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.

[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:

[ QUOTE ]
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.


[/ QUOTE ]

Last sentence is a typo right.


[ QUOTE ]
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.

[/ QUOTE ]

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.

Um, people in this thread have said its 6:1 to hit an overcard to make a pair. This is simply not true. First of all, 6.7:1 is a lot different than 6:1 in the long term. This is epecially true once you discount outs.

[TTChamp]

[ QUOTE ]
Um, people in this thread have said its 6:1 to hit an overcard to make a pair. This is simply not true. First of all, 6.7:1 is a lot different than 6:1 in the long term. This is epecially true once you discount outs.

[/ QUOTE ]

Yea that was my fault. I wrote this while I was tired.

Very good responses so far (although the someraulting one seemed a little cocky for some one with 10 posts).

BTW, some one asked about donk bet. A donk bet is when a player who is out of posiiotn bets into the in postiion player who bet or raised the last round. The idea is that the more "standard" play would be to c/r the guy because he will most likely bet.


I think I have an idea of where my reaoning was bad, but I wanted to write this post to get a better idea of how to exploit inforamtion to get the most money out of a hand.

In this case, we are in good shape because based on the information villian has he should bet the flop. We know that his informaiton is wrong (because we know our hand more precisely than he does), and we exploit that to get two bets in the pot by c/r vice donking.

I still stand by my postion that the villian played "mistake free" poker when he bets the flop. He is acting in a manner that shows the most profit based on the information he has at the time. The same is also true when he calls the flop c/r.

[sweetjazz]

[ QUOTE ]
I still stand by my postion that the villian played "mistake free" poker when he bets the flop. He is acting in a manner that shows the most profit based on the information he has at the time. The same is also true when he calls the flop c/r.

[/ QUOTE ]

The relevant defintion of "mistake" for the analysis of this hand according to the FTOP is playing other than you would have if you could see your opponent's hand. In that sense, betting with AJ in the hand you described is a mistake.

As Sklansky CLEARLY POINTS OUT in TOP, this is an unusual sense of the word "mistake". The villain did not play badly here, because there is no way he could determine your hand. He played well, but he made a mistake from the point of view of the TOP. Just as calling with four of a kind on the river when your opponent has a royal flush is a mistake, though obviously not a bad play in general.

Nowhere in TOP does Sklansky ever suggest it is possible to play mistake-free poker. The goal is simply to make less mistakes than your opponents do.

In keeping with the way Sklansky discusses TOP, your "paradox" is resolved by noting that while the villain made a theoretical mistake from the point of view of people who know what the hole cards are, his play was obviously not a bad one given the information available to him.

[TTChamp]

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I still stand by my postion that the villian played "mistake free" poker when he bets the flop. He is acting in a manner that shows the most profit based on the information he has at the time. The same is also true when he calls the flop c/r.


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The relevant defintion of "mistake" for the analysis of this hand according to the FTOP is playing other than you would have if you could see your opponent's hand. In that sense, betting with AJ in the hand you described is a mistake.

As Sklansky CLEARLY POINTS OUT in TOP, this is an unusual sense of the word "mistake". The villain did not play badly here, because there is no way he could determine your hand. He played well, but he made a mistake from the point of view of the TOP. Just as calling with four of a kind on the river when your opponent has a royal flush is a mistake, though obviously not a bad play in general.

Nowhere in TOP does Sklansky ever suggest it is possible to play mistake-free poker. The goal is simply to make less mistakes than your opponents do.

In keeping with the way Sklansky discusses TOP, your "paradox" is resolved by noting that while the villain made a theoretical mistake from the point of view of people who know what the hole cards are, his play was obviously not a bad one given the information available to him.

[/ QUOTE ]

Here is how I see poker: the best you can do is put your opponent on a range of hands and act in a manner that shows the most profit (or least loss) based on the range you have put him on. This process has both scientific and artistic aspects.

The "art" of poker is being able to put your opponent on a range (including the possiblity that he is bluffing) based on his past actions. This requires experience, observation, and good judgement and is extremely challenging. I would guess that most players at my limit (including me) only superficially understand this aspect of poker.

The "science" of poker is being able to choose the best action based on your opponent's hand range. The science of poker is deterministic and mathematical, but challenging from a calculational point of view. There is always one correct action based on a given hand range.

When a given hand is viewed from the point of view of the fundamental TOP, there is one right play, and it is impossible for two players to both play a hand correctly. For example, in a HU NL game, the sb goes all-in with AA, the BB looks down and has KK. By the fundamental TOP the BB is making a "mistake" by calling.

But this is useless from a practical point of view (I know that is heresy, hopefully I don't get banned). From a practical point of view, the BB puts the SB on a range of hands (e.g. TT-AA, and AQ-AK, and a 5% chance of a bluff) and notes that KK is profitable against this range. Therefor the BB should call. Let's use the words "error free" to describe the BB's play with KK since there seems to be a lot of objection to the words mistake-free.

It is possible for both players to play a hand "error-free". In the context of the FTOP, it is not possible for both players to play "mistake free" poker (save split situations).

I would like to see replies from anyone who disagrees with the last paragraph (including you David!).

[Nomad84]

[ QUOTE ]
When a given hand is viewed from the point of view of the fundamental TOP, there is one right play, and it is impossible for two players to both play a hand correctly. For example, in a HU NL game, the sb goes all-in with AA, the BB looks down and has KK. By the fundamental TOP the BB is making a "mistake" by calling.

But this is useless from a practical point of view (I know that is heresy, hopefully I don't get banned). From a practical point of view, the BB puts the SB on a range of hands (e.g. TT-AA, and AQ-AK, and a 5% chance of a bluff) and notes that KK is profitable against this range. Therefor the BB should call. Let's use the words "error free" to describe the BB's play with KK since there seems to be a lot of objection to the words mistake-free.

It is possible for both players to play a hand "error-free". In the context of the FTOP, it is not possible for both players to play "mistake free" poker (save split situations).

I would like to see replies from anyone who disagrees with the last paragraph (including you David!).

[/ QUOTE ]

You are failing to consider the difference between strategic and mathematical mistakes that Xhad mentioned in his post. As a result, you are entirely wrong that it is impossible for both players to play a hand mistake free. If I have AA and I know that my opponent has KK, then I am going to bet. No FTOP mistake so far. If I have KK and my opponent just pushed all in preflop, ordinarily, I would call for the reasons you mentioned. My hand is better than the range I put him on. That would be a FTOP mistake if he had AA. However, if I knew he had AA, then I would fold. That would be the mathematically correct play unless I was BB and he (or I) had a very small stack. While it isn't likely for someone to lay down KK in this spot, it would be the correct play. If BB lays down the KK, then neither player has made a FTOP mistake. It wouldn't lead to a very interesting game, and it certainly wouldn't be a profitable game, but it is possible for two players to both make the correct play in a hand.

[pzhon]

Getting your opponents to make mistakes is only part of outplaying them. The other half is to avoid making mistakes yourself.

If you are a 3:1 favorite with a pair versus overcards, and can choose whether to put in 1 or 2 bets on the flop, then you are making a mistake if you choose to put in 1 bet rather than 2.

hi, my first post so please be nice.

the "paradox" arises from your mixing of correct move for specific events with ev for the long run--the right move for the long run may not be the right move for a given event. you can play perfect long-term poker and still get beat, sometimes.