The paradox of making money from opponents mistakes

[AaronBrown]

I think it helps to simplify the problem.

Suppose you have a fair coin that everyone knows is fair. Someone agrees to bet $2 against your $1 that it will come up heads. If you take this bet you profit, regardless of what happens on the coin flip. You make $0.50 expected value when you make the bet, heads you lose $1.50, tails you make $1.50.

Now suppose you draw a random card from the deck and keep it face down. Another person offers to bet you $1 even money that he can draw a higher card. The minute he says this, you are ahead $0.1267. Now you look at your card. If it is an 8 or lower, you lose the $0.1267 back, because you'll turn down the bet. If it is 9, you lose $0.0189, because you'll accept the bet but your positive expectation is only $0.1078 instead of $0.1267. If you drew a 10, you win an additional 0.0596; up to an Ace where you win an additional $0.3733.

Whatever card you have, and whatever card the other guy draws, you make $0.1267 expected value when he offers you the bet. Over the long run, you'll win and lose a lot of bets, but you'll collect that $0.1267. If you drew a 2 the offer is worthless to you, if you drew a King and he draws an Ace it costs you money, but in the long run, you win and he loses. What happens after he offers the bet depends on luck, but the positive expected value stays with you.