  We examine the long-run behavior of a wide range of dynamics for learning in
nonatomic games, in both discrete and continuous time. The class of dynamics
under consideration includes fictitious play and its regularized variants, the
best-reply dynamics (again, possibly regularized), as well as the dynamics of
dual averaging / "follow the regularized leader" (which themselves include as
special cases the replicator dynamics and Friedman's projection dynamics). Our
analysis concerns both the actual trajectory of play and its time-average, and
we cover potential and monotone games, as well as games with an evolutionarily
stable state (global or otherwise). We focus exclusively on games with finite
action spaces; nonatomic games with continuous action spaces are treated in
detail in Part II of this paper.
