Jump to content

Calculating Bluff Equity And Breakeven Points


Recommended Posts

There are 3 cases:1. Check / Check2. Bet / Call3. Bet / FoldQuestion - How often does the fold in case 3 happen to make the bet you make in cases 2&3 the same value as not betting, which is case 1.1 is easy to figure. We win 5.5 BB in 18/46 cases, or an EV of 2.15BB, and lose nothing (since we made no more bets) in the other 28/46 cases.2. We win 6.5 BB in 18/46 cases, for 2.54 BB, but lose the 1BB we bet in 28/46 cases, or -0.60BB, for a total EV of 1.94BB.3. We win 5.5BB every time in this case.So, how often is case 3? Call it X, and solve 2.15 = X(5.5) + (1-X)1.94.Distributing X gives us: 2.15 = 5.5X + 1.94 - 1.94XSimplifying: 0.21 = 3.56XX = .21/3.56 = 0.059.6% of the time.Peace,Opie
Ship it ALLLL to Opie. You guys are banging your heads against the wall, and this clown with 75 posts makes you all look afoolHere are the two ways I came up with to figure the answer. Our answers might be slightly different, due to rounding error.This is the way that's really intuitive for me:There are 5.5BB in the pot. 18/46 of that is ours, and 28/46 is our opponent's. That is to say:Our Equity = 2.15BBOpponent's Equity = 3.35BBWhen we're bluffing, we are ONLY bluffing at our Opponent's share of the pot, right? Part of that money is ours. So when I think about it, I thinK- What am I bluffing at?- How much does a bluff cost me?We've already figured the first part, so let's figure the second. How much does it cost to fire a bluff here, with 18 outs? Easy. The cost of a bet is simply My Equity - Opponent's Equity. 18/46 times I win 1BB when he calls and I get there, and 28/46 times I lose a BB when he calls and I don't get there. Simply, the EV of a called bet here is -10/46BB, or about -.218BB.For me, the easiest way to visualize the situation is as follows:"How many times can I bluff at my opponent's share of the pot and get called before I spend EXACTLY the amount of money in my opponent's share of the pot? In otherwords, how many times will he have to call me for a single successful bluff to make up for it?"Easy:Opponent's Equity/BetCost3.35BB/ (10/46) = 15.4So I can bet 15.4 times before I bluffed off his share of the pot. He folds to the next bluff -- the 16.4th -- and then we're dead even.So, he has to fold 1 in 16.4 times. Or about 6.1% of the time. I have a second, more simplified model that I sent in a PM to Peter. It's probably going to be more intuitive for most.WangEDIT- The whole point of this exercise was to show that bluffing with a lot of outs is often the BEST PART of having all those outs. There aren't many situations anymore where I don't fire the turn if my opponent isn't a total calling station. There are obviously more considerations, like being check/raised, not knowing how many outs are good, meta-game stuff, etc., but let this be a small lesson for you.
Link to post
Share on other sites
  • Replies 64
  • Created
  • Last Reply

Top Posters In This Topic

Opie - I think your step 2 is incorrect. By your calculation betting is +EV even if we're always called and that's clearly not true. The EV of that bet is only the additional amount that you win based on that bet or .4 BB. That bet doesn't help (or hurt) us win the $$$ already in the pot.
Nope. Compare his calculated equity when we bet (1.94BB) vs the equity of checking (2.15BB). The difference, there, is the EV of betting vs checking (obviously a negative number).EquityBet - Equity Check = 1.94 - 2.15 = -.21BBBetting costs us about .21 BB when we're called, and is (obviously) -EV.Wang
Link to post
Share on other sites

Psujohn, you were on the right track, just got a little mixed up with some of the things you were equating.I rounded, and assuming I didn't **** up any of the algebra, here's the answerLet p(x) = probability of your opponent folding i.e. your semi-bluff works.If you bet and your opponent calls, your EV:18/46*6.5bb -28/46*1bb = 2.54bb -0.58bb = 1.96bbIf you bet your opponent folds, your EV:5.5bbIf you take the free card, your EV:18/46*5.5bb = 2.15bbSo, if you want to know how often your semibluff has to work for it to be break even, you set your EV from a free card equal to your EV from betting, and solve for p(x):2.15bb = p(x)*5.5bb + (1-p(x))*1.96bb2.15 = p(x)5.5 + (1-p(x))1.962.15 = 5.5p(x) + 1.96 - 1.96p(x)0.19 = 5.5p(x) -1.96p(x)0.19 = 3.54p(x)p(x) = 0.0537If your semibluff here works more than about 5.4% of the time, it is +EV

Link to post
Share on other sites
Ship it ALLLL to Opie. You guys are banging your heads against the wall, and this clown with 75 posts makes you all look afoolThe whole point of this exercise was to show that bluffing with a lot of outs is often the BEST PART of having all those outs. There aren't many situations anymore where I don't fire the turn if my opponent isn't a total calling station. There are obviously more considerations, like being check/raised, not knowing how many outs are good, meta-game stuff, etc., but let this be a small lesson for you.
Yeah, I guess I am a "clown" - because I should post more. I'll try. I'm good at math - it is poker that I need to learn better!Thanks for the fun math puzzle, Wang. But more importantly, thanks for the valuable poker lesson! I guess it is related to the statement - the aggressive player more often wins the pot.Peace,Opie
Link to post
Share on other sites
Yeah, I guess I am a "clown" - because I should post more. I'll try. I'm good at math - it is poker that I need to learn better!Thanks for the fun math puzzle, Wang. But more importantly, thanks for the valuable poker lesson! I guess it is related to the statement - the aggressive player more often wins the pot.Peace,Opie
No offense meant by the little "clown" remark. It was more a barb directed at the regular posters here who I enjoy poking at, from time to time, even though I'm pretty sure most of them have higher equity than I do at most any table.That being said, you should post here, more. I hardly do anymore, but you seem to have a pretty solid understand of some the underlying concepts involved. Some pretty good, winning cardplayers don't possess the basic tools you seem to. What games do you play, and where? Bring a little something to the table, Clown.Wang
Link to post
Share on other sites

> x*(5.5)+(1-x)*(18/46*(5.5+1)-28/46*(1)); 3.565217391 x + 1.934782609> 18/46*(5.5); 2.152173913> solve(3.565217391*x+1.934782609 = 2.152173913,x); 0.06097560966That's how I'd solve it, which is pretty much exactly what opie did. You should start posting more man, we need more math guys around here!

Link to post
Share on other sites
That being said, you should post here, more. Bring a little something to the table, Clown.
You should start posting more man, we need more math guys around here!
OK. I'll try. I've actually been looking for an interesting hand to post for quite a while and haven't been able to find one. Some of the sort of interesting ones are because I didn't use my normal strategy based on a read of an opponent. But it takes so long to describe why I had that read (unless I use PT stats, which I haven't set up yet). Another interesting type of hand is when multiple players raise at different points in a hand, but if I'm in a game that aggressive, I play very tight and my decisions get a lot easier and the hands I'm in aren't post-worthy. I'll keep looking.Peace,Opie The Happy Clown :-)
Link to post
Share on other sites
OK. I'll try. I've actually been looking for an interesting hand to post for quite a while and haven't been able to find one. Some of the sort of interesting ones are because I didn't use my normal strategy based on a read of an opponent. But it takes so long to describe why I had that read (unless I use PT stats, which I haven't set up yet). Another interesting type of hand is when multiple players raise at different points in a hand, but if I'm in a game that aggressive, I play very tight and my decisions get a lot easier and the hands I'm in aren't post-worthy. I'll keep looking.Peace,Opie The Happy Clown :-)
Just find one where you think there's a close decision, or where an opponent takes a line that confuses you, or you feel like you called down but it didn't feel right or that you were losing value. Or post a boring one so I can contradict Zach.
Link to post
Share on other sites

before reading repliesPot 5.5 18 outs x = % he folds (1-x) = % he calls we win 18/46 times he calls we win 6.5 when he calls and we hit we lose 1 when he calls and we miss we win 5.5 when he folds -1 * (1-x) * (28/46) + 6.5 * (1-x) * (18/46) = 5.5 * (x) -1 * (1-x) * 28 + 6.5 * 18 * (1-x) = 5.5 * 46 * (x) 89 * (1-x) = 253 * (x) 89 = (253 + 89) * (x) x = 89 / (253 + 89) x = 0.260234

Link to post
Share on other sites
Just find one where you think there's a close decision, or where an opponent takes a line that confuses you, or you feel like you called down but it didn't feel right or that you were losing value. Or post a boring one so I can contradict Zach.
OK, here's one I thought about posting from a $4/$8 live game in Atlantic City earlier this summer.I am in cut-off with QQ.2 limpers, I raise, button calls (huh?), SB folds, BB calls, 2 limpers call. 5 to the flop (actually probably below average at the table for players to the flop - there were a few bad players who limped most hands)Flop: 9TJ (I don't recall suits, let's say rainbow)Not a great flop for me, but not bad - I have an overpair and a straight draw.Check, Check, Check, Opie bets, Button raises (hmmm), BB re-raises (wow!), fold, fold, Opie ...???My answer at the table was to call the 2 bets, figuring I still have a straight draw even if I am behind, which I probably am (guessing someone had TJ, but otherwise confused). Button called (luckily he didn't reraise).Turn: blankBB bets, Opie (fearing a reraise from button and being stuck in between 2 raisers, and realizing I'm probably behind) folds, ...As the hand turns out, that the fold saved me money, but perhaps I should have folded to the flop check re-raise? Button had 78s for the straight, BB had JJ for trips. The only way it could have been worse was if someone had KQ. It turns out I did have 7 outs (minus the full house possibility for the BB), but I had to call 2 bets with a chance button would cap it for another.But here's the thing - it depends on the opponents. The BB was a quiet guy, so his raise was intersting. But the BB was an older lady who was a regular and definitely knew what she was doing. Without a read, maybe (but doubtful) that my call of 2 bets was OK. But with the read I had, it should have been an easy lay-down - it only took me 2 days of thinking about it to realize it!See? Was that interesting enough to post? I initially thought so, but eventually figured out the answer, which is hard to understand without knowing the opponents.Peace,Opie
Link to post
Share on other sites
before reading repliesPot 5.5 18 outs x = % he folds (1-x) = % he calls we win 18/46 times he calls we win 6.5 when he calls and we hit we lose 1 when he calls and we miss we win 5.5 when he folds -1 * (1-x) * (28/46) + 6.5 * (1-x) * (18/46) = 5.5 * (x) -1 * (1-x) * 28 + 6.5 * 18 * (1-x) = 5.5 * 46 * (x) 89 * (1-x) = 253 * (x) 89 = (253 + 89) * (x) x = 89 / (253 + 89) x = 0.260234
Actuary,There are 2 sides to the equation. One side should represent when you check behind on the turn, the other should represent when you bet. Therefore, the equality you made should actually be on the same side, with the other side containing an equation to show the EV from simply checking behind on the turn.
Link to post
Share on other sites
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.
Link to post
Share on other sites

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

Link to post
Share on other sites

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) = csemibluffing.png

semibluffing.xls

Link to post
Share on other sites
To try to acquire some nerd cred, I solved it symbolically to avoid further algebra. semibluffing.png
Consider your nerd-cred earned in full. Nice job there, Martin. Hey, didn't you get shot or something?Wang
Link to post
Share on other sites

Nice work nerds. Sorry I was away for the weekend and didn't get a chance to play.

Link to post
Share on other sites

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

Link to post
Share on other sites
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.
Link to post
Share on other sites

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

Link to post
Share on other sites
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.
Link to post
Share on other sites

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.

Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now

Announcements


×
×
  • Create New...