Postflop Coach — free heads-up postflop poker trainer中文 · 日本語 · Deutsch

Polarization & Bluff-catching

Lesson 7 · MDF in polarized spots, shown on a toy model · 24 practice questions

Polarized betting

On the river there are no more cards, so a bet can only do two things: get called by worse or make better fold. Medium hands can do neither well — the hands that call them usually beat them, and the hands that fold were already behind — so they check. What is left in a betting range is the very strong hands, which want to be called, and the bluffs, which want a fold. That is a polarized range: two ends and no middle. Bluffs and value are linked by the price the bet offers. If you bet pot, the caller needs 33% equity, so to make his bluff-catchers indifferent your betting range should be about one-third bluffs — one bluff for every two value hands. Bet smaller and you can bluff less; bet bigger and you can bluff more.

Example. This course uses a toy model, not a solver run on real ranges: on the river 2♣7♦9♥J♠4♠ with 100 chips in the pot, the bettor holds either J♣J♦ (a set that beats everything) or Q♥T♥ (air), half the time each, and the caller holds A♦A♣, a pure bluff-catcher. The bettor bets pot with every set and with half of his Q-T: out of 100 hands he bets 50 sets and 25 bluffs, so bluffs are one-third of his bets. The caller calls 100 to win 200 and wins 1 time in 3 — exactly the price, so calling and folding are worth the same.

Common mistake: betting medium hands on the river 'for value'. If only better hands call, it is not a value bet; check and let the showdown decide.

Bluff-catching

A bluff-catcher is a hand that beats every bluff and loses to every value bet — beyond that, its strength does not matter. Against a polarized bet all your medium hands are bluff-catchers, so the decision is no longer 'how good is my hand' but 'how many of these hands should continue'. Two numbers answer it. The price says what share of the time a call must win: bet/(pot + 2 × bet). MDF says what share of your range must continue so that villain cannot bluff with any two cards: pot/(pot + bet). Against a pot-sized bet you need 33% and must defend 50%; against a half-pot bet you need 25% and must defend 67%. If villain bluffs more often than the price, every bluff-catcher calls; if he bluffs less, they all fold; at balance, you call with the best half and fold the rest.

Example. In the toy model on 2♣7♦9♥J♠4♠, A♦A♣ faces a pot-sized bet of 100 into 100. It needs 33%, and the bettor bluffs exactly one-third of the time, so calling breaks even — and if A♦A♣ folded too often, every Q♥T♥ would become a profitable bluff. That is why the default answer is call. Against 50 into 100 the price drops to 25% while MDF rises to 67%: call again. In a real hand, choose the bluff-catchers that hold blockers to villain's value hands (a blocker is a card you hold, which he therefore cannot hold) and do not hold cards from his bluffs; course 14 covers blockers in detail.

Common mistake: raising a bluff-catcher. A raise folds out the bluffs it beats and is called only by the value hands it loses to.

Toy model: pot bet MDF = 50%

The toy model strips a river spot down to its skeleton so the frequencies can be solved by hand. One player is polarized — the nuts or nothing — and the other holds a single bluff-catcher. Two conditions fix the equilibrium. The bettor bluffs just often enough that the bluff-catcher is indifferent: against a pot-sized bet, one bluff for every two value hands. The caller defends just often enough that a bluff is indifferent: against a pot-sized bet, 50% — exactly MDF. Neither side can gain by changing alone. Solving a game this small reproduces the textbook formulas; it is a teaching model, not a solver run on real ranges, and real rivers have medium hands, blockers and several bet sizes.

Example. On the low river 8♣7♦5♥3♠2♣ (pot 100) the bettor's value is Q♠Q♦ and his bluffs are 9♠9♦, and the caller always holds J♦J♣. Q♠Q♦ bets every time — it beats the jacks and gets called half the time. 9♠9♦ looks like an overpair but loses to J-J, so it is a bluff: bet it at the frequency that keeps one bluff per two value hands, and check the rest. Change one fact — the bettor's range is only J-J and you hold K♥T♥ on 2♣7♦9♥J♠4♠ — and nothing is left to catch: fold.

Common mistake: treating the model's numbers as a rule for real hands. Use its logic — polarized bets, a bluff ratio tied to the size, defense near MDF — and adjust when the real opponent bluffs more or less than the model.

Key takeaways

• River, polarized range (nuts or air), betting pot → bet every value hand; bluff about one hand for every two value hands.
• Bluff-catcher facing a pot-sized polarized bet → you need 33%; defend about 50% of your range (MDF), keeping the hands that block value (blockers; see course 14).
• Bluff-catcher facing a half-pot bet → you need 25%; defend about 67%.
• Medium hand first to act on the river → check; a bet is called only by better.
• Opponent who never bluffs → fold your bluff-catchers; opponent who bluffs too much → call them.

Sample drill from this lesson

FAQ

Key terms in this lesson

MDF (Minimum Defense Frequency) · Equity · Range (Poker Hand Range) · Polarized Range · Bluff Catcher · Indifference (Mixed Strategies) · Value Bet · Blockers

All 29 lessons