// pot odds · concept

Minimum defense frequency (MDF)

Minimum defense frequency (MDF), worked out for every bet size. The derivation first, then the numbers for every common bet size, then where the idea stops being true.

The minimum defence frequency is how often you must continue against a bet so that the bettor cannot profit by betting any two cards. It comes from the bluff, not from your hand. A bluff with no equity risks the bet B to win the pot P:

EV(bluff) = fold × P − continue × B
set to 0 with fold = 1 − continue:
(1 − c) P = c B   =>   c = P / (P + B)

half pot (P = 2, B = 1): MDF = 2 / 3 = 66.67%

The fold frequency that makes a bluff break even, B / (P + B), is often called alpha; MDF is 1 − alpha. Both are on every bet size:

Pot odds, required equity and MDF by bet size
BetPot oddsCall needsMDFBluff must workRiver value : bluffOuts to call a turn bet
Quarter-pot bet5 to 116.7%80.0%20.0%5 : 18
Third-pot bet4 to 120.0%75.0%25.0%4 : 110
Half-pot bet3 to 125.0%66.7%33.3%3 : 112
Two-thirds-pot bet2.5 to 128.6%60.0%40.0%5 : 214
Three-quarter-pot bet2.33 to 130.0%57.1%42.9%7 : 314
Pot-sized bet2 to 133.3%50.0%50.0%2 : 116
1.5× pot overbet1.67 to 137.5%40.0%60.0%5 : 318
2× pot overbet1.5 to 140.0%33.3%66.7%3 : 219
3× pot overbet1.33 to 142.9%25.0%75.0%4 : 320

What defending too little gives away

Each cell is what a zero-equity bluff earns, in pots, if you continue that often against that size. Green means the bettor is printing money with air. The zero crossing in every row is that size’s MDF.

EV of a pure bluff by bet size and defence frequency
Betdefend 30%defend 40%defend 50%defend 60%defend 70%defend 80%
Quarter-pot bet+0.63+0.50+0.38+0.25+0.130.00
Third-pot bet+0.60+0.47+0.33+0.20+0.07−0.07
Half-pot bet+0.55+0.40+0.25+0.10−0.05−0.20
Two-thirds-pot bet+0.50+0.33+0.170.00−0.17−0.33
Three-quarter-pot bet+0.47+0.30+0.13−0.05−0.22−0.40
Pot-sized bet+0.40+0.200.00−0.20−0.40−0.60
1.5× pot overbet+0.250.00−0.25−0.50−0.75−1.00
2× pot overbet+0.10−0.20−0.50−0.80−1.10−1.40
3× pot overbet−0.20−0.60−1.00−1.40−1.80−2.20

What MDF does not say

  • It is a range frequency, not a hand rule. It says how much of your range continues, not that a given hand must. Which hands make up that share is an equity question.
  • It assumes the bluff has no equity. A bluff with outs loses less when called, so it breaks even on fewer folds, and stopping it takes more defence than MDF, not less.
  • It ignores what the bettor actually holds. If a player never bluffs, defending to MDF only pays off their value bets. MDF is the line against an opponent you cannot read; against one you can, the opponent ranges and the spot check beat it.
  • Raises count as defence, and on early streets so do calls that plan to fold later. MDF is cleanest on the river, where nothing else happens.

Every bet size

The ideas behind the numbers

// what is computed and what is assumed
  • The prices are exact. Pot odds, required equity, MDF and the bluff frequencies are arithmetic on the pot and the bet, printed as reduced fractions. Nothing is rounded until it is displayed.
  • Draw odds are counted. The chance of hitting comes from the outs chart, which deals every turn and river. We assume clean outs: every out wins and nothing else does. Real draws are dirtier, and against a stated range the spot check counts the real equity instead.
  • No rake, no future betting unless a page says otherwise. Implied odds are the one place we price a later street, and that page states its model.
  • MDF and the river ratio are a toy game, not a solver output: a bettor with only the nuts or air, a caller with only bluff-catchers. They say what stops the other player profiting automatically. They are not a strategy for any single hand.