  We prove that the determinacy of Gale-Stewart games whose winning sets are
accepted by real-time 1-counter B\"uchi automata is equivalent to the
determinacy of (effective) analytic Gale-Stewart games which is known to be a
large cardinal assumption. We show also that the determinacy of Wadge games
between two players in charge of omega-languages accepted by 1-counter B\"uchi
automata is equivalent to the (effective) analytic Wadge determinacy. Using
some results of set theory we prove that one can effectively construct a
1-counter B\"uchi automaton A and a B\"uchi automaton B such that: (1) There
exists a model of ZFC in which Player 2 has a winning strategy in the Wadge
game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game
W(L(A), L(B)) is not determined. Moreover these are the only two possibilities,
i.e. there are no models of ZFC in which Player 1 has a winning strategy in the
Wadge game W(L(A), L(B)).
