Evolutionary stability implies asymptotic stability under multiplicative weights

We show that evolutionarily stable states in general (nonlinear) population games (which can be viewed as continuous vector fields constrained on a polytope) are asymptotically stable under a multiplicative weights dynamic (under appropriate choices of a parameter called the learning rate or step size, which we demonstrate to be crucial to achieve convergence, as otherwise even chaotic behavior is possible to manifest). Our result implies that evolutionary theories based on multiplicative weights are compatible (in principle, more general) with those based on the notion of evolutionary stability. However, our result further establishes multiplicative weights as a nonlinear programming primitive (on par with standard nonlinear programming methods) since various nonlinear optimization problems, such as finding Nash/Wardrop equilibria in nonatomic congestion games, which are well-known to be equipped with a convex potential function, and finding strict local maxima of quadratic programming problems, are special cases of the problem of computing evolutionarily stable states in nonlinear population games.