The profitablity of a bluff
 

I was thinking about how to calculate the EV of a bluff. I came up with this formula. Do you think it is correct?

P = Pot before you bluff
B = Bet-size of the bluff
FE = The percentage of the time you guess all opponents will fold to your bluff
EV = Exptected value

EV = -(B -(B * FE)) + (P * FE)

This formula assumes you will never win any part of the pot in any other way if the bluff is called.

Example:

Party Poker
No Limit Holdem Ring game
Blinds: $5/$10
10 players
Converter

Stack sizes:
SB: 1000$
Hero: 1000$

Pre-flop: (10 players) Hero is BB with 2[img]/images/graemlins/club.gif[/img] 3[img]/images/graemlins/diamond.gif[/img]
8 folds, SB calls 5$, Hero checks.

Flop: A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/diamond.gif[/img] Q[img]/images/graemlins/spade.gif[/img] ($20, 2 players)
SB checks, Hero bets $15, SB calls $15.

Turn: 8[img]/images/graemlins/heart.gif[/img] ($60, 2 players)
SB checks, Hero bets $25, SB calls $25.

River: 7[img]/images/graemlins/heart.gif[/img] ($100, 2 players)
SB checks, Hero bets $50

In this hand, Hero thinks the opponent will have a busted flush draw 50% of the time,
in this case he will fold to Heros bluff.

The remaining 50% of the time Hero thinks villain has been slow-playing a monster, and the bluff will be
at least called.

The opponent will therefore fold to a bet 50% of the time here.

Here are the calculations:

P = 100$
B = 50$
FE = 50% = 0.5

EV = -(B -(B * FE)) + (P * FE)

EV = -(50$ -(50$ * 0.5)) + (100$ * 0,5)

EV = -(50$ -(25$)) + (50$)

EV = -(25$) + (50$)

EV = 25$

In the example, your bluff will on average win you 25$, if I have gotten things right here.

I just came up with this forumla and I'm not really confident with it yet. I'd appreciate if anyone in this forum has any thoughts on it.

Re: The profitablity of a bluff
 

It's correct. It's more intuitive to rewrite it:

Your formula:
EV = -(B -(B * FE)) + (P * FE)

Rewrite:
EV = P*FE - B(1-FE)

The first term is the probability that she will fold, times the amount you win if she does. The second term is the probability that she will call (call probability = 1 - fold probability) times the amount you lose if she does. The second term could also be interpreted as the sum of the call probability and the raise probability if you won't be calling a raise.

edited to add:

When calculating expectations of anything, you always take the probability of each possible event and multiply it by the corresponding amount should that event take place. Then you add all of these up to get the expectation.

So here there are two events, that she folds and that she doesn't (again, I'm assuming either you or her are all-in or that you will fold to any raise). She folds with probability FE using your notation. That means she calls/raises with probaibility 1-FE. If she folds you win P, if she calls or raises you lose B. So to find your expected value for this play, add P*FE and -B*(1-FE) as you did.

Re: The profitablity of a bluff
 

All nice and smooth, but this line here is the one I have a problem with.

[ QUOTE ]
FE = The percentage of the time you guess all opponents will fold to your bluff

[/ QUOTE ]

Why try to make precise math if you don't know what you are talking about? If you start guessing at what percentage all opponents might fold (/random 0 100), why not try another shortcut and just guess if the whole play might work or not?

By just going "yay" or "nay" you save yourself a lot of calculations.

Re: The profitablity of a bluff
 

I agree with Shandrax.

Why not keep it a simple equity/pot odds formula? For a very rough guess that's a lot of calculation.

Re: The profitablity of a bluff
 

I liek the idea and I agree with the math but I don't think it's practical. the FE is just too big of a variable, too hard to calculate correctly.

Re: The profitablity of a bluff
 

With the way you bet that pot you are giving the villain 3:1 on the end. According to you, you are giving the villain better odds than yourself. This is more about your villains pot odds than your EV. You really need to make it too expensive for a mid to low hand to call. I would gather to say you are going to get called way more than 50% on that type of hand. Just my 2 cents...

Re: The profitablity of a bluff
 

I disagree with Shandrax, Gonso and iblufftomuch. It's true that you cannot compute expected value precisely, but doing it is still an important discipline. Most bad poker players do things that make no sense given any guess about what might happen. If you make a calculation you are sure you have a plan that works at least under some assumptions. That puts you in at least the middle rank of players.

Also, it allows you to correct mistakes, which is the only way to improve to the top rank. If you put money in the pot expecting 25% folds, and notice that you're getting only 10% folds, you'll get better. If you put money in the pot because it feels like a good time to bluff, how will you know if you are right? Some players react too much to the result of the last hand, others never react even after consistent losses. Math helps you steer a middle course.

I have a different objection, that this calculation has nothing to do with bluffing. If you think about things this way, you should do this with a marginal calling hand, so you still have some chance of winning if called. Bluffs are done with your worst hands.

The main payback for bluffing is that people will notice you do it and therefore call your good hands. Poker theorists differ, traditional ones think you should lose money on bluffs in order to maximize your overall EV. Modern ones, following Sklansky, generally recommend breaking even or better on the bluffs (which means they either bluff less frequently or semi-bluff). But everyone agrees that the main reason to bluff is not this hand but future hands.

That's why it doesn't pay to bluff bad players. They don't notice what you're doing. You can bet with weak hands if they don't call enough, but use your strongest weak hands, not your weakest hands. And there's generally no problem getting bad players to call your good hands, so no reason to risk bluffs.

The good thing about a bluff, done correctly, is you gain whatever happens. If you get called, you more than make up the loss on your next good hands. If you don't, you make money now.

Re: The profitablity of a bluff
 

Wanna thank everybody for the answers, especially the very good posts of JaredL and AaronBrown.

[ QUOTE ]

[ QUOTE ]

FE = The percentage of the time you guess all opponents will fold to your bluff


[/ QUOTE ]

Why try to make precise math if you don't know what you are talking about? If you start guessing at what percentage all opponents might fold (/random 0 100), why not try another shortcut and just guess if the whole play might work or not?



[/ QUOTE ]

I obviously use the word guess because you can never know this amount for certain. However, most good hand readers usually have a reasonable feeling on this number.

Saying that if you can't be quite sure about this calculation, you don't wanna use it at all: I think that's a bit off. I find it pretty useful when working out bluffs. "If I risk this amount on my bluff, how often does my opponent have to fold for it to be a break-even or profitable move"


[ QUOTE ]

With the way you bet that pot you are giving the villain 3:1 on the end. According to you, you are giving the villain better odds than yourself. This is more about your villains pot odds than your EV. You really need to make it too expensive for a mid to low hand to call. I would gather to say you are going to get called way more than 50% on that type of hand. Just my 2 cents...


[/ QUOTE ]

This is just an example hand which I made up, the numbers aren't necesarilly reasonable.

Thanks again guys.

Re: The profitablity of a bluff
 

I believe the small blind calls and shows you A7 and takes down the pot. I think the bluff on the river doesn't work 50% which drops the EV. I like the triple shelled bluff courage, but think the hero could find a much better spot to bluff.

Re: The profitablity of a bluff
 

I agree with what Aaron Brown has said here.

Those saying that a formula as this, which is correct btw, are unlikely to succeed in moving up and becoming winners are medium or higher stakes games unless they change this mindset. The problem in poker is that you don't get to see the other guys' cards so you basically will have to make some guesses (perhaps estimate is a more accurate word than guess) about what they have.

At some stage this will be based on hand reading. Based on your opponents' play on previous streets you have put them on a range of hands. Decide what hands will call your bet and which will fold. Then you have a probability estimate for the probability that they will fold.

The closer your guess is for the probability the better and you will improve as a poker player by improving you ability to estimate this.

Also, it might be easier to think in terms of odds. If there is a 1/3 chance she will fold to your bluff then it is profitable if you the pot is more than 3 times the size of your bet.