On the determinacy of games on ordinals

Let λ be an ordinal and A ⊆ λ^ω x λ^ω. As usual we associate with it the game G(A;λ): I II I and II alternatively ξ0 η0 play ξ0, η0, ξ1,η1,.... from λ; ξ1 I wins iff (ξ, ή) eA. η1 ξ ή.