The property on the filter in Definition 1, a kind of large cardinal property, suffices for the proof in Liu Shelah [LiSh484] and is proved consistent as required there (see Conclusion 6). A natural property which looks better, not only is not obtained here, but is shown to be false (in Claim 7). On earlier related theorems see Gitik Shelah [GiSh310]. On such games see e.g. [Je], [Sh-b], [Sh-f].