Card-removal and river frequencies
Friday, 19 July 2013
High-level thinking from Ike Haxton
If you’ve spent any time at all talking strategy with serious poker players, you’ve almost certainly heard someone justify a play they made on the basis of “blockers”. It’s highly likely they were talking about having the nut flush blocker: bluffing or calling a potential bluff because they hold the bare A? on a three spade board, for example. Or maybe they had T-T on a J-8-7-3-2 board. When you and/or your opponent represents a very strong hand, the value of greatly reducing or even entirely eliminating the possibility that your opponent has the nuts is fairly apparent and pretty widely understood.
The importance of top-pair blockers, and even kicker blockers, in scenarios where both players have relatively wide ranges is greatly underestimated, however. Let’s examine an illustrative example in close detail and see how it differs from a solution that ignores card-removal effects. Don’t worry if you don’t agree with some of the assumptions about how each side is playing up until the river; they are deliberately oversimplified. The point is what happens on the river given these assumptions.
Suppose you’re playing heads up no limit and your opponent raises from the button, you three-bet and he calls. The flop comes Ks7c2d, you bet and he calls. The turn is an off-suit 3h, you bet and he calls. The river pairs the two and you go all-in for a pot-sized bet. Let’s make some assumptions about the ranges each player reaches this spot with:
Our range to shove the river for value is A-A (6 combos), K-K (3 combos), A-K (12) and K-Q (12), for a total of 33 value-bet combinations. We always play these hands exactly like this. We also always play A-Q (16 combos) and Q-J (16 combos) the same way up until the river but at this point in the analysis we haven’t yet decided if we bluff with them or how often.
We’ll assume that our opponent always four-bets pre-flop with A-K and pairs 9-9+ and folds pre-flop with kings worse than K-8o, K-5s and sevens worse than A-7o and 8-7s. We also assume he always raises the flop or turn with K-J or better and folds anything worse than second pair to our turn bet. Thus, our opponent reaches the river with the following hands:
K-T (12)
K-9 (12)
K-8 (12)
K-6s (3)
K-5s (3)
8-8 (6)
A-7 (12)
Q-7s (3)
J-7s (3)
T-7s (3)
9-7s (3)
8-7s (3)
This is 75 total combinations, of which 42 are K-x.
So, we have a pure nuts or air versus bluff-catcher scenario here. All of the hands in our range are either effectively the absolute nuts – ie, they beat 100% of the hands in the opponent’s range, or pure air – ie, they beat 0% of the hands in our opponent’s range.
If we were to check, our opponent will always check back. It’s impossible for him to value-bet or bluff because we will always call with our nut hands and never with our air hands. Thus, it’s clear we should always bet our nut hands. It’s less clear what we should do with our air hands. Surely we want to bluff sometimes, but how often? Does it matter whether we use A-Q or Q-J as our bluff? How should our opponent respond?
Of course, if we think our opponent will almost never call we should go ahead and bluff, and if we think he will nearly always call we should never bluff. However, if we expect the opponent to play very well, it makes sense to think in terms of what we would do if we had to reveal our plan to our opponent and allow him to play the best possible response. The best possible strategy given this assumption is sometimes called the “game theoretically optimal strategy”, and a pair of such strategies, one for each player, is called a Nash Equilibrium. If these concepts are totally unfamiliar to you, I recommend spending a few minutes reading about them on Wikipedia.
Typically, when I’ve seen similar situations analysed in poker books or discussed on forums, the analysis ignores card removal and proceeds as follows:
Our opponent always has a bluff-catcher so we need to choose our bluffing frequency such that he breaks even by calling. If we make calling profitable, he will always call and we can never bluff. If we make calling unprofitable, he will never call and we should always bluff. At equilibrium, we will bluff at a frequency where his calls will break even and he will call with a frequency that makes our bluffs break even.
We’re going all-in for the size of the pot, so we lay our opponent 2:1 odds on a call, so he breaks even if he wins a third of the time. Therefore, we need to construct our range to contain one combination of bluff for every two combinations of value. We have 33 combinations of value, so we choose any 16.5 combinations that we like from our range of 32 air hands and use them as bluffs. In other words, we bluff just slightly more than half the time we get to this spot with air.
We lay ourselves 1:1 on a bluff, so in order to make our bluffs break even, our opponent needs to call us half the time. Since all of his hands are pure bluff-catchers it doesn’t matter which ones he calls with. He can flip a coin or he can call with his best 50% of hands.
However, consider the situation from our opponent’s perspective when he holds Kc5c. Now there’s one fewer king in the deck. Where previously there were three ways for us to have K-K (KcKd, KcKh, KdKh), now there’s only one (KdKh). Similarly A-K is reduced from 12 combinations to eight: AcKc, AdKc, AhKc, and AsKc are no longer possible, leaving only AcKh, AdKh, AhKh, AsKh, AcKd, AdKd, AhKd, AsKd. In the exact same way, K-Q is reduced from 12 combinations to eight. A-A is unaffected, so, in total, we are reduced from 33 value combinations to only 23. At the same time, he blocks none of our bluff combinations. So, if we’re bluffing with 16.5 combos, as our card-removal-naïve game theorists would have us do, when our opponent calls us with a K-x bluff-catcher, he doesn’t break even at all. He wins nearly 42% of the time and shows a profit of more than 25% of the pot! And our opponent has a king more than half the time, so if he calls us with all of them our bluffs lose a lot of money! This can’t be equilibrium at all.
How can we fix this? The trick is that we need to make our opponent indifferent to calling with the actual bluff-catchers he has, accounting for card removal. So, we assume our opponent does have K-x when constructing our range. Thus, from his perspective, we only have 23 combos of value and only get to have 11.5 combos of bluffs.
Does it matter whether we bluff with A-Q or Q-J? Consider the blocker effects on our opponent’s range. A-Q kills three combos of A-7 and one combo of Q-7s. Q-J, meanwhile only kills one combo of Q7s, and one combo of J-7s. Neither hand blocks any of our opponent’s calling hands, but A-Q blocks more folding hands so it seems that Q-J is the better choice. Since we have 16 combos of it and only get to shove 11.5, we shove it just under three quarters of the time we have it and never shove A-Q.
Now, how does our opponent respond? Just as we needed to account for card removal and shove a range that makes his actual bluff-catchers indifferent, he needs to call so that Q-J specifically is indifferent to shoving. Accounting for the fact that Q-J kills two of his combos, he has 73 combos and 42 of them are K-x. He needs to call 36.5 of these. They’re interchangeable in terms of both hand value and blocker value, so it’s irrelevant which ones he chooses. He should pull out his random number generator and call 87% of the time that he has a king. Notice that this range of 36.5 combos is slightly less than half of the card-removal-naïve counting of 75 combos in our opponent’s range.
Let’s double check what happens if we shove A-Q into this response: accounting for card removal, our opponent now holds 71 combos, including the same 42 of K-x as before. If he pays off the same 36.5 combos of K-x, now he’s calling 51.4% of the time, so A-Q loses slightly while Q-J breaks even.
Let’s also check what happens if our opponent decides to pay off with 8-8: this hand blocks none of our range, so we have 33 combos of value and 11.5 of bluff. He wins just under 26% of the time. This mistake costs him nearly half the size of the pot!
So, in summary, the Nash Equilibrium for this river scenario is this: we shove all of our value and we bluff just less than three quarters of the time with Q-J but never with A-Q. Our opponent pays off with about 87% of his K-x hands but never with worse. The biggest takeaway from this example is that we get to bluff significantly less than we would conclude we should when examining this spot without accounting for card removal. It also drives home the point that not all bluffs and bluff-catchers are created equal. The subtle ways in which card removal affects our opponents’ ranges is frequently relevant and sometimes very large. If you think this seems too detailed to apply in real time as you play, you’re right. And, in fact, this scenario is simpler than most that will actually occur in-game. However, doing this sort of exhaustive analysis will train your intuition to identify relevant information quickly at the table. For example, if a scenario like this one comes up while I’m playing, I don’t “solve” it, but little guiding principles pop into my head immediately like: “He has a lot of kings, card removal forces me to bluff less often,” or “He can have A-7 but he can’t have A-K, it’s better to bluff without an ace in my hand.” In today’s tough games, that sort of intuition is a big part of what separates the best players from the rest of the pack.