The "Flip-Flop" game described by Stathis Papaioannou strikes me as a 
version of the old Two-Envelope Paradox.

Assume an eccentric millionaire offers you your choice of either of two 
sealed envelopes, A or B, both containing money.  One envelope contains 
twice as much as the other.  After you choose an envelope you will have the 
option of trading it for the other envelope.

Suppose you pick envelope A.  You open it and see that it contains $100. 
Now you have to decide if you will keep the $100, or will you trade it for 
whatever is in envelope B?

You might reason as follows: since one envelope has twice what the other one 
has, envelope B either has 200 dollars or 50 dollars, with equal 
probability.  If you switch, you stand to either win $100 or to lose $50. 
Since you stand to win more than you stand to lose, you should switch.

But just before you tell the eccentric millionaire that you would like to 
switch, another thought might occur to you.  If you had picked envelope B, 
you would have come to exactly the same conclusion.  So if the above 
argument is valid, you should switch no matter which envelope you choose.

Therefore the argument for always switching is NOT valid - but I am unable, 
at the moment, to tell you why!

Norman Samish

----- Original Message ----- 
From: "Stathis Papaioannou" <[EMAIL PROTECTED]>
Sent: Monday, October 04, 2004 5:43 PM
Subject: RE: Observation selection effects

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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