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Calculating Bluff Equity And Breakeven Points


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#41 Actuary

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Posted 13 August 2007 - 06:35 PM

View PostActuary, on Monday, August 13th, 2007, 6:22 PM, said:

Looks like a set up problem, I need to look over what he said
I see wher putting it all on the same side and setting = to 18/46 * 5.5 gives me about 6%; but I'm too thick right now to see what is wrong with mine.************I see now what the "correct" eqation is caculating, I'm not sure what mine is, though, and I need to figure that out.

#42 Shimmering Wang

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Posted 13 August 2007 - 07:40 PM

Here is the formula I reverse engineered from my intuitive model:We have to model it like this:Total Equity = (Current Pot equity) + (Opponent's Pot Equity * Opponent Fold %) - [Bet Cost * (1- Opponent Fold %)]or, in this case, when we're trying to solve for Opponent Fold %Totel Equity = (Current Pot Equity) + (Opponent's Pot Equity * x) - [Bet Cost * (1-x)]where Bet Cost is simply: (Our Chance to Win - Opponent's Chance to win) x Bet Amount [in this case, we're just using the 1 as a bet amount, since we're dealing with BB)To solve this in a break-even bet scenario, simply set the left "Total Equity" side of the equation equal to our current Pot EquityAs an example (because I am bored, and into stuff like this):We've got 1/3 chance of winning a pot on the turn, and the pot is 10BB.Total Equity = 1/3(10) + 2/3(10)(x) - [(2/3 - 1/3) * (1-x)]where x = the chance our opponent folds to a betTotal equity = 10/3 + 20/3(x) - 1/3(1-x)= 10/3 + 20/3(x) - 1/3 + 1/3x= 9/3 + 21/3(x)Total Equity = 3 + 7xSo, set our total equity as our current equity, then solve for x10/3 = 3 + 7xx = 1/21I am the biggest. nerd. ever.Wang

#43 David_Nicoson

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Posted 14 August 2007 - 05:44 AM

To try to acquire some nerd cred, I solved it symbolically to avoid further algebra. I did this wrong a few times before I matched Wang's answer for the original case, so if this is right it's only because I had an existing answer to cheat off of.c = fraction of time villain callsp = pot sizew = fraction of the time we win a showdownEV of checking = EV betting0 = EV of winning outright + EV of getting called0 = (chance he folds)(whole pot - our share) + (chance he calls)(equity change - our investment)0 = (1-c)(p-pw) + c (2w-1)0 = p - pw - cp + cpw + c(2w-1)0 = p - pw + c(2s-1-p+pw) (pw-p) / (2w-1-p+pw) = c p (w-1) / (2w-1-p+pw) = cPosted Image

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#44 Shimmering Wang

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Posted 14 August 2007 - 08:34 AM

View PostDavid_Nicoson, on Tuesday, August 14th, 2007, 9:44 AM, said:

To try to acquire some nerd cred, I solved it symbolically to avoid further algebra. Posted Image
Consider your nerd-cred earned in full. Nice job there, Martin. Hey, didn't you get shot or something?Wang

#45 Frez

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Posted 14 August 2007 - 10:03 AM

Nice work nerds. Sorry I was away for the weekend and didn't get a chance to play.
Progress is made by lazy men looking for easier ways to do things. (Robert Heinlein).

#46 Shimmering Wang

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Posted 14 August 2007 - 10:35 AM

For those of you looking to do this on the fly, it's usually not all that tough for some common situations.Remember, I always think about it like this: "How much does a bet 'cost' me in terms of equity? How much of the pot is my opponent's? How many times can I make a -EV bet until I've basically spent my opponent's share of the pot? If he folds the next time, it's break even."If you're looking to do this in a pinch, you probably can. "Let's see here. I've got a live flush draw, which is 9 outs, which is about a 1/5 shot. There are 4 bets in the pot. 3.2 of those are my opponent's. A bluff costs me the difference in our equity, so 4/5 -1/5 = 3/5 = .6BB. .6 into 3.2 is a little more than five. He's got to fold more than than 1/6+ times to make a bet profitable.""Okay, I've got an Open Ender and a flush draw. 15 outs. I get there 1/3 times. The pot is 7BB, and 4.6666 of those are my opponent's. A bluff costs me 2/3 - 1/3 = 1/3. A third goes into 4 and 2/3... let's see here.... 12, 13, 14 times. 1/15."This make sense to people? After thinking about it, these things obviously become pretty intuitive, but when I'm playing live I do think about stuff like this. Just having the general ability to calculate a quick and dirty EV of a bet is helpful, and you can extrapolate all sorts of uses and shortcuts to different solutions.Wang

#47 David_Nicoson

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Posted 14 August 2007 - 02:36 PM

View PostShimmering Wang, on Tuesday, August 14th, 2007, 12:34 PM, said:

Consider your nerd-cred earned in full. Nice job there, Martin. Hey, didn't you get shot or something?Wang
Yes, but luckily my brain, at least the part of it responsible for doing excel, appears to be intact.
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#48 Actuary

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Posted 14 August 2007 - 05:40 PM

wang you got me excited and nice graph daveI always thought betting was better with less outs, or that ther was a corridor, depending on the lielihood of getting c/r'd too.But I'm intriguedWish my head was more clear

#49 Frez

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Posted 14 August 2007 - 06:20 PM

View PostActuary, on Tuesday, August 14th, 2007, 7:40 PM, said:

I always thought betting was better with less outs, or that ther was a corridor, depending on the lielihood of getting c/r'd too.
This was a super interesting exercise, but the original conditions did state there was no chance of getting raised, and no action to consider on the river. Rigid conditions need to be set to get good answers to a problem like this, however in real life there is that risk of being raised.
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#50 Zach6668

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Posted 14 August 2007 - 06:30 PM

Getting raised when we have a million outs isn't that bad though.Getting raised out of a pot where we have to fold our outs, is bad.You should be betting when you have more outs, more often.
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#51 Actuary

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Posted 14 August 2007 - 07:34 PM

View PostZach6668, on Tuesday, August 14th, 2007, 6:30 PM, said:

Getting raised when we have a million outs isn't that bad though.Getting raised out of a pot where we have to fold our outs, is bad.You should be betting when you have more outs, more often.
But with less outs we need him to fold.Theres less reason to bet when we have more outs,.Until I have more time I'm just thinking off the top of my headAs seen in David's graphThe more outs we have the less often villain needs to fold; therefore the less outs we have the more we need him to fold.I'm still mad at myself for my botched EV equationI'm still not sure what I calculated and how it relates

#52 Zach6668

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Posted 14 August 2007 - 07:45 PM

I know Act, I hear that a lot, about betting with less outs.I tend to do the opposite though.Am I a trailblazer? lol
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QUOTE (Acid_Knight @ Monday, March 10th, 2008, 4:15 PM) <{POST_SNAPBACK}>
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#53 Actuary

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Posted 14 August 2007 - 07:57 PM

View PostZach6668, on Tuesday, August 14th, 2007, 7:45 PM, said:

I know Act, I hear that a lot, about betting with less outs.I tend to do the opposite though.Am I a trailblazer? lol
no..lolI think Wang agrees with you; but I'm not sureI think the "problem" is in the phrasing of the question and I am not saying I'm right...for sure...only that it's a mult-vector problem

#54 Shimmering Wang

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Posted 14 August 2007 - 09:35 PM

View PostFrez, on Tuesday, August 14th, 2007, 10:20 PM, said:

This was a super interesting exercise, but the original conditions did state there was no chance of getting raised, and no action to consider on the river. Rigid conditions need to be set to get good answers to a problem like this, however in real life there is that risk of being raised.
See, that's kinda not true. I set the problem up to limit the amount of math that had to be done to solve it, but you can add in all the other variables, too. We can easily set up a model where our opponent check-raises a third of the time, and then decide how often a bet has to win the pot right there. It's not hard.If the pot is 10BB and we win the pot 1/3 of the time, a bet "costs" us 1/3 of a BB every time he calls. Assume he check/raises half the time. In these instances, it costs us double, or 2/3 of a BB. So, our actual cost is: .5(2/3) + .5(1/3) = .5BBNow we can just plug .5BB in for cost instead of 1/3BB. 6 and 2/3 / .5 = 13 and 1/3. He folds the next time, so he has to fold 1/(14 and 1/3) times for a bet to be breakeven. Or a little less than 7% of the time. These things are very adaptable.

View PostActuary, on Tuesday, August 14th, 2007, 11:34 PM, said:

But with less outs we need him to fold.Theres less reason to bet when we have more outs,.Until I have more time I'm just thinking off the top of my headAs seen in David's graphThe more outs we have the less often villain needs to fold; therefore the less outs we have the more we need him to fold.I'm still mad at myself for my botched EV equationI'm still not sure what I calculated and how it relates
Well, I think if you set up an equity function, you'll find that when you own a larger portion of the pot (say, a full third), then a semibluff has less risk (because it's cheaper, and also must be successful less often to show a profit), but also has less reward (because you're earning a much smaller share of the pot, since a big part is yours already.)For example:Say the pot is 10BB on the turn, and our equity is 1/3. Like above, a bet costs us 1/3 of a BB, and must only be successful 1/21 times when he never check/raises. {Math: 1/ (6.666/.333 +1) }If we have no equity, on the other hand, he must fold 1/11 times. Let's assume he folds a full HALF of the time. When we're drawing very live with a full third of the deck on our side, and he folds half the time, we earn half of his share of the pot, or .5 x 6.66666. So we earn 3 1/3 BB.When we're drawing stone dead and he folds half the time, we earn HALF OF THE ENTIRE POT (because all of it is his and none of it is ours), or a full 5BB.So that's the rub. If you bet when you're drawing live, it's cheaper and more often earns you SOME profit, but if you bet when you're drawing dead it's much more expensive, but earns you much more profit each time it's successful.Ya heard? David, you should put together a graph that shows how earned equity increases when a bluff is more or less successful, depending on your number of outs. I'm not EXACTLY sure what I'm looking for.Let me babble a little...A graph which contains the variables: Opponent Fold Rate (total rate, not breakeven)Turn Pot Equity And then we need to be able to determine how this affects: Total Equity gained at each fold %age and Turn Equity %age. I think I could figure out how to do this, but I'd have to sit down in front of a whiteboard and just start drawing graphs and shit. You're much better suited for this than I am.Wang

#55 Shimmering Wang

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Posted 14 August 2007 - 10:02 PM

I guess what I was trying to say earlier is this:If your opponent is folding x% of the time (where x% is often enough to show a profit), it's more important to bet when drawing dead than it is to bet when you're drawing live. BOTH cases are profitable, but it's much more profitable when you're drawing dead.A corollary:An opponent must fold MUCH MORE OFTEN when we're drawing live to show the same Earn Rate (ie, increased equity) than when we're drawing dead. We show SOME profit when we semibluff with a lot of outs much more often, but the marginal returns for every extra fold are lesser compared to when we're drawing to fewer outs.Y'all heard?Wang

#56 Ouch-8s

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Posted 14 August 2007 - 10:14 PM

View PostShimmering Wang, on Tuesday, August 14th, 2007, 10:35 PM, said:

Well, I think if you set up an equity function, you'll find that when you own a larger portion of the pot (say, a full third), then a semibluff has less risk (because it's cheaper, and also must be successful less often to show a profit), but also has less reward (because you're earning a much smaller share of the pot, since a big part is yours already.)
but aren't you also increasing the size of the pot, making it harder for him to fold when you do hit and causing him to pay off when you make the (earlier agreed upon not to happen) value bets on later streets?
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#57 Ouch-8s

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Posted 14 August 2007 - 10:15 PM

View PostShimmering Wang, on Tuesday, August 14th, 2007, 11:02 PM, said:

If your opponent is folding x% of the time (where x% is often enough to show a profit), it's more important to bet when drawing dead than it is to bet when you're drawing live. BOTH cases are profitable, but it's much more profitable when you're drawing dead.
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#58 Actuary

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Posted 14 August 2007 - 10:37 PM

Wang, you had me at "equity"so less outs, be more apt to bet the turn (I was on track)- .. and I'll guess the more we put him on a c/r the more we should CHECK with less outs. (like we say, check more vs tricky players)

#59 Shimmering Wang

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Posted 14 August 2007 - 11:54 PM

View PostOuch-8s, on Wednesday, August 15th, 2007, 2:14 AM, said:

but aren't you also increasing the size of the pot, making it harder for him to fold when you do hit and causing him to pay off when you make the (earlier agreed upon not to happen) value bets on later streets?
Yeah. This is an added benefit of betting when getting called is -EV.If he's 25% more likely to call a river bet when we make a hand -- which we will do 1/3 of the time -- our turn bet improves our equity by .25 BB the 1/3 times we do get there. That's an equity boost of about .08BB, which we could really "write off" from the cost of the turn bet. So, instead of costing us 1/3 of a BB, it really costs us 1/3 - .08 = .25BBObviously this will affect our cost-benefit on a turn semibluff calculations (ie, the "how often does a turn bet have to win the pot to be successful?" question this thread is addressing). So now, instead of costing 1/3 BB everytime the turn goes bet/call, it REALLY costs .25BB. Which means we have to be successful with a turn semi-bluff even less often to break even or show a profit.But this is a very extreme and simple example, since we're still ignoring a lot of factors, like the likelihood of a check/raise, etc. Plus, increasing the size of the pot by 2BB on the turn rarely means we'll improve the odds of getting called on the river by a full 25%. It'll be much lower, and usually negligible.

View PostActuary, on Wednesday, August 15th, 2007, 2:37 AM, said:

Wang, you had me at "equity"so less outs, be more apt to bet the turn (I was on track)- .. and I'll guess the more we put him on a c/r the more we should CHECK with less outs. (like we say, check more vs tricky players)
Not necessarily. The only real issue is: "Is betting here +EV?" Doesn't matter HOW positive the EV is, really. If we show a quarter-bet EV boost by betting when we have a ton of outs, and a 2BB boost when we have few outs in the same situation, we should still bet in both cases. It's just that when we're semibluffing with a lot of outs, our opponent must fold much more often to show the SAME equity boost when compared to semibluffing with few outs. It's also likely that we're rarely going to LOSE equity by semibluffing the turn with a lot of outs (like when he only has to fold 6% of the time for us to break even), but we may find ourselves repeatedly betting the turn with 8 outers where we're getting called -- or raised -- and making bets that are long-term losers.WangPS- I'd like to add that this has easily been the strangest series of posts I've ever made in the LHE strat thread. Do any of you guys remember being this analytically solid in the past? I think quitting the booze has added like 15 IQ points. Since when did I get all... mathy and shit? Dig it? Betting may be

#60 Ouch-8s

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Posted 15 August 2007 - 12:08 AM

View PostShimmering Wang, on Wednesday, August 15th, 2007, 12:54 AM, said:

I think quitting the booze has added like 15 IQ points. Since when did I get all... mathy and shit? Dig it? Betting may be
Is this deliciously ironic?
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