# On a Connection between Differential Games, Optimal Control, and Energy-based Models for Multi-Agent Interactions

arXiv: https://arxiv.org/abs/2308.16539 · HTML: https://mlsanity.com/paper/2308.16539/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 77 references that are in the map.
- First posted on arXiv 31 Aug 2023.

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Social GAN: Socially Acceptable Trajectories with Generative Adversarial Networks](https://mlsanity.com/paper/1803.10892/) (2018)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Large Scale Interactive Motion Forecasting for Autonomous Driving : The Waymo Open Motion Dataset](https://mlsanity.com/paper/2104.10133/) (2021)
- [VectorNet: Encoding HD Maps and Agent Dynamics from Vectorized Representation](https://mlsanity.com/paper/2005.04259/) (2020)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [TNT: Target-driveN Trajectory Prediction](https://mlsanity.com/paper/2008.08294/) (2020)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [MultiPath++: Efficient Information Fusion and Trajectory Aggregation for Behavior Prediction](https://mlsanity.com/paper/2111.14973/) (2021)
- [Convolutional Social Pooling for Vehicle Trajectory Prediction](https://mlsanity.com/paper/1805.06771/) (2018)
- [Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion](https://mlsanity.com/paper/2203.13777/) (2022)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [PRECOG: PREdiction Conditioned On Goals in Visual Multi-Agent Settings](https://mlsanity.com/paper/1905.01296/) (2019)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [A Connection between Generative Adversarial Networks, Inverse Reinforcement Learning, and Energy-Based Models](https://mlsanity.com/paper/1611.03852/) (2016)
- [Human Trajectory Forecasting in Crowds: A Deep Learning Perspective](https://mlsanity.com/paper/2007.03639/) (2020)
- [Implicit Latent Variable Model for Scene-Consistent Motion Forecasting](https://mlsanity.com/paper/2007.12036/) (2020)
- [Reward (Mis)design for Autonomous Driving](https://mlsanity.com/paper/2104.13906/) (2021)
- [What the Constant Velocity Model Can Teach Us About Pedestrian Motion Prediction](https://mlsanity.com/paper/1903.07933/) (2019)
- [Explainability of deep vision-based autonomous driving systems: Review and challenges](https://mlsanity.com/paper/2101.05307/) (2021)
- [Differentiable Integrated Motion Prediction and Planning with Learnable Cost Function for Autonomous Driving](https://mlsanity.com/paper/2207.10422/) (2022)
- [Theseus: A Library for Differentiable Nonlinear Optimization](https://mlsanity.com/paper/2207.09442/) (2022)
- [DSDNet: Deep Structured self-Driving Network](https://mlsanity.com/paper/2008.06041/) (2020)
- [Cooperative Training of Descriptor and Generator Networks](https://mlsanity.com/paper/1609.09408/) (2016)
- [A Tale of Two Flows: Cooperative Learning of Langevin Flow and Normalizing Flow Toward Energy-Based Model](https://mlsanity.com/paper/2205.06924/) (2022)
- [Learning Energy-Based Model with Variational Auto-Encoder as Amortized Sampler](https://mlsanity.com/paper/2012.14936/) (2020)
- [JFP: Joint Future Prediction with Interactive Multi-Agent Modeling for Autonomous Driving](https://mlsanity.com/paper/2212.08710/) (2022)
- [Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games](https://mlsanity.com/paper/1909.04694/) (2019)

## Built on by (most cited first, 4 of 4)

- [Energy-based Potential Games for Joint Motion Forecasting and Control](https://mlsanity.com/paper/2312.01811/) (2023)
- [DiffTORI: Differentiable Trajectory Optimization for Deep Reinforcement and Imitation Learning](https://mlsanity.com/paper/2402.05421/) (2024)
- [LoRD: Adapting Differentiable Driving Policies to Distribution Shifts](https://mlsanity.com/paper/2410.09681/) (2024)
- [Distributed Stackelberg Strategies in State-based Potential Games for Autonomous Decentralized Learning Manufacturing Systems](https://mlsanity.com/paper/2408.06397/) (2024)

## Abstract (arXiv)

Game theory offers an interpretable mathematical framework for modeling multi-agent interactions. However, its applicability in real-world robotics applications is hindered by several challenges, such as unknown agents' preferences and goals. To address these challenges, we show a connection between differential games, optimal control, and energy-based models and demonstrate how existing approaches can be unified under our proposed Energy-based Potential Game formulation. Building upon this formulation, this work introduces a new end-to-end learning application that combines neural networks for game-parameter inference with a differentiable game-theoretic optimization layer, acting as an inductive bias. The experiments using simulated mobile robot pedestrian interactions and real-world automated driving data provide empirical evidence that the game-theoretic layer improves the predictive performance of various neural network backbones.