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Sep 12 at 16:52 comment added Dmytro Taranovsky @JoelDavidHamkins I clarified that player 2 strategy is required to win with probability ≥p for all choices of $P$.
Sep 12 at 16:52 history edited Dmytro Taranovsky CC BY-SA 4.0
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Sep 12 at 16:04 comment added Joel David Hamkins Your player 2 strategy has some affinity with the solution of a hat puzzle of Andreas Leitz's. See his paper with Jeroen Winkel here: andreas-lietz.github.io/resources/PDFs/hat_problems.pdf
Sep 12 at 15:31 comment added Joel David Hamkins I'm unsure what it means exactly to win this game with probability $p$, since the choice of the distribution is also part of the game, but we don't have a distribution for how that distribution is picked. Only part of player 1's first move, after all, involves random variables. (Presumably you want to block the observation that for any given $P$ there is a winning player 2 strategy of playing the finite support set itself.)
Sep 12 at 15:12 history asked Dmytro Taranovsky CC BY-SA 4.0