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In the book of Bill Chen, "The mathematics of poker", there is a formula to calculate the right frequency of a bluff action.

The formula has been calculated in a simplified version of poker.

Even further, in his poker lessons at MIT, Will Ma states that most players bluff too much or don't bluff enough.

How can we know the right frequency of a bluff action in real poker?

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I don't believe there are any formula about bluff action. – Soner Gönül Jan 16 '13 at 16:32
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@SonerGönül A "bluff" frequency calculation is just a weighted equity calculation where you would manipulate the folding frequency of the opponent to figure out if your action was +EV. So, yes, there is a formula. – Toby Booth Jan 16 '13 at 16:35
    
In an ideal game with perfect opponents, considering a single bet that your opponent will either call or fold, there is a simple formula: bet so as to make your opponent indifferent to calling: in other words, make his pot odds exactly the same as his odds of winning, so his EV for the call and fold are both 0. If you bluff more than this, he will win by calling more; less, he wins by folding more. Unfortunately, this situation NEVER occurs in actual play, so the guys below giving you more subjective advice are actually right. – Lee Daniel Crocker Feb 4 at 23:12
up vote 6 down vote accepted

The correct bluffing frequency is a subjective measure. It all depends on the perceived probability that your opponent will call your bluff, and the estimated equity you have at that moment, whichever street you're on.

Similar to the question How Do I Calculate EV Of Shoving..., you can work out how often bluffing would be a profitable play by manipulating the times you think your opponent would fold. Of course, you're bluffing so you'd prefer them not to call or raise!

  • Jeffrey Blakes answer here gives a detailed breakdown of the equation to do this, using mathematical formula.

  • My answer here uses the same process but I decided to take a semantic look, and put it into plainer language.

In each instance, you are looking to manipulate the first section of each formula. The part where your opponent folds. This has a subsequent effect on how often you assume your opponent will call, described in the latter part of the formula.

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Those links are not the same question. EV of single hand with a stone cold bluff has a w% of zero. – Paparazzi Mar 12 at 19:13
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@Frisbee Which is why I used the word "similar". – Toby Booth Mar 12 at 19:22

--Even further, in his poker lessons at MIT, Will Ma states that most players bluff too much or don't bluff enough.--

This is a ludicrous statement. If you have calculated an optimum bluff frequency then half the players will be higher and half lower minus the very few who nail the frequency exactly.

As several other posters have noted, the actual optimum frequency of bluffing depends greatly on the table dynamics.

If you are at a table full of calling stations, the optimum bluff frequency is near zero. You might want to throw in a cheap bluff at some point with the idea of getting called so that they will call a bigger bet in the future.

If you are at a table full of nits, you will want to be bluffing a lot.

I don't see where some theoretical number for optimal bluffing frequency has much value in the real world.

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Yes there is a mathematical number from Bill Chen, "The mathematics of poker". See my answer. Do you also see no needs for a theoretical pot odds calculation? – Paparazzi Feb 5 at 18:17
    
Pot odds are essential to playing the game and are not "theoretical". I don't see any valid comparison between that and the metric we are discussing. – Philip Hall Feb 5 at 23:27
    
Pot odds is theoretical calculation and this theoretical calculation is just as mathematically valid. – Paparazzi Feb 5 at 23:30
    
Not even close. Pot odds is a strict mathematical calculation. The odds of a successful bluff need to factor in the probability that someone will fold and that is not a real mathematical calculation because it depends on a factor that cannot be determined by math alone. – Philip Hall Feb 6 at 12:30
    
OK, but, it is not the odds of a successful bluff. It is a bluff rate the results in 0 EV to call of fold. That is real math. Did you read the math in my answer (from Chen). – Paparazzi Feb 6 at 14:02

Lol at the answers saying there isn't a correct frequency to bluff.

I'll give a simple example where we have a polar vs bluff catcher range. This means that in the polar players range there are hands that are better than all of the bluff catching players hands and always win at showdown and hands that are worse than all of the bluff catching players hands at never win at showdown. Therefore the only option for the polar player is to go all in as the bluff catching player will never bet as he/she never gets called by worse.

So let's do a basic example. In this river spot there is $50 in the pot. Each player has $100 left behind. We will call the polar player P(P) and the bluff catching player P(BC). We will say that the P(P) gets to this spot with his range comprised of 30% value hands and 70% air(hands that never win at showdown). P(P) will bet $100 with his entire betting range, for reasons we have already established. The frequency that the P(P) bluffs with is (Bet/(Bet+Pot)). In this case this is (100/250) or 40%. This 40% is not 40% of his entire range however, but 40% of his value range, or 40% of 30%, so 12%, or a total of 42% of his entire range. So this comes down to the P(P) betting his entire value range, and then another 12% of his air (which turns out to be around 17% of his air) and he gives up with the rest. This may seem odd to someone as these hands always lose when they check, but if P(P) bet with all of these hands then P(BC) could happily call with all of hands and expect to do better, on average, than if he folded any one of these hands.

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The idea is to bluff at a frequency that gives your opponent 0 EV
f is frequency
s is the fraction of the pot you bet
What f gives your opponent an EV of 0?
0 = f(1 + s) - (1-f)s
0 = f + sf - s + sf
0 = f + 2sf - s
s = f(1 + 2s)
s / (1 + 2s) = f
The mathematical number of s / (2s + 1) is a good number
That is a bigger bluff frequency than most people would guess
A pot size raise should mathematically bluff 1/3 of the time

The most important part of a bluff is selling it. Your betting history on the hand needs to support the hand you are trying to represent. And the hand you are representing needs to beat the hand they have. If you did not bet out on the flop or turn and blank turns up on the river it is pretty hard to sell that bluff. The best time to bluff is when a scare card comes up and opponents have not shown strength.

Let say you played 89 suited and totally miss the flop. Decide the range of hands you want to represent. Bet enough to try and take down the pot right then. Or put in a value bet to make your later bluff believable. You may need to fire three times at the bluff.

Early position is in some ways better. Open early with 1/2 or full pot bet is a strong statement. In late a position a 1/2 or full pot bet is also a strong statement but if you are behind bet(s) it is also more expensive as pot is bigger.

Also make sure you have a player capable of laying down a hand. You get some poor players that just don't lay down hands. And you get some poor players that lay down too easily.

Related question is how often should player 2 call?
Answer: Often enough to make player 1 indifferent to checking or bluffing.
Check has 0 EV - player 1 is on a stone cold bluff
In this case fc is frequency of call
0 = (1-fc) - fcs
fc + fcs = 1
fc = 1 / (1 + s)
So mathematically you should call a pot size bet you know could be a bluff 1/2 the time

Mathematically a big raise should not get called as often. The problem there is you are putting more money at risk. The player can be on a monster. You have to convince them you are at least representing a better hand.

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Someone made an edit that call fraction for a pot size bet should be 1/3. No - according to 1 / (1 + s) with s = 1 then 1/2. That equation is based on bluff bettor EV. That is not the only factor in your decision to call. That is frequency to call that will take the bettor bluff EV to 0 on check versus bluff. Run the numbers. If I bluff pot size and get looked up exactly 1/2 the time it is a wash. I win the pot via fold 1/2 the time and I lose a pot size bet 1/2 the time. – Paparazzi Feb 4 at 17:04

There really isnt a correct frequency to bluffing. Its all dependant on the reads you have on the players around you. If you think you can get a lot of bluffs through to steal pots, then do it. If however the table is quite loose and call with a wide range then you should not bluff as much.

I guess if you keep getting caught bluffing then you are doing it too much. If you never get caught you arent bluffing enough.

Also it can be a good thing to get caught bluffing in certain situations as it can set you up to win a big pot later on when you make a similar move but this time with a monster.

Position is also key. If you are out of position then bluffing is a bad idea. If you are in good position such as on the button, you can pretty much play any two cards you like.

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