Game representations of classes of piecewise definable functions

We present a general way of defining various reduction games on \omega\ which "represent" corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for \Gamma-measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10].