  We consider zero-sum games on infinite graphs, with objectives specified as
sets of infinite words over some alphabet of colors. A well-studied class of
objectives is the one of $\omega$-regular objectives, due to its relation to
many natural problems in theoretical computer science. We focus on the strategy
complexity question: given an objective, how much memory does each player
require to play as well as possible? A classical result is that finite-memory
strategies suffice for both players when the objective is $\omega$-regular. We
show a reciprocal of that statement: when both players can play optimally with
a chromatic finite-memory structure (i.e., whose updates can only observe
colors) in all infinite game graphs, then the objective must be
$\omega$-regular. This provides a game-theoretic characterization of
$\omega$-regular objectives, and this characterization can help in obtaining
memory bounds. Moreover, a by-product of our characterization is a new
one-to-two-player lift: to show that chromatic finite-memory structures suffice
to play optimally in two-player games on infinite graphs, it suffices to show
it in the simpler case of one-player games on infinite graphs. We illustrate
our results with the family of discounted-sum objectives, for which
$\omega$-regularity depends on the value of some parameters.
