R. Laraki and P. Mertikopoulos. SIAM Journal on Control and Optimization, vol. 53, no. 5, pp. 3141–3170, October 2015.
Aiming to provide a new class of game dynamics with good long-term convergence 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 states in symmetric (single-population) games.
arXiv link: https://arxiv.org/abs/1305.0967