Games with Filters

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^\kappa$ is equivalent to $\kappa$ being measurable. We show that for games of intermediate length $\gamma$, II winning implies the existence of precipitous ideals with $\gamma$-closed, $\gamma$-dense trees. The second part shows the first is not vacuous. For each $\gamma$ between $\omega$ and $\kappa^+$, it gives a model where II wins the games of length $\gamma$, but not $\gamma^+$. The technique also gives models where for all $\omega_1< \gamma\le\kappa$ there are $\kappa$-complete, normal, $\kappa^+$-distributive ideals having dense sets that are $\gamma$-closed, but not $\gamma^+$-closed.