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Old 12-13-2005, 03:09 AM
TTChamp TTChamp is offline
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Default 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!

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Old 12-13-2005, 04:51 AM
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Default 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.
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Old 12-13-2005, 12:03 PM
Cezar Cezar is offline
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Join Date: Apr 2005
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Default 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.

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Old 12-13-2005, 07:57 PM
cpk cpk is offline
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Location: Seattle, WA
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Default 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.
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Old 12-13-2005, 08:26 PM
StinkWater StinkWater is offline
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Default 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?
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Old 12-13-2005, 08:41 PM
college kid college kid is offline
Join Date: Dec 2003
Posts: 40
Default 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.
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Old 12-13-2005, 09:29 PM
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Default 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
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Old 12-13-2005, 09:44 PM
Xhad Xhad is offline
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Join Date: Jul 2005
Posts: 205
Default 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.
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Old 12-13-2005, 11:27 PM
AaronBrown AaronBrown is offline
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Join Date: May 2005
Location: New York
Posts: 505
Default 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.
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Old 12-14-2005, 12:12 AM
Posts: n/a
Default 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.
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