A game on partial orderings

We study the determinacy of the game Gκ(A) introduced in Fuchino, Koppelberg and Shelah (to appear) for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ<κ. In particular we obtain a characterization of κ<κ = κ in terms of determinacy of the game Gκ(L) for linear orders L.