The paradox of making money from opponents mistakes

[TTChamp]

Aaron, thanks for the response. I have always liked your posts on here.

I'm not seeing a direct connection to poker in these examples. These are deterministic examples. There is no information defecit in a coin flip or a card draw. I think you are simply addressing the aspect of chance in poker.

In other words you are adressing the fact that you must not consider only what the outcome of any given trial is, but instead you must consider what is correct over every possible trial.

In the examples you gave, the other guy is making a bad bet. He may get lucky here or there, but in the long run he will lose money. In my example the villian will make money over many trials when he bets his AJ on the flop. He will also make money when he calls the c/r on the flop. So the coing flip guy has made a bad choice, while the AJ guy has made a profitable choice.

Also, I have a couple questions on your scenarios:

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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.


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Last sentence is a typo right.


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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.

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Because we are winning when he draws the same card????

[AaronBrown]

Thanks for the kind words.

You're right that the first example is not like poker, there is no hidden information. The last sentence is correct in my accounting. You make $0.50 when he takes the bet, then you have an even $1.50 win or loss on the coin toss. If you win the toss, you get $0.50 from his mistake, and $1.50 from the luck of the flip, $2.00 total payout. If you lose the toss, you get $0.50 from his mistake but lose $1.50 from the luck of the flip, -$1.00 total.

In the second example, I assume the bet is a push if he draws the same card (although I worded it wrong). The reason you are ahead $0.1267 when he makes this offer is you have the option to accept it (which you do with a 9 or higher) or decline it (which you do on a 7 or lower, with an 8 it's a fair bet).

This case is very much like poker. He makes a bet, you can call or fold. It's simpler because it depends only on one card, there is no ante or blind, and you cannot raise. The point is that you make money from his bad bet, regardless of what you hold or what he draws. Those are random events that will add ot or subtract from your initial expectation.