# Maximum-Entropy Multi-Agent Dynamic Games: Forward and Inverse Solutions

arXiv: https://arxiv.org/abs/2110.01027 · HTML: https://mlsanity.com/paper/2110.01027/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Oct 2021.

## Built on by (most cited first, 22 of 22)

- [Deep Interactive Motion Prediction and Planning: Playing Games with Motion Prediction Models](https://mlsanity.com/paper/2204.02392/) (2022)
- [Learn the Ropes, Then Trust the Wins: Self-imitation with Progressive Exploration for Agentic Reinforcement Learning](https://mlsanity.com/paper/2509.22601/) (2025)
- [MIMIC-D: Multi-modal Imitation for MultI-agent Coordination with Decentralized Diffusion Policies](https://mlsanity.com/paper/2509.14159/) (2025)
- [Blending Data-Driven Priors in Dynamic Games](https://mlsanity.com/paper/2402.14174/) (2024)
- [Multi-Agent Inverse Reinforcement Learning in Real World Unstructured Pedestrian Crowds](https://mlsanity.com/paper/2405.16439/) (2024)
- [On a Connection between Differential Games, Optimal Control, and Energy-based Models for Multi-Agent Interactions](https://mlsanity.com/paper/2308.16539/) (2023)
- [Auto-Encoding Bayesian Inverse Games](https://mlsanity.com/paper/2402.08902/) (2024)
- [Mixed Strategy Nash Equilibrium for Crowd Navigation](https://mlsanity.com/paper/2403.01537/) (2024)
- [Integrated Decision Making and Trajectory Planning for Autonomous Driving Under Multimodal Uncertainties: A Bayesian Game Approach](https://mlsanity.com/paper/2409.13993/) (2024)
- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [Soft-Bellman Equilibrium in Affine Markov Games: Forward Solutions and Inverse Learning](https://mlsanity.com/paper/2304.00163/) (2023)
- [On Convex Data-Driven Inverse Optimal Control for Nonlinear, Non-stationary and Stochastic Systems](https://mlsanity.com/paper/2306.13928/) (2023)
- [Active Inverse Learning in Stackelberg Trajectory Games](https://mlsanity.com/paper/2308.08017/) (2023)
- [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)
- [NashFormer: Leveraging Local Nash Equilibria for Semantically Diverse Trajectory Prediction](https://mlsanity.com/paper/2305.17600/) (2023)
- [When Should a Leader Act Suboptimally? The Role of Inferability in Repeated Stackelberg Games](https://mlsanity.com/paper/2310.00468/) (2023)
- [Learning Collective Dynamics of Multi-Agent Systems using Event-based Vision](https://mlsanity.com/paper/2411.07039/) (2024)
- [PACE: A Framework for Learning and Control in Linear Incomplete-Information Differential Games](https://mlsanity.com/paper/2504.17128/) (2025)
- [Peer-Aware Cost Estimation in Nonlinear General-Sum Dynamic Games for Mutual Learning and Intent Inference](https://mlsanity.com/paper/2504.17129/) (2025)
- [Bayesian Inverse Games with High-Dimensional Multi-Modal Observations](https://mlsanity.com/paper/2601.00696/) (2026)
- [Matching Multiple Experts: On the Exploitability of Multi-Agent Imitation Learning](https://mlsanity.com/paper/2602.21020/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of multiple stochastic agents interacting in a dynamic game scenario with continuous state and action spaces. We define a new notion of stochastic Nash equilibrium for boundedly rational agents, which we call the Entropic Cost Equilibrium (ECE). We show that ECE is a natural extension to multiple agents of Maximum Entropy optimality for single agents. We solve both the "forward" and "inverse" problems for the multi-agent ECE game. For the forward problem, we provide a Riccati algorithm to compute closed-form ECE feedback policies for the agents, which are exact in the Linear-Quadratic-Gaussian case. We give an iterative variant to find locally ECE feedback policies for the nonlinear case. For the inverse problem, we present an algorithm to infer the cost functions of the multiple interacting agents given noisy, boundedly rational input and state trajectory examples from agents acting in an ECE. The effectiveness of our algorithms is demonstrated in a simulated multi-agent collision avoidance scenario, and with data from the INTERACTION traffic dataset. In both cases, we show that, by taking into account the agents' game theoretic interactions using our algorithm, a more accurate model of agents' costs can be learned, compared with standard inverse optimal control methods.