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Poker Optimal Bluffing Frequency

 

By Andrew Brokos

Introduction

Poker Optimal Bluffing Frequency Definition

I had a nightmare last night that I was playing high-stakes heads up no-limit hold ’em with Phil Ivey himself. I knew he had picked up a tell on me that revealed the approximate strength of my hand as strong, marginal, or weak, but I didn’t know what it was or how to stop doing it.

Bluffing in poker has always been a key ingredient to the game. The ability to keep your opponents guarded or cautious about your intended play, is extremely useful, and can increase your frequency of winning hands. Alice may decide to penalize a possible bluff (with a frequency α), and Bob may have decided to bluff (with frequency β). If Bob has a weak hand, the choice for him is between bluffing or pass.

The river had just completed a possible flush, and the final board read 5 [spade] 8 [diamond] T [spade] Q [heart] 2 [spade]. I was holding A [spade] T [heart] and checked. Phil gave me that look, like he’d just spotted my tell, and then announced, “All in.” The dealer counted the bet down: $14,000 even, into a pot of just $6000. Somehow, I managed to have the Great One covered. But could I call this bet?

Optimal Calling Frequency

If you've never played poker, you probably think that it's a game for degenerate gamblers and cigar-chomping hustlers in cowboy hats. That's certainly what I used to think. It turns out that poker is actually a very complicated game indeed. Poker originated in Europe in the middle ages. The early forerunners of poker originated in Europe in the middle ages, including brag in England and pochen. What is the Optimal Bluffing Frequency? The optimal frequency for attempting bluffs on opponents is completely dependent on a variety of situational factors. A few questions to keep in mind when deciding to make a bluff: What is my knowledge of my opponent’s potential hand range?

November 7, 2019 Beginner Poker Strategy, Complete Guide, Live Poker, Online Poker, Poker Articles Comments off 10538 Views 3 Bluffing is one of the most important skills a poker player can have. Anybody can wait around for a big hand and hope to get paid off – that takes almost zero skill.

OK, I don’t really dream about poker. At least not that vividly. But it’s a good example of a nightmare situation, facing a big bet on the river when your hand is clearly defined as good but not great. Unless you have some exploitable read on your opponent that he either bluffs too much or not enough, then your best defense in a situation like this is to use game theory to make your decision.

Let’s assume that this river overbet represents either a flush or a bluff. The real Ivey is probably good enough that his game can’t be pigeonholed so neatly, but this is my nightmare, and I make the rules. Is he going to bluff all of his air to make me fold one pair? Is he never going to bluff because he knows I know he knows I only have one pair and he expects me to expect him to bluff? He’s Ivey and I’m lowly old me, so I’m going to abandon any pretense of outthinking or outplaying him.

In a situation where I beat all of his bluffs and none of his value hands, I’m going to call with a frequency such that it doesn’t matter what he does. In fact, I could show him my hand, tell him what percentage of the time I’m going to call, and there would still be nothing he could do to take advantage of me. I need to find the calling frequency such that whether he bluffs 100%, 0%, or anywhere in between, it makes no difference to my bottom line.

To do this, I have to figure out what calling frequency will make Ivey indifferent to bluffing with this bet. He is risking $14,000 to win $6000, so his Expected Value (EV) for a bluff is equal to -14000 (x) + 6000 (1-x), where x is my calling frequency. We want to solve for x such that his EV will be 0, so

Poker Optimal Bluffing Frequency Examples

0 = -14000 (x) + 6000 (1-x)
0 = -14000x + 6000 – 6000x
0 = 6000 – 20000x
20000x = 6000
x = 6000/20000, or 30%.

One way to prevent Ivey from exploiting me with a bluff in this situation is to use a random number generator to call with an arbitrary 30% of my bluff-catching range. Dan Harrington recommends the second hand of a watch for this purpose. Any time I have a hand that can only beat a bluff, I check my watch. If the second hand is at 18 or lower, I call. Otherwise, I fold.

Again, even if Ivey knows that I am doing this, there is nothing he can do to exploit me. If he bluffs more, I catch him just often enough. If he bluffs less, then he misses out on just enough pots that he could have stolen from me.

Blockers

That’s one method, anyway. If I know that I need to call 30% of the time, then I can call with each of my bluff-catchers 30% of the time.

But not all bluff-catchers are created equal. In this example, there is a big difference between my hand, which is A [spade] T [heart], and the nearly identical A [heart] T [heart]. Can you see what it is?

When I have the A [spade], Ivey has fewer flush combinations that he could be value betting. The equation we looked at above is just the EV of Ivey’s bluffs. Since I never have a hand stronger than a flush, his value bets are always going to be profitable. My EV on the river is going to be equal to the amount I win by catching his bluffs minus the amount I lose by calling his value bets.

The A [spade] in my hand removes twelve combinations of flushes from my opponent’s range. When I call with A [spade] T [heart], I will run into a flush a lot less often than when I call with A [heart] T [heart]. Thus, even though both hands beat all bluffs and lose to all flushes, one of them will be shown a flush far less often and is thus a far superior candidate for bluff-catching.

I will have the A [spade] 25% of the time that I have AT. Since it is a better bluff-catcher than my other AT combinations, I want to call with it over the others whenever possible. Thus, I should call 100% of the time that I have A [spade] T and use a random number generator to call 5% of the time that I have any other AT combination, so that I am still catching bluffs 30% of the time but paying off value bets as infrequently as possible.

Hand Strength

Poker optimal bluffing frequency calculator

This, then, is one of the characteristics of a good bluff-catcher: it has blockers to my opponent’s value betting range.

Another important characteristic is that a bluff-catching hand should be able to beat all of your opponent’s bluffs. That may seem obvious, but I’ve had a river bluff called by a hand that I beat on more than one occasion.

In this example, since we don’t expect Ivey to be value betting one-pair, it may seem like AT and 33 are functionally the same hand. The catch is that Ivey could be bluffing one-pair. What a disaster it would be to “correctly” snap off a bluff only to find that he was turning 66 into a bluff and just took you to Valuetown, completely by accident!

Stronger hands are also better if there’s any chance of beating a hand that your opponent is betting for value. As I said before, Ivey is an extremely good player, so he might try to confound all of this reasoning by betting a hand like KT for value. Even if I don’t think that’s likely, all other things being equal, I might as well call with AT rather than 33 just in case.

Practice Avoidance

The best tactic of all for dealing with a situation like this is to avoid it altogether. You never want to be in a spot where your hand is as clearly defined as mine is in this example. Hopefully you do not regularly compete against opponents with reads as rock-solid as those of Nightmare Phil Ivey, but you should still be careful about avoiding situations where your range contains nothing stronger than bluff-catchers.

We don’t know the action leading up to the river in this hand, but let’s say that I bet the turn with my top pair, top kicker, and then checked the scare card on the river. That’s a fine way to play it as long as I’m also capable of checking a strong hand like the nut flush in the same spot. Doing so won’t prevent Ivey from value betting or bluffing, but it will make both of these plays less profitable.

By the way, if I were capable of showing up with a value hand when Ivey shoves the river, I would need to adjust my bluff-catching frequency accordingly. For example, if 10% of my range were flushes and the rest were AT, then I would only need to call with AT 20% of the time, since my overall calling frequency still needs to be at 30% to prevent exploitation from bluffing. That means I’d never want to call with any non-spade AT, and even with the A [spade], I’d only need to call 89% of the time.

Where did that number come from? When flushes are 10% of my range, AT is the other 90%. One-fourth of those AT combinations include the A [spade], so overall A [spade] T is 22.5% of my range. But I only need another 20% worth of calls, so I don’t want to call every time I have the A [spade], and 20/22.5 is approximately 89%. To translate that into seconds on a wristwatch, multiply by 60 to get approximately 53.

Real-Time Decision Making

You’re probably wondering what good all of these calculations are going to do you at the table. Well, we practice this kind of mathematical precision away from the table so that our understanding and our instincts are better when tough spots arise in live games. Even if we aren’t able to be quite so precise in the real world, we can use our understanding to make good approximations.

If I really found myself in this situation, the first question I’d ask myself is how the hand I’m holding compares to all of the other hands I would have played in the same way. If I rarely or never check a hand stronger than AT on the river, then I know that I have to call sometimes with AT or a comparable bluff-catcher to avoid being exploited by bluffs.

The math behind my optimal bluff-catching frequency isn’t hard: it’s just the size of the pot divided by the sum of the pot plus the river bet, or Pot/ (Bet + Pot). Once I know that I need to call 30% of the time, I think about my range and try to decide what are the best 30% of hands that I could have in this situation for catching a bluff?

Poker Optimal Bluffing Frequency Analyzer

Remember our criteria for a good bluff-catcher: (1) able to beat all of the hands he could be bluffing with; (2) blocks some portion of the opponent’s value betting range; (3) possibly even ahead of a thin value bet. If all I can ever have in this spot is AT, then even without doing any math I can recognize that a hand with a spade is a much better bluff-catcher than the alternatives. Calling when I have a spade and folding when I don’t would be a very close approximation to the optimal solution, costing me only about $300 in EV for the 5% of the time that he gets away with stealing a $6000 pot.

Playing high-stakes heads up no-limit hold ’em with Phil Ivey and losing no more than $300… now that’s a dream come true!

I’ve been toying around with this situation for a while, and although I want to retain some measure of caution simply because I have a history of making mistakes with this sort of thing, I’m pretty sure I’ve determined that the optimal bluffing frequency for Hero doesn’t change even if Villain has some monsters in his range.

The critical thing is that Villain have enough non-nut hands in his range on the turn to make bluffing profitable for Hero. In the hand that I posted, double-barrel bluffing for pot-sized bets on the turn and river will see Hero risking about 2800 to win about 700, or 4 units to win 1. To make Hero indifferent to bluffing, Villain needs to check-call twice with 25% (1 out of every 4 combos) or his range.

If Villain calls with more bluff-catchers than that, Hero can exploit him by never bluffing, in which case Villain will pay off Hero’s nuts too often. If he calls with fewer bluff-catchers, Hero can exploit him by bluffing all of his non-nut hands. The latter requires that Hero actually have bluff candidates in his range, a point to which we’ll return in a moment.

Because this bluff stretches over two streets, Villain needs to make Hero indifferent to two distinct bluffing lines: the single barrel and the double barrel. This requires Villain to call a pot-sized bet with at least half of his range on the turn, and then half again of that range on the river. So not only must Villain take 25% of his range to showdown, but he must take an additional 25% to the river or risk exploitation from a player who bluffs the turn and then gives up on the river.

The bottom line here is that if the nuts make up more than 25% of Villain’s range, Hero should never bluff. Similarly, Villain should never bluff-catch. This makes intuitive sense: I’ve often advised that you should fold bluff-catchers when you have many stronger hands in your range. Your opponents can’t profitably, and won’t generally try to, bluff into you when you could easily have a monster.

If Hero can polarize his range such that more than 25% of Villain’s range becomes bluff-catchers, then he can profitably bluff even if 24% of Villain’s range were nutted. The optimal frequency for doing so is just as Sklansky described in his article. On the river, Hero can bluff half as many hands as he has combos of the nuts (the nuts here referring to any hand with 100% equity against Villain’s bluff-catching range). On the turn, Hero should bet all of his nut hands, all of the hands he will bluff on the river, plus another x hands, where x is one-half the total number of combos he will bet on the river. The more combos of the nuts Villain has in his range, the less profitable this strategy will be for Hero, but as long as Villain doesn’t have the nuts more than 25% of the time, Hero is better off bluffing in a balanced way than not bluffing at all.

Finding More Bluffs

The complication in my example is that Hero may actually not have enough non-nut hands in his range to execute this strategy. If we assume that Hero’s range after seeing the turn card is {A9s,Q9s,J9s,T9s,98s,97s,96s,87s,66}, ie 7 nutted combos, 4 open-ended straight draws, and 3 counterfeited boats, he should bet 100% of his range on the turn, and Villain should nonetheless fold whenever he doesn’t have the nuts. (You might notice another complication in this last sentence, that I’ll return to at the end of this post.)

If Hero plans on shoving the river optimally, meaning with 7 combos of the nuts and 3.5 combos of bluffs, then he can profitably bet 6 additional combos of bluffs on the turn with which he will give up on the river. However, the above range gives Hero only 3.5 additional combos with which to bluff the turn. Thus, Hero should strongly consider turning hands with showdown value into bluffs if he has any. If, for instance, Hero cold calls the flop raise with a hand like 88, he should occasionally turn these hands into bluffs rather than trying to check them down and win against Villain’s flop bluffing range. (The question of how, if at all, to balance Hero’s checking range is beyond the scope of this post.)

Another minor point I’ve ignored is that some of Hero’s bluffs provide blockers to Villain’s nut combos. For example, when Hero has 8d 7d, his opponent can’t have 9d 8d. Bluffing somewhat more often with this hand would probably be profitable, but it’s a minor effect and I haven’t taken the effort to calculate it.

I’ve also ignored the possibility of Villain doing anything other than checking and either calling or folding, since betting or check-raising both seem to be dominated strategies, and the possibility of Hero using any sizing other than pot-sized bets on both streets, which I’ll address in a moment.

The main point here is to demonstrate the importance, against good players/hand readers, of being able to show up with weak hands in unexpected spots.

An Unresolved Problem

There’s one thing I still haven’t resolved, that I’ll try to sort out and post here when I do. If Villain is correct, against Hero’s extremely nutted range, to fold all of his bluff-catchers on the turn, then Hero can exploit this by bluffing the turn and giving up on the river. However, Villain could exploit that by calling the turn and folding the river with all of his bluff-catchers, which Hero could exploit by double-barreling all of his bluffs, which Villain could exploit by folding all of his bluff-catchers on the turn…. Basically I’m not sure yet what would be the equalibrium strategy in the event that the nuts comprise more than 4/9 of Hero’s range, which is the point at which he can profitably bet-shove his entire range.

My suspicion is that Hero can make the most of this situation by betting less than pot on the turn. This enables him to lose less on his bluffs while still putting sufficient pressure on Villain’s bluff-catching range. If he can do is in a balanced way, though, Hero’s optimal line is to pot both the turn and river.