# Inertial game dynamics and applications to constrained optimization

arXiv: https://arxiv.org/abs/1305.0967 · HTML: https://mlsanity.com/paper/1305.0967/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2013.

## Built on by (most cited first, 2 of 2)

- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [Accelerated regularized learning in finite N-person games](https://mlsanity.com/paper/2412.20365/) (2024)

## Abstract (arXiv)

Aiming to provide a new class of game dynamics with good long-term rationality properties, we derive a second-order inertial system that builds on the widely studied "heavy ball with friction" optimization method. By exploiting a well-known link between the replicator dynamics and the Shahshahani geometry on the space of mixed strategies, the dynamics are stated in a Riemannian geometric framework where trajectories are accelerated by the players' unilateral payoff gradients and they slow down near Nash equilibria. Surprisingly (and in stark contrast to another second-order variant of the replicator dynamics), the inertial replicator dynamics are not well-posed; on the other hand, it is possible to obtain a well-posed system by endowing the mixed strategy space with a different Hessian-Riemannian (HR) metric structure, and we characterize those HR geometries that do so. In the single-agent version of the dynamics (corresponding to constrained optimization over simplex-like objects), we show that regular maximum points of smooth functions attract all nearby solution orbits with low initial speed. More generally, we establish an inertial variant of the so-called "folk theorem" of evolutionary game theory and we show that strict equilibria are attracting in asymmetric (multi-population) games - provided of course that the dynamics are well-posed. A similar asymptotic stability result is obtained for evolutionarily stable strategies in symmetric (single- population) games.