Fold Equity & Bluffing (x-post)

[GrunchCan]

Introduction

A couple of non-2+2 friends of mine were asking me what I was talking about when I used the term "fold equity." This is my response, and it includes a general discussion the EV of bluffing.

A bluff is a bet that we hope isn't called becasue our hand can't win a showdown and can't take any more action. Bluffing *is* a part of poker, but it is one of, if not the most overused actions in all of poker. Conversely, one of the popular advices given on 2+2 is, "never bluff". While this advice might not be -EV, it also isn't optimal. There are times when bluffing is right becasue it will show a long-term profit. But those times are very situational dependant, and don't come nearly as often when we should just check and fold.

The term 'equity' refers to how much of the pot that 'belongs' to us. In the case of a draw to the nuts that will come in 33% of the time, our equity is 33% of the pot. If the pot is $15, our equity is $5. Over one million hands, we expect to win $5 per hand on average. Will win some pots, lose many more, and overall pull in $5 per hand. If we spend less than $5 to win those hands on average, we'll come out ahead after the million hands. If we spend more, we're losing money on this draw.

In the case of a bluff, the only equity we have in the pot is based on the chance that the opponents will all fold to our bluff. (The case of a semi-bluff is more complex) So as you can see, the profotability of bluffing depends on:

1) How much we bet
2) How big the pot is
3) The chance that everyone will fold.

The first two are easily deduced, and the third is a key poker skill that takes experience and study to develop.

The Variables Aren't Independant

There are two critical things to note here. First, and way the most important, is that the chance that everyone will fold goes down exponentially with each additional opponent. In many cases heads-up it will be clearly +EV to bluff. In the same hand with 2 opponents it becomes a close decision, and with 3 opponents it is often clearly -EV to bluff.

The second thing to note is that the 3 variables aren't independant. When we bet more in relation to the size of the pot, the chance that everyone will fold probably goes up. The bigger the pot on the other hand, the more likely opponents are to call to try to win a big pot.

Computing Fold Equity & the EV of Bluffing

If you have a good handle on the value of the 3 variables, we can easily & quickly compute the EV of bluffing. There's a more rigorous mathematical approach that isn't easy to use at the table, and another intuitive approach that you can use at the table. The later is what I'll discuss next, and it has the added benefit that you don't need to know ahead of time the answer to the harder question: what is the chance that the opponents will all fold? Instead, it presents you with a question in the form of, "Will all my opponents fold 75% of the time here?" which is often a very easy question to answer.

Suppose you have QJ of hearts, and the flop comes Ace-high no draws. You plan to either check & fold or make a continuation bet*. In small stakes no-limit, a pot sized bet is a reasonable continuation bet becasue it isn't as likely to be called as a smaller bet. So you plan on making a continuation bet of pot-size if you continue. You are betting $1.00 to win $1.00. You can intuitively see that in order for this bet to show a profit, you must win the $1.00 that's in the pot more often than you are called. In other words, you must have a fold equity of 50% or more. All your opponents must fold 50% of the time when you make this bet in order for it to be +EV.

In limit poker, the continuation bet can't be the size of the pot. Usually it's something like $1.00 to win $4.00. In this case, the chance that your opponents will call goes up becasue the bet is small, but you also have to win less often for the bet to be +EV. Specifically, you must win 1 $4.00 pot for every 4 $1.00 bets you make and lose. You must bet 5 times and win one of those times, or have a fold equity of 20%.

So here's what you do when you are considering a bluff. Look at the pot size and compare it to the size of your bet. In this case, we're betting $1 to win a $4 pot. Ask yourself how many times you can lose the bet for each time you win in order to break even. In this case, we can lose 4 $1 bets for every $4 pot we win. That's 5 bets total (4 losses, 1 win), and we won 20% of the time when we break even. Now ask yourself, will the opponents *all* fold more than 20% of the time? The longer you play the better you will get at answering this question correctly.

Bluffing Multiple Opponents

In the introduction I said that bluffing is one of the most-overused lines in all of poker. I believe that this is becasue people tend to vastly underestimate the importance of not bluffing a multi-way pot. If you need some hard & fast rule to follow regarding bluffing, try this: Only bluff when heads-up. This again isn't optimal, but it's pretty close.

In the example above where we had QJ and missed, we came to the conclusion that if the opponent would fold 50% of the time then our bluff was +EV. This might be a reasonable estimate, but just to make a point, let's say that the opponent will fold 75% of the time. Now it's clear that bluffing is very +EV. In fact, it's so +EV that it would be a pretty big error to not bluff here.

But now let's say we have 2 opponents who will fold 75% of the time. Now ask yourself: if we bet, how often will we win the pot?

If you said 75% of the time, you're wrong. It's actually closer to 56% of the time. (.75 * .75). Now we're betting 1 to win 1, and we are a 56:44 favorite to win. This is still +EV, but it's pretty close. If our estimate of how often the opponent will fold was off by even 5%, bluffing is now -EV. (Compute the EV of bluffing if the each opponent will fold 70% of the time.) How sure are you that your estimate of 75% is so precise?

Now, if we have 3 opponents that will fold 75% of the time each, bluffing becomes clearly -EV. I know that this conclusion was very surprising to me when I first realized it, but it's true. Bluffing in to 3 people is virtually always -EV.

No-Limit versus Limit

You might be asking yourself, how does this apply to limit versus no-limit? What other things can I think about?

The answer is, nothing changes. Take a look at the list of variables above: bet size, pot size, % chance opponents will fold. All the differences between no-limit and limit that apply to this decision are wrapped up in the variables already. You'll punch in different numbers, but use the same formula. One word of warning however. Even if you become an expert at deducing how often the opponent will fold if you bet at one game, once you switch to another game you'll need to re-learn this skill.

Conclusion

Bluffing is over-used by the opponents we want to play against, but that doesn't mean that never bluffing is the way to go, either. Sometimes bluffing is +EV, but you have to be careful. You need to know how often the opponent will fold, and you need to recognize that the chance that bluffing being +EV goes down incredibly fast with additional opponents.

There is much more to bluffing, too. Semi-bluffing, bluffing in position, bluffing when a scary flop comes, bluffing whiffed overcards or small pocket pairs, and hand quizzes are all things that I'm hoping others will elaborate upon.

Good luck.
__________________________________________________

Continuation Bet: A bet you make on the flop after you were the pre-flop aggressor but our hand missed the flop completely. A continuation bet is a bluff, and we hope to win the pot right now. If you have a draw, then your bet isn't a CB; it's a semi-bluff.

[vulturesrow]

Great post. One thing I think you couldve brought in your discussion of estimating how often people will fold is the type(s) of opponents. To us it seems obvious that there are some opponents that we ill nearly always bluff at and other where we will hardly ever bluff at (although in the case of opponent B, the bluff may be a value bet!)

[GrunchCan]

You're abolutely right. That's one of those things I was hoping for elaboration on.

There's 2 reasons why I didn't include a lot of specifics like that. First and most important, they are generally things that we can consider to refine our estimate that the pot will be folded to us. Second, this was a cross-post (with micros) and that was getting in to some differences that are hard to discuss in a generalized post.

One example is bluffing in to 2 opponents where one is weak-tight and the other is loose. The WT opponent is a dream come true for us, becasue he'll fold a lot. You might think that this offsets the frequent calls by the loose player, but I've found that this just isn't true.

Remember that with 2 opponents, the chance that the pot will be folded to us is A * B where A is the chance that one folds and B is the chance that the other folds. Now, A (the weak-tight) is folding a lot, but B is still calling a lot. It turns out that B has a bigger effect on our equity than A. If A is .75 and B is only .25, then the chance that the pot will fold to us is only .25 * .75 = .1875! That was a pretty surprising result for me. And again, the more opponents we mix in here, the worse the situation gets for us.

[BobboFitos]

[ QUOTE ]
One example is bluffing in to 2 opponents where one is weak-tight and the other is loose. The WT opponent is a dream come true for us, becasue he'll fold a lot. You might think that this offsets the frequent calls by the loose player, but I've found that this just isn't true.

Remember that with 2 opponents, the chance that the pot will be folded to us is A * B where A is the chance that one folds and B is the chance that the other folds. Now, A (the weak-tight) is folding a lot, but B is still calling a lot. It turns out that B has a bigger effect on our equity than A. If A is .75 and B is only .25, then the chance that the pot will fold to us is only .25 * .75 = .1875! That was a pretty surprising result for me. And again, the more opponents we mix in here, the worse the situation gets for us.

[/ QUOTE ]

this isn't entirely true, as people (tend) to tighten up more in regard to flop or turn calls in multiway pots (this aint limit)

but yeah, does show how fruitless bluffing multiway pots can be.

Wow. Nice post. Very insightful and usefull. I just found out how much bluffing multiway pots sucks. Thanks.

[TheWorstPlayer]

[ QUOTE ]
Wow. Nice post. Very insightful and usefull. I just found out how much bluffing multiway pots sucks. Thanks.

[/ QUOTE ]
Only thing is that in multi-way pots a)the pots are usually bigger so the reward is greater and b)people give more credit to bets so probability of people folding is higher on a stand-alone basis. So it is all just cost-benefit analysis. Bluffing multi-way pots can be great. On the MHFL board, gonores has a few posts about multi-way turn bluffs that he thinks are great. I would recommend reading them for some interesting discussion.

[GrunchCan]

Just observing an irony...

[ QUOTE ]
Loc: Losing money bluffing

[/ QUOTE ]

[img]/images/graemlins/smile.gif[/img]

[ QUOTE ]
people (tend) to tighten up more in regard to flop or turn calls in multiway pots (this aint limit)

[/ QUOTE ]

This is definitely true and to take it a step further, the position of the players can make a big difference too. Take the example of Player A who will fold 75% of the time and Player B who will fold 25% of the time in heads-up situations. Let's say they're both in the same pot with you and you bet into them. If Player A is first to act, his fold % may climb to 85% because (a) you've shown more strength by betting into 2 players, and (b) he's afraid of Player B still left to act behind him. If Player A folds, Player B's fold % remains close to his original 25%, maybe a little higher due to reason (a). The true percentage chance of getting both to fold in this case is around:
.85 * .30 = .255

Now take the same situation but reverse the positions. Player B is first to act and let's say his fold % increases to 50% for the same reasons. If he folds, A's fold % is around 80%. The percentage chance of getting both to fold in this case is:
.5 * .8 = .4

Now these are all just numbers I pulled out of the air, but the point is it's not enough to just say that you have one loose player and one tight player. You also need to account for which of the two is going to get sandwiched between your raise and the other player and to what extent that will cause him to tighten up. I think generally speaking, unless he's a total maniac, the loose player will tighten up far more simply because he's otherwise playing a lot of very marginal hands, where the tight player is only calling with solid hands anyway. That would suggest that if you were to bluff in a multiway pot, you'd probably prefer the looser players to your immediate left.

[TheWorstPlayer]

I was speaking theoretically. Obviously practically speaking all fold equity equals zero because the probability of anyone folding equals zero. And once I realised that my win rate went up astronomically. You won't believe how good it is for the discipline and ego and win rate when you check down a hand in position heads up only to lose to a monster. It is my new favorite thing in the world. Yesterday I had a hand that went something like:
Folded to SB who completes and I check in the BB. AAQ flop. Check, check. Q turn. Check, check. Blank river. Check, check. I lose to AK.

In a previous life I would have probably bet the flop, definitely bet the turn, and probably would have raised the river all-in after calling his check/raise on the flop or turn.

[GrunchCan]

Just to be clear, I was being facetious.

I respect TWP's opinion as much or more so than anyone else's.