  We consider a real-valued function $f$ defined on the set of infinite
branches $X$ of a countably branching pruned tree $T$. The function $f$ is said
to be a \textit{limsup function} if there is a function $u \colon T \to
\mathbb{R}$ such that $f(x) = \limsup_{t \to \infty} u(x_{0},\dots,x_{t})$ for
each $x \in X$. We study a game characterization of limsup functions, as well
as a novel game characterization of functions of Baire class 1.
