Here is another version of the paradox, where the way an individual chooses does not change the initial probabilities:
In the new casino game Flip-Flop, an odd number of players pays $1 each to individually flip a coin, so that no player can see what another player is doing. The game organisers then tally up the results, and the result in the minority is called the Winning Flip, while the majority result is called the Losing Flip. Before the Winning Flip is announced, each player has the opportunity to either keep their initial result, or to Switch; this is then called the player's Final Flip. When the Winning Flip is announced, players whose Final Flip corresponds with this are paid $2 by the casino, while the rest are paid nothing.
The question: if you participate in this game, is there any advantage in Switching? On the one hand, it seems clear that the Winning Flip is as likely to be heads as tails, so if you played this game repeatedly, in the long run you should break even, whether you Switch or not. On the other hand, it seems equally clear that if all the players Switch, the casino will end up every time paying out more than it collects, so Switching should be a winning strategy, on average, for each individual player.
I'm sure there is something wrong with the above conclusion. What is it? And I haven't really thought this through yet, but does this have any bearing on the self sampling assumption as applied in the Doomsday Argument etc.?
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