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.