We study the determinacy of the game G_kappa (A) introduced in [FKSh:549] for uncountable regular kappa and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that G_kappa (A) is undetermined. For the class of linear orders, the existence of such A depends on the size of kappa^{ < kappa}. In particular we obtain a characterization of kappa^{ < kappa}= kappa in terms of determinacy of the game G_kappa (L) for linear orders L .
Categories: math.LO