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Nice question. What happens if we change the condition from <1/16 to <1/4 or <1/2? I think U can be simply connected if the condition is <1/2+epsilon. But maybe it also works for a stricter condition...?
I see. I don't really know anything about infinite games, and I'm not a mathematician but a software engineer, so I guess I was thinking about this too "constructively", in a sense. In this case: We can prove that for every alleged winning strategy sG for G, there exists a strategy sC(sG) for C which defeats it. The probablity that C will actually play that strategy may be zero (because C doesn't know sG and thus can't choose sC(sG)), but that doesn't matter. Correct?
"suppose the Glutton will follow a fixed strategy ... then the Chocolatier can present these pairs {π,π0}, {π,π1}, {π,π2}, in turn" β This requires that the Chocolatier knows the Glutton's strategy, doesn't it? (I think we should require that neither player knows the other's strategy. Otherwise, both could try to adapt their strategy to the other's strategy, and we'd run into paradoxes...)
The four axioms remain true if you replace "touches" by "is element of". Does this mean that "is element of" is a special case of "touches"? Or is this a silly idea?
