Here is a similar paradox to the traffic lane example:

 

In the new casino game called Flip-Flop, an odd number of players pay $1 each to gather in individual cubicles and flip a coin (so no player can see what another player is doing). The game organisers tally up the results, and the result which is in the minority is said to be the Winning Flip, while the minority result is said to be the Losing Flip. For example, if there are 101 players and of these 53 flip heads while 48 flip tails, tails is the Winning Flip and heads is the Losing Flip. Before the result of the tally is announced, each player must commit to either keep the result of their original coin flip, whether heads or tails, or switch to the opposite result. The casino then announces what the Winning Flip was, and players whose final result (however it was obtained) corresponds with this are paid $2, while the rest get nothing.

 

The question now: is there anything to be gained by switching at the last step of this game? From the point of view of typical player, it would seem that there is not: the Winning Flip is as likely to be heads as tails, and if he played the game repeatedly over time, he should expect to break even, whether he switches in the final step or not. On the other hand, it seems clear that if nobody switches, the casino is ahead, while if everbody switches, the players are ahead; so switching would seem to be a winning strategy for the players. This latter result is not due to any cooperation effect, as only those players who switch get the improved (on average) outcome.

 

Stathis Papaioannou

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