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The maths of bluffing: alpha and MDF
How often a bluff must work for its size, how much of a range has to keep going against a bet, and where those two numbers stop applying.
A reference page: an idea the course assumes, without practice or progress. Lessons link here the first time they use the term.
On this page
- Work out how often a bluff of a given size has to succeed.
- Work out what share of a range must continue so a pure bluff cannot profit.
- Relate a bigger bet to more folds needed by the bettor and fewer calls needed by the defender.
- Explain why both numbers are only guides before the river and against players who rarely bluff.
How often a bluff must work
A pure bluff wins the pot when your opponent folds and loses the bet when he calls. It breaks even when the two balance out, and that fold rate has a name: alpha.
Alpha equals the bet divided by the pot plus the bet.
- pot
- the pot before your bet
Bet 10 bb into 20 bb and you risk 10 to win 20: the bluff needs folds 10 / 30 of the time, about 33%. If he folds more, any two cards profit from betting; if he folds less, the pure bluff loses.
The defender's side: MDF
Turn it around. If you fold more often than alpha, your opponent can bet any two cards and profit. To stop that, a defender has to continue, by calling or raising, with at least the rest of the range: the minimum defence frequency.
MDF equals one minus alpha, which is the pot divided by the pot plus the bet.
Against a half-pot bet of 10 bb into 20 bb, MDF is 20 / 30, about 67%. That is not the caller's price: the call needs 25% equity, as the pot-odds lesson showed. Alpha and MDF answer "how often", pot odds answer "with what".
- MDF: defend at least
- 66.7%
- Alpha: folds needed
- 33.3%
- Caller needs
- 25%
Sizes worth knowing
Like pot odds, alpha and MDF depend only on the bet as a share of the pot:
- A third of the pot: alpha 25%, MDF 75%.
- Half the pot: alpha about 33%, MDF about 67%.
- Two thirds of the pot: alpha 40%, MDF 60%.
- The size of the pot: alpha 50%, MDF 50%.
- Twice the pot: alpha about 67%, MDF about 33%.
How many bluffs to bet
From the bettor's side, the question is how many bluffs to put with your value bets on the river. With the right mix, a hand that beats only bluffs gains nothing by calling or folding. That share of bluffs is the bet divided by the pot plus twice the bet.
For a pot-sized bet that is 1 / (1 + 2), about 33%: one bluff for every two value hands. For a half-pot bet, 10 / (20 + 20) = 25%: one bluff for every three.
If you feel you bluff too much, count. Add up your value combos first, using the last lesson's counting, then allow bluffs in that ratio. Against players who call too much, use fewer.
Where the baselines stop
Alpha and MDF are exact only for pure bluffs on the river, against an opponent you know nothing about. Everywhere else they are guides.
- Before the river, bluffs have equity, so they need fewer folds, and defending hands can be pushed off later.
- Against a player who rarely bluffs big, defending to MDF just pays off value. Fold more.
- Against a player who bluffs too much, defend more than MDF.
- MDF counts calls and raises together, and with several players left the defence is shared between them.
Rules of thumb
- Alpha = bet ÷ (pot + bet): how often a pure bluff must work.
- MDF = pot ÷ (pot + bet): how much of a range must continue against a bet.
- Half pot: alpha about 33%, MDF about 67%. Pot: 50% each.
- On the river, bluff in the ratio bet ÷ (pot + 2 × bet) of your bets.
When they break
- Against players who under-bluff, fold more than MDF; against over-bluffers, defend more.
- Before the river both sides have equity still to come, so the numbers are a starting point, not an answer.