Duparc introduced a two-player game $G_f$ for a self-map $f$ of the Baire space such that Player II has a winning strategy iff $f$ is Baire class 1. We define a game $G'_f$ for an arbitrary function $f$ between arbitrary Polish spaces such that Player II has a winning strategy in $G'_f$ iff $f$ is Baire class 1. We also show that $G'_f$ is always determined.