Blockers and Picking Bluffs

Four cards do not make your blockers stronger

River T♥9♦8♣4♠2♥. The best possible hand is a queen with a jack — 16 specific two-card combinations. Two questions, and most players get the second one backwards.

How often does somebody have it? An opponent with four cards holds 6 different two-card combinations, so he has six chances at those sixteen: 7.73% of all possible hands are the absolute nuts. With two cards it is 1.48% — one combination and no second chance. That is 5.2 times as often.

How much does holding one queen take away? Hold Q♣5♦3♠2♣ and the number drops from 7.73% to 7.03% — a cut of 9.1%. Run the identical calculation against an opponent holding 2 cards instead of four and the same single queen cuts 1.48% down to 1.33%, a cut of 10.2%.

A single blocker removes roughly a tenth of the danger. Treat it as a tie-breaker between two hands you were already close to indifferent about, never as a reason to fire.

Any river where the best possible hand needs two specific cards — which is every river, because you must use exactly two.

"Four cards means my blockers are stronger" is exactly wrong: 9.1% here against 10.2% with two cards. The four cards help the other player, not your removal. What is genuinely different is the denominator — 7.73% against 1.48% — so you need blocking more and get it less.

Both columns are complete enumerations by our rule layer on the same board, differing only in whether the opponent is dealt four cards or 2. Nothing here is copied from anywhere.

The best blocker sits on the worst hand

Same river T♥9♦8♣4♠2♥. Pairs of blockers do move the needle. Here is the whole ladder, and then the problem with it:

• One queen: cuts 9.1%.
• Q♣J♦5♠3♣: 30.1%.
• Q♣Q♦5♠3♣: 37.8%.
• Q♣Q♦J♠3♣: 52.2%.
• Q♣Q♦J♠J♥: 67.3%.

Now look at what those hands are. Queen-jack, queen-queen-jack and queen-queen-jack-jack all contain a queen and a jack, so all three are ranked 1 on this board: they are the nuts. You do not bluff with the nuts. The only hand on that list you would ever want to bluff with is the pair of queens, ranked 21, which is a pair on a board where a pair is worth nothing — and it cuts 37.8%.

Pick bluffs from the hands that block a lot and win nothing. On this board that is one specific shape: a pair of the card the nuts needs, with no jack.

River, heads up, when the best possible hand is a straight that needs two named ranks. On a flush board the same logic runs through the two highest cards of the suit instead.

Do not read the ladder as "more blockers, better bluff". The rungs that cut the most are the rungs you cannot bluff with, and the one you can bluff with is not the top. Ranking bluff candidates by removal alone picks the nuts every time.

Our own thresholds, so you can see where the line is drawn: a hand is called a blocker at 30% relative removal and at least 1 point of absolute removal, weak from 15%. The absolute floor matters — one queen removes 0.71 points and the pair removes 2.93, and on a board where the nuts is rare a large percentage of almost nothing is still almost nothing.

Block what calls, not just the nuts

River K♥8♥3♥2♠7♣. Three hearts, so "the ace of hearts blocks the nuts" — everybody says it. Count it against the single best hand and the ace is no better than the queen. Count it against the hands that would actually call you and the ace is far better. Which of those two you count is the whole lesson.

The best possible hand here is the ace and the queen of hearts: exactly 1 two-card combination out of everything, which is 0.56% of all opponent hands. So:

• A♥2♣7♦9♠ removes 100% of it.
• Q♥2♣7♦9♠ removes 100% of it — the identical number.
• J♥2♣7♦9♠ removes none of it. Its removal is 19.7 points in the other direction, because taking four cards out of the deck slightly concentrates what is left.

Against that one combination the ace and the queen are identical: the nuts is a pair of cards and either one kills it. But nobody calls a river bet here only with that one combination. What calls is any ace-high flush — the ace of hearts plus any other heart. Before your cards come out, an opponent holds one 4.15% of the time. Hold the queen and it becomes 4.45% — higher, because the queen takes out a single one of those hands while your four cards concentrate the rest. Hold the ace and it is 0%: every ace-high flush needs that one card.

Before asking whether you block the nuts, name the class of hands that would call your bet, then count how much of that class your cards remove.

Any river bluff where you can say which hands would call. On a flush board that class is usually the top flushes, and they all share the one card at the top — which is why the ace matters there and a lone middle card does not. Straight boards work the same way with ranks instead of suits.

Do not stretch this into "always bluff with the ace". Removing a class is only worth something if that class is a big part of what calls. And the single-combination numbers above still fail our rule layer: both hands remove 100% of the nut combinations, but 0.56% was all there was to remove and our floor is 1 point of absolute removal. Which hands would call is your read of this opponent; the counting is the part we can do exactly.

Every number on this page is a complete enumeration over every opponent holding on that board. The relative threshold is 30%; the absolute floor exists because relative removal goes wild whenever the target is rare.

What to take away

• A single blocker is weak, and it is not weaker here for the reason you think. 9.1% against four cards, 10.2% against two.
• The difference is the denominator. The nuts turns up 7.73% of the time instead of 1.48%, which is 5.2 times as often.
• The blocker you can use is a pair. A pair of queens cuts 37.8% and is ranked 21 — the only hand on the ladder that is both a real blocker and a real bluff.
• Block what calls, not just the nuts. Against the one best combination the ace and the queen both remove 100% — of a hand so rare it fails our 1-point floor. Against the ace-high flushes that would actually call, the queen leaves 4.45% and the ace leaves 0%.

All of it comes from complete enumerations by our own engine, and the thresholds are rules of this course, not an equilibrium solution — nobody solved anything to produce them. Next: the street where these decisions get expensive.