Since the seminal work by Wadge in the 70s and 80s, a tradition has been established in descriptive set theory of using games to characterize certain important notions and classes of objects. Particular attention has been devoted to characterizing classes of functions in Baire space by games, with Wadge's game for continuous functions and Duparc's eraser game for the Baire class 1 functions as two important examples. In his PhD thesis, Brian Semmes introduced his tree game which characterizes the Borel measurable functions, and a restriction of the tree game which characterizes the Baire class 2 functions.

In this work, we show how to restrict Semmes's tree game in order to obtain games characterizing each finite Baire class, in a uniform way. The Wadge and eraser games are particular cases of our construction, but interestingly enough our construction for Baire class 2 gives a different -- though of course equivalent -- game than Semmes's.

The author would like to acknowledge that Alain Louveau and Brian Semmes proved the main result independently with a different proof.