Bluff Catching

By Andrew Brokos

Introduction

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.

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

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

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

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?

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!