  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.
