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.?
Stathis Papaioannou
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Received on Mon Oct 04 2004 - 20:47:24 PDT