Bluffing frequencies and MDF: why bet sizing decides everything

Most bluffing advice is about nerve. This page is about arithmetic, because at equilibrium bluffing is not a matter of nerve at all — it is a ratio, and the bet size you choose decides it.

Three quantities, all of them easy to compute from the pot P and the bet size b:

• MDF (minimum defence frequency) = P ÷ (P + b) — how much of your range must continue so that an opponent cannot profit by bluffing any two cards.
• alpha = b ÷ (P + b) — the share of the pot an opponent takes down for free when you fold too much.
• Bluff share of a betting range = b ÷ (P + 2b) — how much of your own betting range has to be bluffs to make your opponent's bluff-catchers indifferent.

The three sizes you actually use

Plug the sizes in and the whole picture falls out. The table is short enough to memorise, and it is worth checking each row against the formulas above rather than trusting the table:

• Half pot — your opponent should fold 33.3%, you must defend 66.7%, and about 25% of your betting range is bluffs (a ratio of 3 value to 1 bluff).
• Full pot — fold 50%, defend 50%, bluff share 33.3% (2 to 1).
• Twice the pot — fold 66.7%, defend only 33.3%, and bluffs must be 40% of your betting range (1.5 to 1).

The last row is the one people get wrong by instinct: doubling the bet does not double the bluffs, it takes them from a quarter to two fifths. The relationship is not linear, and that is most of what there is to know about sizing.

Why MDF is P ÷ (P + b)

Here is the reasoning, and it is worth having because it is what turns a formula into a decision. Your opponent bets b into a pot of P. If their bluff gets a fold, they win P. If they get called, they lose b. Bluffing any two cards breaks even when:

P × (how often you fold) = b × (how often you call)

Rearrange it and you get the fold frequency that makes a pure bluff indifferent — that is alpha, b ÷ (P + b). Defending more than the remainder, P ÷ (P + b), means bluffing any two cards loses them money. So MDF is not a suggestion about being brave; it is the point at which their worst bluffs stop being free.

A half-pot bet makes alpha 33.3%, so folding more than a third of your range there is exploitable by a player who simply bets everything. That is the entire origin of the phrase "he bets too much, I have to call more".

A river example, both sides of it

The pot is 20bb. Your opponent bets 10bb — half pot. You hold a bluff-catcher: a hand that beats their bluffs and loses to their value bets.

• Defence: MDF is 20 ÷ 30 = 66.7%. Folding more than a third of your range means their worst hands profit by betting; calling every bluff-catcher is over-defending, but not by as much as people assume.
• Price: your call needs 10 ÷ 40 = 25% equity, which is the pot-odds number from the other guide.
• Attack: if you were the one betting half pot for 20bb, about 25% of that betting range should be bluffs — roughly three value combinations for every bluff. Which hands you pick matters less than the ratio: choose hands that block their calling range and unblock their folding range.

Notice that the defender's decision and the attacker's ratio are two views of one equilibrium. If the attacker bluffs less than 25%, the defender can profitably fold everything; if the attacker bluffs more, the defender calls everything. The frequencies exist because both players are watching each other.

Sizing and bluffs move together

The trade-off in the table is real and runs in both directions. Bigger bets need fewer value hands to be credible, let you fold out more of your opponent's range — and demand a larger share of bluffs, which means the bluffs have to be chosen well. Smaller bets let you bet a wide range cheaply, but they leave your opponent a comfortable calling decision and take less of the pot when they do fold.

That is why big bets are paired with polarised ranges: hands strong enough to want a huge pot, plus selected bluffs, with the middling hands checking instead of betting. A merged range making a huge bet is giving away its own weakness.

One street is not the whole hand

Everything above is a single-street calculation, and that is the main way people misapply it. When betting continues on the next street, the real bluffing requirement is higher than the table says: your bluffs still have to survive a card, and your opponent gets another chance to raise or call. As a rule of thumb, treat the table as a floor and expect multi-street lines to need more bluffs, not fewer — how much more is a solver question, not a mental-math one.

When MDF does not apply

MDF is the answer to "I do not know what this opponent does". It is not a law:

• Against someone who never bluffs, there is nothing to defend against — fold your bluff-catchers and let them have the pot while they show you the nuts.
• Against someone who bluffs far too often, calling more than MDF is correct; the extra calls are pure profit.
• On the river, if you can see that your hand beats no value bet at all and your opponent has no bluffs in this line, folding is right no matter what the frequency says.

This is the same dividing line as everywhere else on the site: equilibrium is the default when you have no read, and deviation is how you take money off a player who has a leak. The reasoning is in what GTO actually means.

Practising it in the trainer

The panel gives you the equilibrium frequencies and the MDF for the spot in front of you, and the grade measures how far your action was from them. The productive way to use that is to look for a pattern in the misses: if most of your bad grades are calls against large bets, you are over-defending; if they are folds facing small bets, you are being pushed around by exactly the sizes that need the most defence.

One drill worth doing: play a stretch where you always bet half pot or a third of pot when you bet, and watch how the required bluff share changes with the size. Sizing stops being a matter of taste the first time you see the ratio move underneath it. Panel mechanics are in how to use this trainer, and the call-side arithmetic is in pot odds, equity and EV.