The paradox of making money from opponents mistakes
 

Warning:long post that might make your brain hurt.

Let me first say that I have a conceptual error that is leading me to a paradox. Help me find my way out! This post is written form the point of view of limit HE, but applies in NL or other games as well.

The statemet "over the long haul, you make money in poker when your opponents make mistakes" makes perfect sense to me. A mistake in this context is not refering to table selection or BR management (though that is also true), but instead is refering to putting money into the pot when they don't have the right odds to do so. That may be an imprecise definintion, but there can be no arguing with the mathematical basis of this statement.

So say you are in the BB with Q9. The CO raises with AJ and you call. The flop comes 952r. It is fair to say that at this point you are in good shape against your opponets range. In other words, over many hands with this scenario, you will profit from the money going into the pot on the flop. The specific example is not important, if you don't like it just choose some other scenario where we are ahead of the opponets range on the flop.

So we now have 2 choices: we can donk the flop or check. Sometimes he will check the flop and other times he will raise our donk, but lets zero in on two specific (and probably the most likely) lines:we c/r and he calls and we donk and he calls.

Say we donk; he is making a mistake by calling because he has pot odds of 4:1, but is only about 6:1 to hit his pair. Again, the details of the scenario are not important, just the fact that he is not getting the right odds to call. So when he calls our donk, he is making a mistake therefore we profit.

Now when we check and he bets, his bet is +EV against the range of hands that we have when we call a pf raise in the BB. In otherwords, his flop bet in response to our check is profotable over the course of many hands (because we frequently fold). Again, you could argue that this particular board is actually profitable to us if we bluff enough. If you want to make that arguement, just think up a different board where against our range his flop bet is profitable. The point is that given the information that the villian has, his flop bet is not a "mistake".

When he bets we c/r him and he calls. Has he made a mistake when he calls? He is getting 7:1 to call and he is about 6:1 to hit his pair. So he has not made a mistake.

So in conclusion his when we donk and he calls, he made a mistake. But when we c/r him, niether his bet or his call is a mistake.

Now for the paradox: when we c/r him, he has put 2 bets into the pot when he is behind. When we donk him he has but 1 bet into the pot when behind.

So when he plays "mistake free" poker he loses more than when he makes a mistake! This seems to contradict the idea that we make money from our opponents mistakes!

HELP!!!!

Re: The paradox of making money from opponents mistakes
 

Both players are capable of playing near-perfectly given their opponents range of hands. This is because poker is a game of imperfect information.

Mistakes happen in poker when players make a play that they wouldnt could they see the other persons cards. However, sometimes one players mistake will lead to the 2nd player making a bigger mistake (ie folding the best hand), which interestingly leads to the firsts players mistake not being a mistake at all.


Overall I think you are confusing what appears to be the best move and what is in reality (if you could see both players hole cards and tendencies) the best move.

Re: The paradox of making money from opponents mistakes
 

You did not clearly define 'mistake', then proceed to use a few different meanings in the same paragraph, no wonder your head hurts.

Here is your way out :
Sklansky makes it very clear that the Fundamental Theorem of Poker refers to one very specific type of mistake - playing your hand differently from the way that would've been mathematically correct if you were to know your opponenet's cards.

In the first instance, that's the definition you use - It is incorrect for AJ to call, as if he is to know that you're not bluffing and his ace-high is no good, and assuming no implied odds (are you check-folding the turn if jack hits ?)

Then you perform a back-flip somersault and claim that if you were to check, his bet is not a mistake, because it is +EV against your range of hands. May be so, but you are using a different definition of mistake now. The fact is, here and now, against that Q9 that you have, his bet is a mistake.

Cezar

Re: The paradox of making money from opponents mistakes
 

His error is putting 2 bets into the pot when he could've checked and put 0 bets into the pot. What you have done by c/ring him is to magnify the effect of his initial mistake. However, he is correct to call the second bet anyway, because of odds.

To put it another way, if he checks he gets infinite pot odds. Even after you check-raise him, he is only getting whatever it is you said he was getting (7:1)? Infinite > 7:1, so he still got a worse outcome. Nevertheless, he should now still call according to FTOP. If he folds here, he's just made another mistake.

It's not really a paradox at all.

Re: The paradox of making money from opponents mistakes
 

you ask for the conceptual error that led to your paradoxical conclusion. here it is:

in the first and third situation you judge the play based on what the bb holds while in the second situation you judge the play based on what the the bb could hold. should the play on the checked flop really be to bet knowing the bb has a pair of nines?

Re: The paradox of making money from opponents mistakes
 

I'm not sure what the hell donking is exactly, but it appears to be some kind of bet, so I'm going to assume that's it. Also, he is not a six to one dog after the flop, he's a three to one dog, so the call there on the flop isn't really that bad, if it's incorrect at all. And your paradox stems from the fact that he IS making a mistake putting in 2 bets on the flop. It's just that with the information available to him, it seems right to bet. He is playing sound poker--no decent player makes a mistake thinking it's a mistake (unless they're on tilt)---but you have more information than he does with regards to ranges of hands and if you are good or not. The FTOP is about mathematical mistakes based on the cards you both have (and here you have more accurate and detailed information, which gives you the edge). The psychology of poker is geared to trying to induce your opponent to act in a manner contrary to how he would if he knew what you had. In this case, while given the information he has, betting is the right move, we know that checking is better. He did not play poorly, you just played BETTER. You can be the second worse player in the world and still make money. If, on the other hand, he checked behind you or called your flop bet, hit the turn, and then got money from you on the turn/river, he would have outplayed you, since your mistake would have cost you more overall than his initial mistake. Or if he bluffd you off your hand on the turn or river, then he would have REALLY outplayed you. A "mathematical disaster." Poker is not about playing good or bad, it's about playing better or worse. Here, let me repeat that.

Poker is not about playing good or bad, it's about playing better or worse.

The fewer plays you make which are different than you would have if you knew what your opponent had (and exactly how he would play in any given situation), the closer to optimal you play. And of course, you also get closer to optimal if you can induce him to make more mistakes in that regard.

Re: The paradox of making money from opponents mistakes
 

I would rather put it like this and before I proceed, I must point out, that I assume, that if Hero donks on flop, then villian with his AJ at most will call the bet.

Villian is getting about 1 to 4 pot odds and is about 1 to 6 to hit his overcards .. so even if Villian was just called and not raised, there is no question, that it is a mistake to bet the flop.

Now we image that Hero checks the flop and after villian bets, our Hero turns over his cards show Villian that he is a dog and raises, thus completing his checkraise scheme

But lo and behold, Villian is getting the right price to call .. what gives ?? .. Can checkraising really be a mistake in this situation ??

No ... the point is, if Hero donks and is called ... or if Villian bets and Hero just called him, then Villian has taken on potodds 1 to 5 on a wager, where the true odds are 1 to 6 against Villian. That is a bad bet, and Villian is losing money over the long run.

But the checkraise is a clever thing, because even though Villian realizes that he is behind, he also knows that he has to stay in the race because for one more bet, he is getting the proper potodds

Has our hero made a mistake, you ask (Tension is building)

No look at what has happened, Villian has been sucked in by hero and milked dry

You see the race is still the same, Villian is 1 to 6 to win, but what you have done is made Villian accept a wager, there he lays to bets against the four bets in the pot and your two bets. That is odds 2 to 6 or as we prefer to say 1 to 3.

And that is a worse bet than if hero just called Villains flop bet.

So no paradox to be found here .. I would rather say, that you have stumbled on some of the beauty of poker.



Now one last thing comes to mind, what if we assume, that if Hero donks then Villian would realize, that he is behind and not getting the proper odds and just lay his hand down, would Hero prefer taking the pot there or is it better to go for the checkraise (yes, we know villian to be an agressive player, that always bets on the flop, so we are totally ignoring, that Villian might check and take a free card)

Now if we bet on the flop and Villian lays down his hand, we have won a pot of five bets, three of these belonging to hero allready, so profit is two bets.

Now if we look at the wager after the checkraise, then the pot is eight bets, and Hero vill win it five out of six times, so that equals (5/6)x8=6.67 and since Hero has put four bets in the pot himself, then profit is 2.67 bets, which is larger that the above scenario.

All in all, there is no paradox here, and given that Hero is sure that Villian will bet the flop, then it is good play.


For simplicity I have not looked at turn and river play, I doubt that it will lead us to a different conclusion (namely that checkraising is a bad idea, and we should look to win the pot right away, but then again, if anyone is up for the challenge, please look into it, I'm just too tired right now)

And I have also chosen to work with the original posters odds on hitting overcards and likewise also I have for simplicity ignored the dead half bet from the small blind in the pot, since it does not change my conclusions.


EDIT .. Two post have been posted while I was writing this one and so my answer comes with less freshness.

As I read the original post is like this:
TT was simply confused, that it can be a mistake to bet on the flop, but to call the checkraise is not a mistake, and therefore questioning if checkraising is powerfull, since Villian has odds to call the raise.

The discussion simply assumed that Villian would bet on the flop and thereby make a mistake.


There really isn't that much psychology in all of this as I see it, if we look at the rest of the play is pretty straight forward, AJ is a good hand against a random hand, so Villian value bets and given the range of hands that Villian could be raising with our Hero is getting proper potodds to call (Since he is on BB and only has to lay one bet against a pot already containing aprox three bets)


So the psychology part is narrowed down to whether Villian will bets the flop after Hero check or simply check. This is no easy answer, it depends on the playertype, that Hero is, I assumed that he was aggresive (Tight aggression is The Main Man here on 2+2 according to Dr. Al), but you can assume differently

What one could do given the time is to model different player types and attach to them a probabillity, that they actually will bet (Give The Maniac a prob of 100% that he will bet, reraise and then call when hero caps it .. and give the Mouse a prob of 100% that he will just check) and then run some computations .. and yeah .. you might as I said involve turn and riverplay as well (allthough this really gets complicated and just might not be feaseble and also might be quite uninteresting)

So my point is (as stated be zillions of pokerplayers before me) go for the checkraise against aggresive player and against a mouse, just bet and take down the pot on the flop instead of giving a free card.

That is my way of taking care of the psychology in the game .. attaching probabillies to different outcomes as a way of modelling different player types, but I am one of those people, who believes that psychology can be expressed with math, in my view it comes down to introducing different distributions of probabilly to different players types .. and in the end unique distributions to every player (Somewhere in these forums there is a discussion about a Mike Caro article, where he attacks math oriented known pokerplayers which many suspect to be well respected 2+2'er, stating that poker is about psychology and math is way overrated)

That thread is so crowded, that I have given up posting there and instead I have jamming it in here for some obscure reason, I do not expect many reading this, since this post really is not much out of the ordinary, and since it was posted many hours ago I suspect that many has moved on since

But IMHO Caro is mistaken, poker is about math .. it is about statistics .. when you get to know people, you learn there patterns, and so you change the distributions that diffine your picture of them .. (He raised his left eyebrow, chances are 90 percent that he hold a premium par) ... you then adjust it as time goes by.. (Ups he noticed me noticing his eyebrow, so next time it proberly signals a bluff)

While math of this type is very heavy to handle, and in the heat of the battle inpossible to handle, I believe that sitting at home going thru small simple computations like these can improve your poker

Poker is much about being observant and noticing patterns ... and while most of this comes at a subconsious level and simple gives you intuition and a feel for the game, basically it is just math

The hard part is sorting which patterns are relavent and which are not. If your opponent scrathes himself it needs not be a tell, he could just have an inch.

Or more typical to online poker, how often does opponent leadbet ... is it probable that he only does this with strength or is he betting to often .. or is he a starkraving maniac ... now if I reraise him, what is the probability, that I hold the better hand ?? (and what if he then reraises me, what is the probability that he now really has strength against him being a maniac, that defends his momentum)


The above is psychology, but since it can be observed given a large enough sample of hands, it does tage a psychiatrist to get into an opponents head

A good player shifts gears, he knows that you have a picture of him, so given the distrubution he figures you have assigned to him, he changes the probabities of his actions, so that you start misreading him (your math is not working) and making mistakes, but then again as the sample grows further, you realize that your distribution is fawlty and you adjust.

Simply in theory, but in praxis we just call it psychology and look away from the math, since the math is difficult and impossible to handle in everyday life


Damn I write too much .. but it is late at night, and my brain has gone haywire beyond overload, sorry about that

Re: The paradox of making money from opponents mistakes
 

The simplest way to put it:

You make and lose money due to mathematical mistakes. Mathematical mistakes are actions that are incorrect according to your EV if you knew your opponents' cards.

A strategic mistake is an action that is likely to produce a mathematical mistake.

Betting the AK in your scenario is not a strategic mistake because you are a favorite against the blind's range of hands, but it is a mathematical mistake.

Re: The paradox of making money from opponents mistakes
 

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.

Re: The paradox of making money from opponents mistakes
 

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.

Re: The paradox of making money from opponents mistakes
 

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.

Re: The paradox of making money from opponents mistakes
 

[ 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.

Re: The paradox of making money from opponents mistakes
 

Excellent post Gelford, although I think you have some math errors.

[ QUOTE ]
Now one last thing comes to mind, what if we assume, that if Hero donks then Villian would realize, that he is behind and not getting the proper odds and just lay his hand down, would Hero prefer taking the pot there or is it better to go for the checkraise (yes, we know villian to be an agressive player, that always bets on the flop, so we are totally ignoring, that Villian might check and take a free card)


[/ QUOTE ]

You hit on what got me asking these questions in the first place. Given what we know about Villina's range, would we prefer to c/r and have him stay in the pot or donk and have him fold?

Can some no limit tourny players chime in on this point? I think it comes up often in no limit where you have the option to bet out and get a guy to fold, or c/r a guy all-in and get a guy tied to the pot because he is getting the right odds to call. The c/r will win you more chips over the long haul, but you will also be eliminated from the tourny more often.

As for limit HE, over the long term we will make more money when we c/r and he stays in, eventhough he is not making a mistake (by any defintion) when he calls the c/r.

Re: The paradox of making money from opponents mistakes
 

[ 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.

Re: The paradox of making money from opponents mistakes
 

[ QUOTE ]
Quote:
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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!).

Re: The paradox of making money from opponents mistakes
 

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????

Re: The paradox of making money from opponents mistakes
 

Nolimit is well nolimit .. given that you suspect the flop just missing Villian your raise would be such, that you would offer him odds, that are wrong with his draw, so that he would make yet another mistake by calling.

Harrington has alway advocated that you should make to large bets in to limit, because you just push villian out, it is better to offer Villian potodds, that while they are wrong they are still tempting, you want him to make another mistake.

Here Hero could make a mistake by paying villian off big time in the end if Villian hits thus giving Villian impied odds to hit.

Re: The paradox of making money from opponents mistakes
 

[ 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.

Re: The paradox of making money from opponents mistakes
 

[ QUOTE ]
Quote:
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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!).


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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.

[/ QUOTE ]

Good point, I should have been more precise. I was speaking of hands that get to SD. There is no way for both players to get to SD playing "mistake-free" in the context of the FTOP (save split possibilities). There is the possibility for both to play "error-free" and get to SD.

Agreed?

Re: The paradox of making money from opponents mistakes
 

[ QUOTE ]
Good point, I should have been more precise. I was speaking of hands that get to SD. There is no way for both players to get to SD playing "mistake-free" in the context of the FTOP (save split possibilities). There is the possibility for both to play "error-free" and get to SD.

Agreed?

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Again, refer to Xhad's distinction between strategic and mathematical mistakes. It is possible to get to showdown without making strategic mistakes, according to Xhad's definition (I think), but it is typically not possible to get to showdown in a non-split pot without someone making a mathematical mistake. Someone has to have a losing hand, and calling with a losing hand is a mathematical mistake.

Of course, it is possible to get to showdown without making any mathematical mistakes if one player is hopelessly shortstacked and the money goes in before the river. If player A has an equity edge, he can bet correctly. If the all-in bet is small enough, player B may still have sufficient pot equity to justify a call, even if he does not currently have the best hand. In this case, the hand could make it to showdown without either player making a mathematical mistake.

Re: The paradox of making money from opponents mistakes
 

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Quote:
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Good point, I should have been more precise. I was speaking of hands that get to SD. There is no way for both players to get to SD playing "mistake-free" in the context of the FTOP (save split possibilities). There is the possibility for both to play "error-free" and get to SD.

Agreed?


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Again, refer to Xhad's distinction between strategic and mathematical mistakes. It is possible to get to showdown without making strategic mistakes, according to Xhad's definition (I think), but it is typically not possible to get to showdown in a non-split pot without someone making a mathematical mistake. Someone has to have a losing hand, and calling with a losing hand is a mathematical mistake.

Of course, it is possible to get to showdown without making any mathematical mistakes if one player is hopelessly shortstacked and the money goes in before the river. If player A has an equity edge, he can bet correctly. If the all-in bet is small enough, player B may still have sufficient pot equity to justify a call, even if he does not currently have the best hand. In this case, the hand could make it to showdown without either player making a mathematical mistake.

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Good point about the non-mathematical mistake SD in all-in situations.

I still don't agree with your first paragraph. Or more precisely I think that saying that betting the AJ in my original scenario is a " mathematical mistake" is a purely academic statement that has no practical application to actually playing poker.

So I guess my question is: who cares if the guy betting his AJ is making a mathematical mistake? It is not any type of error, mistake, or fopa (spelling?) based on the information he has at the time.

The definition of a "mathematical mistake" that Xhad stated involves being able to see the other guys cards. In any real game this isn't true. So if you want to call betting that AJ a "mathematical mistake", I'm fine with that, but I don't see what application that has to actually playing the game.

To me the issue here isn't that the AJ guy is making a mathematical mistake, the issue is that we have an information advantage over the other guy.

We know that our hand is better than his over all of his possible range. We also know that he will think his hand is better than our range of hands when we check to him. Because of this we know he is likely to bet when we check to him. Therefore we are exploiting our information advantage over him to get two bets into the pot instead of one.

When he calls our bet, he knows that against the range of hands we would call a pf raise with and then c/r him with he has the right odds to call. So again he has not made an error of any type based on the information available to him.

The fact that he is actually beat in this particular hand is inconsequential. What is important is that we will profit over the long haul.

Re: The paradox of making money from opponents mistakes
 

You're missing the point. You are trying to minimize mathematical mistakes by playing correctly. However, since you can't see anyone's cards, you can't eliminate them altogether.

You're right that you can't know that betting the AJ is a mathematical mistake this time. But the reason you bet it is that it is less likely to be a mathematical mistake than checking based on the limited information that you have.

Again, strategically correct plays are the plays that reduce the likelihood and impact of your mathematical mistakes while increasing the likelihood and impact of your opponents'.

Re: The paradox of making money from opponents mistakes
 

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.

Re: The paradox of making money from opponents mistakes
 

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.

Re: The paradox of making money from opponents mistakes
 

Only one of the following two scenarios is true:

(1) He knows you have a 9, and is therefore incorrect to call a bet or to make a bet

(2) He doesn't know you have a 9 and must therefore play against your range of hands in which case both betting and calling are close but you must start factoring in things like: what he thinks of you, what he thinks you think of him and so on and so forth.

You can't have it both ways.

Re: The paradox of making money from opponents mistakes
 

Because it is a game of imperfect information, poker isn't really about a particular pair of holdings, but rather about things like reading your opponent, and whatnot. Slanksy's fundemental theorem of poker doesn't explicitly adress exploiting opponent weakness.

One way to look at it:
You have to play the player, as well as the cards. You bluff more against weak-tight players who fold too much, and value-bet mercilessly against calling stations. So, what your opponent's correct move is, depends on your tendencies, and vice versa.

Another way to look at it:
The stacks go up, and down, but if a player does to much of one thing or another in a particular situation against a mathematically ideal (maximally exploitative game theoretically perfect) player, then, in the long run, said mistake will be costly.

A third way to look at it:
Since human players are imperfect, your opponent's weaknesses can make actions that would normally be costly profitable.

Re: The paradox of making money from opponents mistakes
 

One thing I would point out is that villain isn't necessarily making a mistake to call. He doesn't have pot odds to call, but likely has implied odds if he catces an A or J on the turn, assuming hero will bet turn or call at least one bet. Villain could also raise hoping to get a free card on the turn if hero bets.