In a recent paper [19, 20] Serre has presented some decidable winning conditions Ω _{A_1&rtri;&mldr;&rtri;A_n&rtri;A_{n+1}} of arbitrarily high finite Borel complexity for games on finite graphs or on pushdown graphs.

We answer in this paper several questions which were raised by Serre in [19,20].

We study classes C _n (A), defined in [20], and show that these classes are included in the class of non-ambiguous context free ω-languages. Moreover from the study of a larger class C _n^λ (A) we infer that the complements of languages in C _n (A) are also non-ambiguous context free ω-languages. We conclude the study of classes C _n (A) by showing that they are neither closed under union nor under intersection.

We prove also that there exists pushdown games, equipped with winning conditions in the form Ω _{A_1&rtri;A_2} , 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.