Games characterizing limsup functions and Baire class 1 functions

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 $\mathit{\text{limsup}}\mathit{\text{function}}$ if there is a function $u \colon T \to $\mathbb{R}$$suchthat$f(x) = \limsup_{t \to \infty} u(x_{0},\dots,x_{t})$ 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.