# Differentiable Game Mechanics

arXiv: https://arxiv.org/abs/1905.04926 · HTML: https://mlsanity.com/paper/1905.04926/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2019.

## Built on by (most cited first, 30 of 43)

- [A Unified Approach to Reinforcement Learning, Quantal Response Equilibria, and Two-Player Zero-Sum Games](https://mlsanity.com/paper/2206.05825/) (2022)
- [Smooth Tchebycheff Scalarization for Multi-Objective Optimization](https://mlsanity.com/paper/2402.19078/) (2024)
- [Model-Free Opponent Shaping](https://mlsanity.com/paper/2205.01447/) (2022)
- [COLA: Consistent Learning with Opponent-Learning Awareness](https://mlsanity.com/paper/2203.04098/) (2022)
- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [Policy Gradient Methods Find the Nash Equilibrium in N-player General-sum Linear-quadratic Games](https://mlsanity.com/paper/2107.13090/) (2021)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [Adversarial Cheap Talk](https://mlsanity.com/paper/2211.11030/) (2022)
- [Scaling Opponent Shaping to High Dimensional Games](https://mlsanity.com/paper/2312.12568/) (2023)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2211.13668/) (2022)
- [Towards Multi-Agent Reinforcement Learning driven Over-The-Counter Market Simulations](https://mlsanity.com/paper/2210.07184/) (2022)
- [Inverse-Weighted Survival Games](https://mlsanity.com/paper/2111.08175/) (2021)
- [Lyapunov Exponents for Diversity in Differentiable Games](https://mlsanity.com/paper/2112.14570/) (2021)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/2108.00473/) (2021)
- [Adversarial Deep Learning for Online Resource Allocation](https://mlsanity.com/paper/2111.10285/) (2021)
- [Domain Adversarial Training: A Game Perspective](https://mlsanity.com/paper/2202.05352/) (2022)
- [Aligning Individual and Collective Objectives in Multi-Agent Cooperation](https://mlsanity.com/paper/2402.12416/) (2024)
- [Neural Interactive Proofs](https://mlsanity.com/paper/2412.08897/) (2024)
- [Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics](https://mlsanity.com/paper/2203.14129/) (2022)
- [Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games](https://mlsanity.com/paper/2505.19537/) (2025)
- [Adversarial Representation Learning With Closed-Form Solvers](https://mlsanity.com/paper/2109.05535/) (2021)
- [Recursive Reasoning in Minimax Games: A Level k Gradient Play Method](https://mlsanity.com/paper/2210.16482/) (2022)
- [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025)
- [Equilibrium Learning in Combinatorial Auctions: Computing Approximate Bayesian Nash Equilibria via Pseudogradient Dynamics](https://mlsanity.com/paper/2101.11946/) (2021)
- [Consensus Multiplicative Weights Update: Learning to Learn using Projector-based Game Signatures](https://mlsanity.com/paper/2106.02615/) (2021)
- [Polymatrix Competitive Gradient Descent](https://mlsanity.com/paper/2111.08565/) (2021)
- [When is Momentum Extragradient Optimal? A Polynomial-Based Analysis](https://mlsanity.com/paper/2211.04659/) (2022)

## Abstract (arXiv)

Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new tools to understand and control the dynamics in n-player differentiable games. The key result is to decompose the game Jacobian into two components. The first, symmetric component, is related to potential games, which reduce to gradient descent on an implicit function. The second, antisymmetric component, relates to Hamiltonian games, a new class of games that obey a conservation law akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in differentiable games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- while at the same time being applicable to, and having guarantees in, much more general cases.