  We prove that the determinacy of Gale-Stewart games whose winning sets are
infinitary rational relations accepted by 2-tape B\"uchi automata is equivalent
to the determinacy of (effective) analytic Gale-Stewart games which is known to
be a large cardinal assumption. Then we prove that winning strategies, when
they exist, can be very complex, i.e. highly non-effective, in these games. We
prove the same results for Gale-Stewart games with winning sets accepted by
real-time 1-counter B\"uchi automata, then extending previous results obtained
about these games. Then we consider the strenghs of determinacy for these
games, and we prove that there is a transfinite sequence of 2-tape B\"uchi
automata (respectively, of real-time 1-counter B\"uchi automata) $A_\alpha$,
indexed by recursive ordinals, such that the games $G(L(A_\alpha))$ have
strictly increasing strenghs of determinacy. Moreover there is a 2-tape B\"uchi
automaton (respectively, a real-time 1-counter B\"uchi automaton) B such that
the determinacy of G(L(B)) is equivalent to the (effective) analytic
determinacy and thus has the maximal strength of determinacy. We show also that
the determinacy of Wadge games between two players in charge of infinitary
rational relations accepted by 2-tape B\"uchi automata is equivalent to the
(effective) analytic determinacy, and thus not provable in ZFC.
