Inertial Game Dynamics and Applications to Constrained Optimization

Rida Laraki (1) , Panayotis Mertikopoulos (2)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
2 LIG - Laboratoire d'Informatique de Grenoble

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 (multipopulation) 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.

Mots clés: Game dynamics; folk theorem; Hessian–Riemannian metrics; learning; replicator dynamics; second-order dynamics; stability of equilibria; well-posedness

Domaines: Informatique [cs]

Rida Laraki, Panayotis Mertikopoulos. Inertial Game Dynamics and Applications to Constrained Optimization. SIAM Journal on Control and Optimization, 2015, 53 (5), ⟨10.1137/130920253⟩. ⟨hal-01483807⟩