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 from js/content/postflop-rules.js 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.

The ace is not special

River K♥8♥3♥2♠7♣. Three hearts, so "the ace of hearts blocks the nuts" — everybody says it. It is false, and the way it is false is worth more than the rule itself.

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.

Holding the ace and holding the queen are exactly as good, because the nuts is a pair of cards and either one of them kills it. There is no ace privilege. There never was — it is imported from a game where the flush is made from one hole card.

Work out which two cards make the best possible hand, then check whether you hold either one. That is the whole method, and it replaces every rule of thumb about aces.

Any board where a flush is possible. The same reasoning runs on straight boards; it just has two ranks instead of two suited cards.

Blocking everything is not the same as having a good bluff. Both hands above remove 100% of the nut combinations and our rule layer still refuses to call either one a blocker, because 0.56% was all there was to remove and our floor is 1 point of absolute removal. All of the nuts is 100% of very little. That gap between "I block it all" and "blocking it is worth acting on" is exactly where money gets set on fire.

The three rows are complete enumerations over every opponent holding on that board (the postflop rule spec). 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.
• The ace has no privilege. Ace-high and queen-high remove the identical 100%, and 100% of a rare hand still fails our 1-point floor.

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.