Capped Ranges

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In the May issue of this magazine, I introduced a simplified system of hand-reading that works quite well against the majority of no-limit hold ‘em players. The idea is not to put an opponent on a specific hand but rather to put him on one or two of three possible broad categories of hands: monsters with which he wants to play a big pot, moderate-strength hands that figure to be best but that he will generally try to showdown cheaply, and draws or other weak hands that will rarely win without bluffing.
This month, I want to demonstrate one way of using this information to guide your play. On the turn or river, it is often possible to determine, based on how he has played on earlier streets, that your opponent will rarely or never have a monster hand. This is sometimes called a “capped range”, meaning that there is an upper limit to how strong your opponent’s hand can be. If you are able to determine this limit, you can both value bet all of your better hands strongly and represent better hands when you need to bluff, putting your opponent in a very difficult spot.
Often, there are two ways for a bluff to go wrong. Either your opponent happens to have a very strong hand, maybe even the one you are trying to represent, and can easily call you, or he guesses that you are bluffing and calls you down with a weak hand (or re-bluffs you, for that matter). When you are able to cap his range, though, you can all but eliminate the first possibility and put your opponent in the uncomfortable position of never having cards stronger than those you are representing. The best he can hope to do is guess at your bluffing frequency and call you down accordingly.


A Simple Example
We’ll start with an example, probably oversimplified, to illustrate the point. Suppose that you are playing in a €5/€10 9-handed NLHE game. The player first to act opens with a €40 raise. He is extremely tight and predictable, and you know that he would only raise from first position with a big pair, probably tens or better, or a big Ace. Because you have such a precise read, and because the effective stacks are €2000, you call on the button with a suited JT. The rest of the table folds.
The flop comes 982, all different suits, giving you an open-ended straight draw. Your opponent bets €70, and you call. The turn is an off-suit 6, and your opponent bets €150. At this point, just based on his very tight pre-flop raising standards, you are able to identify a clear upper limit on your opponent’s range. He cannot have a hand stronger than one pair, albeit a good pair. You, on the other hand, could have a set, two pair, or even a straight. You raise to €500, and your opponent calls.
The river is a 4. Your opponent checks, and you move all in for a little more than the size of the pot. He scowls at you, fiddles with his chips, mutters something about garbage hands and bad beats, and folds.


A Trickier Example
In a 6-handed €5/€10 NLHE game, the player one off the button opens with a €35 raise. The action folds to you in the big blind, and you call with T [spade] 9 [spade]. The effective stacks are €1000. The flop comes 7 [club] 5 [club] 4 [diamond], missing you entirely. You check, prepared to fold, but your opponent checks behind you.
What can you determine about his hand so far? Many players will open a wide range from late position, so you have very little information about his hand pre-flop. When he checks the flop, though, he is telling you a good deal.
This is not a flop that gets slowplayed very often. If your opponent flopped two pair or a set, he has a lot of incentive to bet. For one thing, there are scary cards that could come on the turn. From his perspective, they may improve you to the best hand, and even if they don’t, they may scare you away from paying off with a second-best hand.
Similarly, because of the coordinated nature of the board, there are a lot of ways you could have flopped a second-best hand. Anything from a worse two pair to a pair and a straight draw to overcards and a flush draw will be willing to put a fair bit of money into the pot on this flop. Thus, if your opponent flopped a strong hand, he would have a lot of incentive to bet it. When he checks, you can be fairly sure he does not have two pair or better.
What could he have? Anything from total air (which in most cases will still beat your hand) to something like Ace-high that has a little bit of showdown value to a strong pair that nevertheless fears a check-raise. Significantly, though, he will very rarely have anything stronger than one pair, possibly with a draw to go along with it.
The turn brings the A [spade]. You could bluff now, but even if you follow it up on the river, this will only put so much pressure on your opponent. It wins nothing more from his bluffs, and many players will call down with a pair after checking the flop. Note that your opponent’s pot control line on the flop has made it difficult for you to bluff him in a conventional way.
If you check, though, your opponent will very often bet. If he has nothing, this is a good card for him to represent. If he hit the Ace, he’ll probably bet for the same reasons that a strong hand would want to bet the flop. Even with a weaker pair, he may feel more confident in his hand, now that you have checked twice, and not want to give a free card. Unless he has A4, A5, or A7, though, it is very unlikely that he will have better than one pair.
You check, and your opponent bets €60 into the €75 pot. You can now make a large raise to represent a strong hand that missed a check-raise on the flop and decided to go for one on this turn.
You make a pot-sized raise to €255. This raise will often take the pot down right away, but suppose that this player makes a stubborn call. It is now more likely than ever that he does not have a strong hand. Given the coordinated nature of the board and the fact that you seem to like your cards, a strong hand would almost certainly put the rest of the money in on the turn.
Thus, you can follow up with a big bluff on most if not all rivers. Your opponent’s flop call probably either represents a skeptical pair that needs a little more convincing or a pair with a draw. Another big bet on blank rivers should show a nice profit.
Bluffing at 8′s, 3′s, and clubs will be a higher variance play but will probably still show a profit. Sometimes you will end up bluffing into a rivered monster, but you may also have better fold equity against pairs that fear you hit the draw yourself.
In this case, your opponent calls the turn raise, and the river brings the K [spade]. You bet €450 into the €585 pot, and he folds. Whereas leading out on the turn may have enabled you to steal a small pot, check-raising the turn and following up with a river bluff won you a much larger one. Essentially, you tricked your opponent into putting a lot of money into the pot when you knew he could not defend it.
Your potential fold equity here is so high that you might choose to make this play even when you have some showdown value, such as when you hold 43. Your pair could be best, but often it won’t be, and you may get outplayed on the river anyway. Turning a weak pair into a bluff like this will often prove more profitable than trying to catch bluffs with it.


Defending Against These Plays
If you are in a game with a player who reads hands this well and has the cojones to makes plays like this, you might do well to find another table. But if you do choose to stay, you should know how to protect yourself.
The first and best strategy is to practice avoidance. When playing with a very competent hand reader, you must go to great lengths to disguise your hands and change up your play. This will require some unconventional play, such as occasionally raising 98s in first position at a 9-handed table or checking back a set on a coordinated flop. These are risky plays, but they are safer than enabling a tricky player to eliminate strong hands from your range altogether.
You may also need to resort to game theory and refuse to fold the best hands in your range, even if they are not objectively all that strong. If you were the Villain in the second example, you might have to call down when you turn a pair of Aces. Sure, your opponent could have a stronger hand, but since you very rarely will, folding a pair of Aces is highly exploitable.


Thin Value Bets
What if your opponent reads this article or just figures out that he’s getting exploited and decides to start calling down with his strong one-pair hands? Believe it or not, you don’t need to stop bluffing him. The occasional tricky play or thin call down are just stop-gap measures that turn an insanely profitable bluff into a marginally profitable bluff. You won’t win nearly as often, but if you can stand the variance, you’ll still show a profit in the long run.
Realize also that capping your opponent’s range will enable you not only to bluff but also to make big, thin value bets. In the second example, if you know your opponent rarely or never has better than one pair, then you could take the same line with a weak two pair hand like 54 or even one strong pair such as AK. There are very few second-best hands that will pay you off, but there are also very few better hands in your opponent’s range. And if he does correctly adapt by occasionally calling down with one pair, he’ll be in for a surprise!


Conclusion
Poker is a battle for information. Any time that you have more information about your opponent’s hand than he has about yours, the potential for profit is there. It is simply a matter of figuring out how to make the most of it. Even many players who read hands well don’t always take full advantage of that information.
Learning to recognize when there is an upper limit on an opponent’s possible hand strength and how you can exploit it can present you with some very profitable opportunities. Hopefully it will also demonstrate the critical importance of mixing up your own play, at least when there are good hand readers at the table, so that no one is able to take advantage of you.







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!