# Newton's Method and Differential Dynamic Programming for Unconstrained Nonlinear Dynamic Games

arXiv: https://arxiv.org/abs/1906.09097 · HTML: https://mlsanity.com/paper/1906.09097/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2019.

## Built on by (most cited first, 5 of 5)

- [Deep Interactive Motion Prediction and Planning: Playing Games with Motion Prediction Models](https://mlsanity.com/paper/2204.02392/) (2022)
- [SOCIALMAPF: Optimal and Efficient Multi-Agent Path Finding with Strategic Agents for Social Navigation](https://mlsanity.com/paper/2210.08390/) (2022)
- [Decentralized Social Navigation with Non-Cooperative Robots via Bi-Level Optimization](https://mlsanity.com/paper/2306.08815/) (2023)
- [How Bad is Selfish Driving? Bounding the Inefficiency of Equilibria in Urban Driving Games](https://mlsanity.com/paper/2210.13064/) (2022)
- [Constraint Learning in Multi-Agent Dynamic Games from Demonstrations of Local Nash Interactions](https://mlsanity.com/paper/2508.19945/) (2025)

## Abstract (arXiv)

Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense, energy systems and etc. However, compared to single-agent control problems, the computational methods for dynamic games are relatively limited. As in the single-agent case, only specific dynamic games can be solved exactly, so approximation algorithms are required. In this paper, we show how to extend a recursive Newton's algorithm and the popular differential dynamic programming (DDP) for single-agent optimal control to the case of full-information non-zero sum dynamic games. In the single-agent case, the convergence of DDP is proved by comparison with Newton's method, which converges locally at a quadratic rate. We show that the iterates of Newton's method and DDP are sufficiently close for the DDP to inherit the quadratic convergence rate of Newton's method. We also prove both methods result in an open-loop Nash equilibrium and a local feedback O(ε2)-Nash equilibrium. Numerical examples are provided.