This is a discussion on From the Vault: Clarkmeister's Theorem Examined within the online poker forums, in the Cash Games section; So since Black Friday, I've played poker a total of once (in Atlantic City). However, back when I was playing everyday, I would regularly start 


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From the Vault: Clarkmeister's Theorem Examined
So since Black Friday, I've played poker a total of once (in Atlantic City). However, back when I was playing everyday, I would regularly start writing articles about various topics I found interesting. Some of them I posted, but others I just saved in a word document to work on later.
This article is one of the ones I saved and forgot about. But since the topic of Clarkmeister's Theorem has come up again, I thought I'd go ahead and post this article that's been rotting in my documents folder since March 2011. =====================Clarkmeister's Theorem Examined===================== Okay, so recently we've had a bunch of (http://www.cardschat.com/f50/10nlhefullringkkoop193497/) threads (http://www.cardschat.com/f50/10nlhe6maxlinecheck193562/) where (http://www.cardschat.com/f50/10nlhe6maxtpgkscary193462/) the hero finds himself in a "Clarkmeister" type situation, where you only need 1 card (or one of many cards) to make a nearnuts hand. Lots of people have been saying "Clarkmeister theorem, bet!" without really knowing why (myself included). So quick background: Quote:
So anyways, why does this theorem work on the river, and why does this work with any two cards? As a Bluff First, lets think of the bet as a bluff. In order for a bluff to be profitable, we need our opponent to fold a certain percentage of the time. For example, if we bet $1 into a $5 pot, we're risking 1 to win 5. EV = 5*F  1*(1F), solve for F = 0.17 or 17% folds for a bet of $1 into a $5 pot to break even. So on a 4 flush board (or a 4straight board like 2KTJQ), we can usually get enough folds for that bet to be profitable. So obviously, betting our complete air on a Clarkmeister board is profitable. In conclusion: We have fold equity on a 4flush or 4straight board. As a Value Bet Now, in order for a value bet to be +EV, we have to be ahead of the range that calls us. Clarkmeister's theorem asserts that if you were going to check and call a river bet, that you are better off bet/folding yourself for the following reasons: 1) If raised, you can safely fold knowing you're beat. This is because it is very easy for us to have a nearnut hand and our opponent is afraid of this fact. 2) More made hands call the river than bet the river themselves. Because your opponent is in position, he will very often take a free showdown with many hands he feel he cannot get value with on this board. So imagine this scenario, where your opponent has the nut flush, or a set on a 4flush board (just those two hands). If we have a 2nd nut flush, obviously bet/folding will show a larger profit than check/calling if our opponent bets all his nuts and checks all his sets. And this is how opponents typically play. When they are one street away from showdown, and they have a decent hand, they typically will check and see a showdown. But since they're only 1 bet from showdown, they'll often call one bet with some of those hands because they get to see a showdown. Being close to the end of the hand and getting to see your cards often affects people's decisions on the river, and you should exploit their desire to see a showdown. So obviously, if we have a near nut hand like a big flush on a 4flush board, we should bet. In conclusion: When you are 1 betting round away from showdown, your opponent will call with more hands than he will bet himself for value. The In Between Range A lot of the controversy on what this theorem implies comes from hands that are "in between". And by "in between" hands, I mean hands that can beat many hands that weren't completed by the scary river card. If we have 6♦6♣ on a 6♥J♠T♠2♠K♠ board, what's the best line? Well the theorem implies that the margin for error in betting is smaller than that for checking. If we bet, irregardless of cards, we know that it will be +EV as a bluff alone. So we KNOW this line is +EV. Don't get caught up in "what worse calls, what better folds", if we bet $X on the river and get X% folds, we always profit, and that X% is almost always obtainable given the board texture. Also, our opponent could call with K♥J♥, or fold A♥Q♥. Both of these results would add to our expectation over checking (since villain likely checks back both of those hands). Now yes, check/calling could possibly be more profitable with these in between hands. Our opponent could bluff X% of the time, and given the pot odds we get, we could pick off those bluffs and profit more than betting ourselves. However, the problem with a Clarkmeister board is that villain very rarely has pure air. Since just about every draw got there on a Clarkmeister board, your opponent does not typically have many hands he can turn into a bluff. For a player to bluff on a Clarkmeister board, he'll typically have to turn hands like top pair or two pair into a bluff. Therefore, we know bet/folding is profitable, and we know our opponent does not have the typical hands (like broken draws) that he will turn into a bluff. Lastly, even if we did know our opponent had some broken draws, we're not certain that he will bluff them more often than bluff catching with his hands that have showdown value. In order for check/calling to be more profitable than bet/folding, we have to be very accurate on those 2 assumptions. And there's a lot of margin for error there. Thus, bet/folding is typically the line we choose since its reliably profitable. The Annoying Caveat Now, if you click the link to that evil forum the mods don't like me linking to, you'll notice the Clarkmeister post was originally created in the limit hold'em forum. So what does this mean? Well playing limit hold'em, you see showdown far more often. Calling the river for 1 bet with any hand that has some showdown value is standard, since the pot is usually large compared to the size of the bet. Also, our opponents are generally less likely to bluff playing LHE than NLHE, because you don't have very much fold equity (because of that tiny bet size). So therefore, Clarkmeister's assertion that many nonflush hands call a bet on the river is probably true for limit hold'em, but may not be true for NLHE where the bet size is a ton larger and our opponents are used to folding a lot more. Additionally, NLHE allows for a lot more bluffing since its easier to get people to fold when you bet out for the pot size rather than 20% of the pot. Therefore, its my opinion that this theorem shouldn't be applied to 100% of rivers playing NLHE. So you should consider the following as reasons to *not* apply the Clarkmeister theorem: 1) Against overly aggressive opponents. 2) In pots where its possible for our opponent to have many "air type" hands in his range. 3) Our hand is on the weaker end of the "in between" range. For example:  Hero: $25.00 MP: $34.82  Fish with a high AFq Pre Flop: ($0.35) Hero is OOP with J♠ 8♦ Hero raises to $0.75, MP calls $0.75, 4 folds Flop: ($1.85) 7♥ 8♠ 9♥ (2 players) Hero bets $1.25, MP calls $1.25 Turn: ($4.35) 2♥ (2 players) Hero checks, MP checks River: ($4.35) 4♥ (2 players) Hero ??  In this hand, our opponent likely has a lot of air in his range (Tx, Jx, 6x, random overs & ace highs). In this scenario, I would check/call the river. We don't really benefit from nonflush hands calling us because those hands likely still have us beat (9x, 8x with a better kicker, TT). And while betting this board is likely +EV, I would say that picking off a bluff is likely more +EV in this specific situation. So what have we learned? I'm not sure I've learned anything other than the fact that "it depends" is still the answer to every question. The more I research Clarkmeister's theorem, the more I start to believe it isn't universally applicable to NLHE. The three key points of Clarkmeister's theorem are still universally true though: 1) We have a lot of fold equity on scary boards like 4flush/4straight boards. 2) Its hard for most of our opponents to raise many hands other than the nuts on these boards. 3) Typically your average player is more likely to convince themselves that you're bluffing, and call with hands that are usually strong (top pair, two pair, sets, straights) than they are to turn showdown value into a bluff when they're just one check away from a showdown. If you want to read more on the Clarkmeister's Theorem, I could link a billion threads, but this post really contains everything: Clarkmeister Theorem: a Review And Discussion (http://forumserver.twoplustwo.com/35/microstakeslimit/1700postclarkmeistertheoremreviewdiscussionlooongbuthassummary430765/) 
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Nice post. I have heard this before, but never explained so far in depth, or at least not in this way.
Isn't there a different name for this on 2p2? I looked it up, and Clarkmeister's theorem is on 2p2, but I thought there was another, different name for it? Or is it just something similar? I can't find it. Maybe I just forgot it was called Clarkmeister's theorem. The name isn't that important after all, it's the idea that's important. 
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Yeah, idk. I wasn't really a fan of this post when I originally wrote it because I thought it was a really long winded way to say "When the board is scary, being the first person to bet it is a profitable situation".
And it was hard to convey that whether we want our opponent to fold, or if we want him to put money in the pot with hands we beat, the best way to accomplish both goals is to bet. So I'm not super happy about how this article turned out, but I figured, meht, its 95% done, just post it. 
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I should probably do some maths of poker, I've literally never done any EV calculations at all.
bet size  "odds"  % to be profitable 50%  50/150  33% 60%  60/160  37,5% 70%  70/170  41% 80%  80/180  44,4% 90%  90/190  47,4% 100%  100/200  50% this is right for the betsize calc and % needed to work right? 
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re: Poker & From the Vault: Clarkmeister's Theorem Examined
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I don't like this theorem at all. Any generalizations which completely ignore villains range should never be used as rules to use 100% of the time.
Say we hold a weak flush, do we bet if: Villains range is polarized to nonflushes (bluffing hands) and strong flushes? Villains range is entirely stronger flushes? 
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re: Poker & From the Vault: Clarkmeister's Theorem Examined
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Example: We're in position on the river, pot is $100, we have 2nd pair and think that if we bet our opponent will call with his entire range which we have 40% equity against. He will never raise our river bet, only call. We decide to bet $10 (small to illustrate point). The EV of just the river bet is $2 ($Bet(%called)(winning%losing%) or (($10*1)(.4.6)). Our overall EV of betting $10 here is $38 (.4*12010). The true EV of checking is better at $40 (100*.4). 
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@Deco: Generally its hard for villains to have such specific ranges on the river, especially given how much the river will change the value of villain's hand. Your "bluff into flushes" example, would typically require villain to let us know in some way that he had a strong flush before the river. And its hard to get there with a bluff, if you *know* he's got a flush. 
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The other scenario I mentioned is very common. Where villains range is polarized to nonflushes or flushes and our flush is weak. A check/call is almost always unknown vs non herocalling or fishy villains. 
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I just said he calls with his entire range because I didn't want someone coming back with well what if he folds his weak hands or bluff raises. It was just there to make it as clear as possible. Clearly it doesn't matter. He could call with half his total range but we still need to be ahead of greater than 50% of the hands he calls with to have a profitable value bet. 
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Deco, I have to agree with C9 here because I cannot see very many cases where villain's range on river is always going to be huge flushes when it's the fourth of a suit. To believe this you would basically be saying that people never get to the river on 3flush boards with worse than a set. Also there are plenty of people who will find a hero fold of a Jhigh/Qhigh flush or worse and surely people who will fold worse. The basic idea is this: There are plenty of hands where people have reasonably wide ranges and you get to a 4flush river and the action goes checkcheck. A huge % of the time a bet by either party will take down the pot, it's an obvious spot where we should polarize our betting range. 
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re: Poker & From the Vault: Clarkmeister's Theorem Examined
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If we have 4s5s flop a flush, bet/bet the river and turn and then the 4th flush card comes in we are not going to get value from betting if villain isn't a station or mad paranoid as nonflushes will fold and only better hands will call. Based on villain and his range we should check/call or check/fold. 5high is an extreme example, I rarely find value betting any 1card flush smaller than queen high vs nonfish on the river OOP. Bear in mind to get to this position we'll have likely fired two barrels already, we look strong, we repped a draw, the pot is big. This situation is further amplified if the board has paired. The worst thing about this rule is that disregards all prior hand reading/ranges is it applies to the river, the street were we have the most information and the best read on villains range. 
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EV = F*P + W*(1F)*(P+B) + (1W)*(1F)(B) Where P is the pot, B is what we're betting, F is the fraction he folds, and W is the fraction we win at showdown. That's why bluffs are profitable, because the first term in the EV equation is bigger than the EV when we're called terms. Your example doesn't account for this because he never folds, so you're neglecting the F*P contribution to your expected value. I'll get to Deco after work. 
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When villain folds, we win $100. This happens half the time, so 0.5*100 is the first term in that equation. If we bet $50, and lose, we lose $50. This happens one quarter of the time. 0.25*(50) is the next term in the EV equation. The other quarter of the time, we win $150. 0.25*150. EV = 50  12.5 + 37.5 = $75 Am I missing something? Because this bet seems hugely +ev, not break even like you suggest... Hell, just look at the EV chart Jurn posted. If you bet half pot, and get 33% folds, a bluff is profitable. Now, you're giving me 50% folds AND I win at showdown half the time! HUGE PROFIT! 
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WV, Deco, or whoever. What's your WTSD when "saw river = true"? Run that filter for me if you could. And take out spots where you're all in on the turn.
I bet its pretty high, and thats prolly what the Clark theorem is getting at. "F" in the EV calculation is abnormally high, and thus most bets are profitable. 
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re: Poker & From the Vault: Clarkmeister's Theorem Examined
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To me there's more assumptions to be made with check/calling than bet/folding. If we have sufficient fold equity (which on these rivers we almost always do) betting can never be EV. However, check/calling has the potential to be a negative EV play, and check/folding is obviously always zero. Anyways, I'll follow up with a graph(s) that I think will explain things better when I get to a real computer. 