonstrations of Local Nash Interactions](https://mlsanity.com/paper/2508.19945/) (2025)
- [LSRE: Latent Semantic Rule Encoding for Real-Time Semantic Risk Detection in Autonomous Driving](https://mlsanity.com/paper/2512.24712/) (2025)
- [An NCAP-like Safety Indicator for Self-Driving Cars](https://mlsanity.com/paper/2104.00859/) (2021)
- [NashFormer: Leveraging Local Nash Equilibria for Semantically Diverse Trajectory Prediction](https://mlsanity.com/paper/2305.17600/) (2023)
- [Learning Nash Equilibrial Hamiltonian for Two-Player Collision-Avoiding Interactions](https://mlsanity.com/paper/2503.07013/) (2025)
- [PACE: A Framework for Learning and Control in Linear Incomplete-Information Differential Games](https://mlsanity.com/paper/2504.17128/) (2025)

## Abstract (arXiv)

Many problems in robotics involve multiple decision making agents. To operate efficiently in such settings, a robot must reason about the impact of its decisions on the behavior of other agents. Differential games offer an expressive theoretical framework for formulating these types of multi-agent problems. Unfortunately, most numerical solution techniques scale poorly with state dimension and are rarely used in real-time applications. For this reason, it is common to predict the future decisions of other agents and solve the resulting decoupled, i.e., single-agent, optimal control problem. This decoupling neglects the underlying interactive nature of the problem; however, efficient solution techniques do exist for broad classes of optimal control problems. We take inspiration from one such technique, the iterative linear-quadratic regulator (ILQR), which solves repeated approximations with linear dynamics and quadratic costs. Similarly, our proposed algorithm solves repeated linear-quadratic games. We experimentally benchmark our algorithm in several examples with a variety of initial conditions and show that the resulting strategies exhibit complex interactive behavior. Our results indicate that our algorithm converges reliably and runs in real-time. In a three-player, 14-state simulated intersection problem, our algorithm initially converges in < 0.25s. Receding horizon invocations converge in < 50 ms in a hardware collision-avoidance test.