Games and general distributive laws in Boolean algebras

The games G(1)(eta)(kappa) and G(<lambda)(eta) (kappa) are played by two players in eta(+)-complete and max(eta(+), lambda)-complete Boolean algebras, respectively. For cardinals eta,kappa such that kappa(<eta)=eta or kappa(<eta)=kappa, the (eta,kappa)-distributive law holds in a Boolean algebra B iff Player 1 does not have a winning strategy in G(1)(eta)(kappa). Furthermore, for all cardinals kappa, the (eta,infinity)-distributive law holds in B iff Player 1 does not have a winning strategy in G(1)(eta)(infinity). More generally, for cardinals eta,lambda,kappa such that (kappa(<lambda))(<eta)=eta, the (eta,< lambda,kappa)-distributive law holds in B iff Player 1 does not have a winning strategy in G(<lambda)(eta) (kappa). For eta regular and lambda less than or equal to min(eta,kappa), lozenge(eta)+ implies the existence of a Suslin algebra in which G(<lambda)(eta) (kappa) is undetermined.