The wager where both players expect to win

Question

Two friends compare the cash in their wallets, without either seeing the other's amount. They agree on a bet: whoever is carrying less money keeps their own cash and also wins the other person's wallet. If they are carrying equal amounts, they replay.

Each friend reasons: "I can only lose the amount I am carrying, but I could win more than that, since the other person might be carrying more. So this bet is favorable to me." Both friends believe the game favors them.

Both cannot have the advantage. Explain the flaw in this reasoning.

Accepted Answer

This is the two-envelope paradox in disguise, and the error is the same: each player treats his own amount as a fixed reference while imagining the other's amount as freely varying above and below it. Those two pictures are not consistent with each other. The two wallets hold fixed amounts, say x and y with x < y. Whoever is carrying the larger pile, y, is exactly the person who will lose, and he loses that larger pile y; whoever carries the smaller pile, x, is the one who wins, and he gains y. The size of the transfer, y, is the same for both, and it is precisely the loser's amount. You cann…

The wager where both players expect to win

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