Compact covering and game determinacy

All spaces are separable and metrizable. Suppose that the continuous and onto mapping f:X --> Y is compact covering. Under the axiom of Sigma(1)(1)-determinacy, we prove that f is inductively perfect whenever X is Borel, and it follows then that Y is also Borel. Under the axiom N-1(L) = N-i We construct examples showing that the conclusion might fail if ''X is Borel'' is replaced by ''X is coanalytic''. If we suppose that both X and Y are Borel,then we prove (in ZFC) the weaker conclusion that f has a Borel (in fact a Baire-1) section g:Y --> X. We also prove (in ZFC) that if we suppose only X to be Borel but of some ''low'' class, then Y is also Borel of the same class, Other related problems are discussed.