## Saturday, August 10, 2013

### Block and Tackle (and Combinatorics)

We all know to assign ranges and not specific hands when hand reading, right? We don't say, "The TAg villain open raised UTG, and then re-raised our LAggy 3bet (with us holding AhKd) on the button. Therefore he must have Aces." Instead, we should be thinking something like, "The TAg villain opened UTG, so his range is TT+, and AQo. Then he saw us 3bet him on the button, but he knows we're LAggy, so he probably thinks we could have a range anywhere from TT+ to AQ+. He then re-raised us, so we can rule out TT and AQ from his range, as he probably wouldn't 4bet with those. So let's put him on JJ+ and AK. So, how does our AKo stack up against this range?"

Big deal, this is standard hand reading in action, right? Yes, but there's an important step that's missing: factoring in so-called "blockers."  A blocker is a card that we know with 100% certainty that the villain does not have in his hand. In other words, these are cards that a) we hold; b) are on board; and/or c) have been inadvertently exposed by another player or the dealer.

In this example, we hold AhKd. This means that these two specific cards are "blockers" to hands our opponent can have. In other words, we know with complete certitude that they cannot be in the villain's hand.

Uh, okay. That's pretty obvious, Bug. What's your point?

My point is that from a combination point of view, it makes a  difference to the range we put villain on, and therefore how we can/should play our own hand

Without factoring in our two blockers, we said villain's range was JJ+ and AK. What this actually means is that villain has one of the following hand combinations:
• Jacks (6 combinations): JcJd, JcJh, JcJs, JdJh, JdJs, JhJs
• Queens (6 combinations): QcQd, QcQh, QcQs, QdQh, QdQs, QhQs
• Kings (6 Combinations): KcKd, KcKh, KcKs, KdKh, KdKs, KhKs
• Aces (6 Combinations): AcAd, AcAh, AcAs, AdAh, AdAs, AhAs
• Ace-King (16 Combination): AcKc, AcKd, AcKh, AcKs, AdKc, AdKd, AdKh, AdKs, AhKc, AhKd, AhKh, AhKs, AsKc, AsKd, AsKh, AsKs
Of these hand combinations, we're in a coinflip with twelve of them (i.e., the Jacks and the Queens), we're getting crushed by twelve of them (Aces and Kings), and there are 16 we're likely to chop with if we get to showdown (Ace-King)*.

Ah, but we forgot to factor in our Ah and Kd blockers. In other words, we can remove all the Ah and Kd cards from the aforementioned combos. If we do this, we see that our villain actually can only have one of the following hands:
• Jacks (6 combinations): JcJd, JcJh, JcJs, JdJh, JdJs, JhJs
• Queens (6 combinations): QcQd, QcQh, QcQs, QdQh, QdQs, QhQs
• Kings (3 Combinations): KcKd, KcKh, KcKs, KdKh, KdKs, KhKs
• Aces (3 Combinations): AcAd, AcAh, AcAs, AdAh, AdAs, AhAs
• Ace-King (9 Combination): AcKc, AcKd, AcKh, AcKs, AdKc, AdKd, AdKh, AdKs, AhKc, AhKd, AhKh, AhKs, AsKc, AsKd, AsKh, AsKs
Said another way, we've reduced the percentage of the villain's range that is crushing us. It's not a huge amount, but it's enough that it could alter how we play our hand back at villain. Poker is a game of identifying and exploiting small edges, which is exactly what this calculation leads to.

Below are a couple of pie charts that shows this effect in graphical form. On the left is the breakdown of hands without factoring in our two blockers, and on the right is the effect with us factoring in the blockers:

So what do we do with our AK in this situation? Well, that depends of course. How "gambly" is our opponent? What is effective stack size? What game format is this? How much fold equity to we have? And so on. For instance, what if this was a deepstack cash game? If so, we might want to just call and reevaluate on the flop**. If this were early in a moderate-stack tournament, however, we might decide that we're too much of a dog to call and that we don't have enough implied odds, so it's time to muck. If we're getting short-stacked in a fast-structure tourney but still have some fold equity, however, we might just 5-bet shove all-in. It doesn't really matter; the point is that we can make much better decisions if we can make better reads, and one step to doing this is to remove "blocker" cards from the range we put our opponent on.

All-in for now...
-Bug
*I'm ignoring the small advantage villain's suited AKs have over our non-suited AK.
**And remember, the three cards on the flop are also blockers, so we can remove them as necessary from our opponent's hand range, too, at that point as we further refine our read. The turn and the river also are blockers.