Inertial game dynamics and applications to constrained optimization

TL;DR

This paper introduces a second-order inertial game dynamics based on Riemannian geometry, demonstrating its stability properties and applications to constrained optimization and equilibrium attraction.

Contribution

It develops a new inertial replicator dynamics framework using Hessian-Riemannian metrics, ensuring well-posedness and stability in game-theoretic and optimization contexts.

Findings

Inertial dynamics are not well-posed under Shahshahani metric.

Alternative Hessian-Riemannian metrics can ensure well-posedness.

Strict equilibria attract solutions in multi- and single-population games.

Abstract

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.