# Riemannian game dynamics

arXiv: https://arxiv.org/abs/1603.09173 · HTML: https://mlsanity.com/paper/1603.09173/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Mar 2016.

## Built on by (most cited first, 19 of 19)

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- [Riemannian stochastic optimization methods avoid strict saddle points](https://mlsanity.com/paper/2311.02374/) (2023)
- [Learning in nonatomic games, Part I: Finite action spaces and population games](https://mlsanity.com/paper/2107.01595/) (2021)
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- [Adaptive first-order methods revisited: Convex optimization without Lipschitz requirements](https://mlsanity.com/paper/2107.08011/) (2021)
- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [No-regret learning in harmonic games: Extrapolation in the face of conflicting interests](https://mlsanity.com/paper/2412.20203/) (2024)
- [Review of Metrics to Measure the Stability, Robustness and Resilience of Reinforcement Learning](https://mlsanity.com/paper/2203.12048/) (2022)
- [Accelerated regularized learning in finite N-person games](https://mlsanity.com/paper/2412.20365/) (2024)
- [The impact of uncertainty on regularized learning in games](https://mlsanity.com/paper/2506.13286/) (2025)
- [Nested replicator dynamics, nested logit choice, and similarity-based learning](https://mlsanity.com/paper/2407.17815/) (2024)
- [Evolution of Societies via Reinforcement Learning](https://mlsanity.com/paper/2410.17466/) (2024)
- [The Bounds of Algorithmic Collusion; Q-learning, Gradient Learning, and the Folk Theorem](https://mlsanity.com/paper/2411.12725/) (2024)
- [Spike-timing-dependent Hebbian learning as noisy gradient descent](https://mlsanity.com/paper/2505.10272/) (2025)
- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.