Determinacy with complicated strategies

Abstract:

For any class of functions $\mathfrak {F}$ from ${\mathbf {R}}$ into ${\mathbf {R}},\operatorname {AD}(\mathfrak {F}{\text {)}}$ is the assertion that in every two person game on integers one of the two players has a winning strategy in the class $\mathfrak {F}$. It is shown, in $ZF + DC + V = L({\mathbf {R}})$, that for any $\mathfrak {F}$ of cardinality $< {2^{{\aleph _0}}}$ (i.e. any $\mathfrak {F}$ which is a surjective image of ${\mathbf {R}}$) $\operatorname {AD}(\mathfrak {F})$ implies AD (the Axiom of Determinacy).