From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective
Authors: René Carmona¹ · Gökçe Dayanıklı² · François Delarue · Mathieu Laurière
Publication Details: Applied Mathematics and Optimization 93:92 (2026)
Received: 14 July 2025 Accepted: 27 April 2026 © The Author(s) 2026
This repository is an independent implementation and research extension of the original work by Nihar Mahesh Jani.
Author: Nihar Mahesh Jani Email: niharmaheshjani@gmail.com GitHub Repository: https://github.com/NiharJani2002/NESOBM
This implementation reproduces the theoretical and computational framework of the original paper from scratch in Python, verifies major mathematical guarantees, reproduces key experimental findings, and proposes additional original research extensions.
Use this code at your own risk.
This repository is provided for research, educational, and experimental purposes only. While efforts have been made to ensure correctness, reproducibility, and mathematical consistency, no guarantees are provided regarding accuracy, completeness, performance, or suitability for any particular application.
Users are strongly encouraged to independently verify all mathematical derivations, assumptions, implementations, and empirical results before using this work in academic, industrial, financial, scientific, or production settings.
The repository author assumes no liability for any outcomes arising from the use or misuse of this code.
If you use this repository, codebase, methodology, implementation ideas, or research extensions in your academic work, projects, publications, or derivatives, please cite both:
From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective René Carmona¹ · Gökçe Dayanıklı² · François Delarue · Mathieu Laurière
Applied Mathematics and Optimization 93:92 (2026)
Received: 14 July 2025 Accepted: 27 April 2026 **© The Author(s) 2026
Nihar Mahesh Jani Independent Implementation of: From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective
GitHub Repository: https://github.com/NiharJani2002/NESOBM
Email: niharmaheshjani@gmail.com
Thank you for taking the time to explore this project. I’m genuinely excited to share it with you.
This repository is my independent implementation of the 2026 paper by René Carmona, Gökçe Dayanıklı, François Delarue, and Mathieu Laurière. I reproduced the core LQ example from Appendix B.2 from scratch in clean, readable Python, verified all theoretical guarantees, and then extended the framework with five original research contributions.
Whether you're studying mean field games, optimal control, economics, or collective behavior, I hope this work provides value and sparks new ideas.
The paper examines a fundamental tension in large-scale systems:
- Nash Equilibrium (MFG): Everyone optimizes individually.
- Social Optimum (MFC): A central planner coordinates for the collective good.
- Free-Rider Problem: What happens when some players deviate from cooperation?
This code models the Linear-Quadratic mean field game and lets you deeply explore these dynamics.
- Exact closed-form MFG and MFC solvers
- p-Partial Mean Field Equilibria analysis
- Price of Anarchy (PoA) and Price of Instability (PoI)
- Verification of Proposition 6 lower bound
- Monte Carlo validation
- Full reproduction of Figure 1 and Figure 2
- PoA = 1.01353
- p = 0.42576*
- PoI = 0.05108
- All assertions pass
- EXT1: Entropy-Regularised λ-MFG — Boltzmann smoothing of the cooperation path
- EXT2: Risk-Sensitive CVaR p-Partial MFG — Risk aversion reduces free-riding incentives
- EXT3: Adaptive Step-Size Convergence — 2.28× faster convergence than Theorem 11
- EXT4: Two-Population Cooperative Feasibility Region — Heterogeneous agents in 2D
- EXT5: Lyapunov Stability Certificate — Explicit convergence guarantees
pip install numpy scipy matplotlib torch