Calculating EV of a semi-bluff

Originally posted by VorpalF2F
Part 1:
In the article "Mathematical Concepts for No-Limit Holdem (3)" is the following:

As bluffs and semi-bluffs play an important role in this article, it is important to introduce the possibility of an opponent's fold to the EV calculation. This formula:

     EV = Equity * (Win + Investment) - Investment

will not be sufficient. There is no variable for an opponent's fold in this formula. We therefore need a more general formula to calculate the EV:

     EV = Pfold * Pot + (1 - Pfold) * (Equity * (Win + Investment) - Investment)

The probability that you win the pot is Pfold; if you don't win the pot immediately we use the EV. Pfold is the probability that your opponent will fold. Under the assumption Equity = 0 the formula can be simplified to this:

     EV=Pfold * Pot - (1 - Pfold) * Investment

Loss in this case is your continuation bet. This is the case for all pure bluffs. Given the definition of a pure bluff, you can only win if your opponent gives up.

The two bold statements seem to attribute two different values to Pfold.
The probability that you win is more than the probability of villain folding, because he could call and you suck out -- even on a pure bluff.

Part 2:
I was actually looking for the calculation that tells you how many times opponent has to fold for your bluff raise to be break-even.

Can someone please point me to it?

Thanks,
--VS

1) Under the assumption Equity = 0 the formula can be simplified to this:

under this assumption both Pfold should be the same. ie villain WILL NOT call you with worse.

2) if equity = 0

EV=Pfold * Pot - (1 - Pfold) * Investment

for break even (EV = 0)

Pfold * Pot - (1 - Pfold) * Investment = 0

gives you Pfold = Investment / ( Pot - Investment )

if equity > 0 ( and if I didn't make mistake in my equations)

Pfold * Pot + (1 - Pfold) * (Equity * (Win + Investment) - Investment) =0

I got

Pfold = (Investment - Equity * (Win + Investment) ) / (Pot + Investment - Equity * (Win + Investment))

The second bolded one is right.

Part 2:

I'm just thinking out loud here to get the formula
Say we have a draw with 40% equity.

We then have a current pot of $300, and semibluff shove all in for $1400. when we win we gain $1700, and when we lose, we lose $1400.

Then our EV if Pfold=0 is 1700*0.4-1400*0.6, or -160
We need the fold equity to equal this -ev sooo
300xPfold=160 Pfold must equal 53.333

To put this in a formula,

(-1x ( (profit*equity%) - (raisesize*(1-equity%)) ) ) /potsize = Pfold needed for breakeven move.

The -1x is there because the value that we divide by potsize must be negative (otherwise move is automatically above breakeven because this number is the quity and if its positive and we have fold equity obviously the play is +EV) and note here Profit = raisesize+current pot.

Also the factor of multiple streets complicates things in terms of changing equity etc. so it only really works for all ins.

EDIT: To simplify this, work out EV equity without folding as an option, THEN if this value is positive you can already semibluff profitably regardless of pfold, but if your EV is negative, balance this negative value with Pfold*Pot, which should be exactly identical to the negative ev (negative ev only when the opponent calls 100% of the time) to balance it.

Interestingly, if there is enough in the pot, even with a Pfold of 0 and an equity of less than 50%, say 40%, a shove can still be profitable. Eg. 700 pot, you shove for 1200, when you win you make 1900 and when you lose you lose 1200 so you actually are +EV even without Pfold factored in.

Originally posted by metza

Interestingly, if there is enough in the pot, even with a Pfold of 0 and an equity of less than 50%, say 40%, a shove can still be profitable. Eg. 700 pot, you shove for 1200, when you win you make 1900 and when you lose you lose 1200 so you actually are +EV even without Pfold factored in.

Just to put metza's last point in more concrete terms. With a Pfold of 0 and 40% equity, shoving 1200 into a 700 pot gives you an EV of:

40% * 1900 - 60%*1200 = 760 - 720 = +40
that is your equity wins you 1900 40% of the time (the 700 in the pot and 1200 from your opponent calling) and the other 60% of the time you lose the 1200 that you shoved.

Originally posted by VorpalF2F
Part 1:
In the article "Mathematical Concepts for No-Limit Holdem (3)" is the following:

As bluffs and semi-bluffs play an important role in this article, it is important to introduce the possibility of an opponent's fold to the EV calculation. This formula:

     EV = Equity * (Win + Investment) - Investment

will not be sufficient. There is no variable for an opponent's fold in this formula. We therefore need a more general formula to calculate the EV:

     EV = Pfold * Pot + (1 - Pfold) * (Equity * (Win + Investment) - Investment)

The probability that you win the pot is Pfold; if you don't win the pot immediately we use the EV. Pfold is the probability that your opponent will fold. Under the assumption Equity = 0 the formula can be simplified to this:

     EV=Pfold * Pot - (1 - Pfold) * Investment

Loss in this case is your continuation bet. This is the case for all pure bluffs. Given the definition of a pure bluff, you can only win if your opponent gives up.

Both bolded parts seem like they are saying the same thing, they just worded it kind of weird which makes it a little confusing. If your opponent folds then your equity does not matter because the hand is over, but if your opponent does not fold then you need to calculate equity since it is very rare that you are getting your money in drawing dead, and this part of the equation accounts for the times that you suck out on your opponent.

For part 2, what exactly do you mean by bluff raise? My interpretation is that you want to bluff raise a continuation bet or something of that nature.

So lets just do a quick example and show how I would calculate this. I am going to make the assumption that your raise is a pure bluff and that your equity is 0.

Since equity does not matter, the best equation to use is

EV=Pfold * Pot - (1 - Pfold) * Investment and we are going to let
x = Pfold just so it looks a little nicer.

So if the pot is 6bbs, and your opponent cbets 5bbs and you raise to 15bbs the eqaution looks like this

Ev = 0 Pot = 11 Investment = 15 Pfold =x

0 = 11*x - (1-x)(15)

0 = 11x - (15-15x)

0 =11x -15 +15x

15 = 26x

(15/26) = x = 0.5769 = 57.69% = Pfold

So in this example you would need your opponent to fold 57.69% of the time for your bluff raise to be break even. Considering your hand probably has some equity, this is most likely a high estimate but trying to calculate your equity when it is not an all-in situation is rather difficult. If you had a flush draw with 9 outs, you would have about 36% equity IF you got to see the river, but only about a 18% equity to improve by next card. And then you can always try to bluff the turn again so the calculation gets even more complex.

Originally posted by fdel15

For part 2, what exactly do you mean by bluff raise? My interpretation is that you want to bluff raise a continuation bet or something of that nature.

So lets just do a quick example and show how I would calculate this. I am going to make the assumption that your raise is a pure bluff and that your equity is 0.

.

Thread title is EV of a semi-bluff though, so equity is obviously not 0.

Originally posted by metza

Originally posted by fdel15

For part 2, what exactly do you mean by bluff raise? My interpretation is that you want to bluff raise a continuation bet or something of that nature.

So lets just do a quick example and show how I would calculate this. I am going to make the assumption that your raise is a pure bluff and that your equity is 0.

.

Thread title is EV of a semi-bluff though, so equity is obviously not 0.

Very true, and at the end of my post I tried describe some of the difficulties of calculating your equity when it is not an all-in situation. If it is an All-in situation and you know your exact equity then you can use the equation

EV = Pfold*Pot + (1-Pfold)*((Equity*(Win+Investment)-Investment))

One benefit of assuming your equity is 0, besides making the math a little easier, is that it gives you a margin of safety. Calculating your opponents folding percentage and even your exact equity vs their hand is not an exact science as you are weighing a bunch of different variables. By assuming you never suck out on your opponent, your break even FE is going to be higher then it actually needs to be. In my example, if our opponent folds exactly 57.69% then instead of breaking even I am actually making a profit depending on how often I suck out. The bigger your draw, and the more you are relying on your showdown equity, the less helpful this assumption is but it is still not completely useless.