  Some decidable winning conditions of arbitrarily high finite Borel complexity
for games on finite graphs or on pushdown graphs have been recently presented
by O. Serre in [ Games with Winning Conditions of High Borel Complexity, in the
Proceedings of the International Conference ICALP 2004, LNCS, Volume 3142, p.
1150-1162 ]. We answer in this paper several questions which were raised by
Serre in the above cited paper. We first show that, for every positive integer
n, the class C_n(A), which arises in the definition of decidable winning
conditions, is included in the class of non-ambiguous context free omega
languages, and that it is neither closed under union nor under intersection. We
prove also that there exists pushdown games, equipped with such decidable
winning conditions, where the winning sets are not deterministic context free
languages, giving examples of winning sets which are non-deterministic
non-ambiguous context free languages, inherently ambiguous context free
languages, or even non context free languages.
