River bluff-to-value ratio
Bluff-to-value ratio on the river, for every bet size. The derivation first, then the numbers for every common bet size, then where the idea stops being true.
The river toy game: the bettor has only the nuts or nothing, the caller has only bluff-catchers that beat the bluffs and lose to the value. The bettor wants the caller to have no good answer, so bluffs just often enough that calling and folding are worth the same. With bluff share s:
EV(call) = s × (P + B) − (1 − s) × B set to 0: s (P + 2B) = B => s = B / (P + 2B) value : bluff = (1 − s) : s = (P + B) : B pot-sized bet (P = B = 1): s = 1 / 3, value : bluff = 2 : 1
The bluff share is the same fraction as the caller’s required equity. That is not a coincidence: when the bettor bluffs exactly that often, the bluff-catcher’s equity is exactly its price.
| Bet | Bluffs in betting range | Value : bluff | Bluffs per 10 value combos |
|---|---|---|---|
| Quarter-pot bet | 16.7% 1 / 6 | 5 : 1 | 2.0 |
| Third-pot bet | 20.0% 1 / 5 | 4 : 1 | 2.5 |
| Half-pot bet | 25.0% 1 / 4 | 3 : 1 | 3.3 |
| Two-thirds-pot bet | 28.6% 2 / 7 | 5 : 2 | 4.0 |
| Three-quarter-pot bet | 30.0% 3 / 10 | 7 : 3 | 4.3 |
| Pot-sized bet | 33.3% 1 / 3 | 2 : 1 | 5.0 |
| 1.5× pot overbet | 37.5% 3 / 8 | 5 : 3 | 6.0 |
| 2× pot overbet | 40.0% 2 / 5 | 3 : 2 | 6.7 |
| 3× pot overbet | 42.9% 3 / 7 | 4 : 3 | 7.5 |
When the bettor gets it wrong
Each cell is what a bluff-catcher earns by calling, in pots, if the bettor’s range is that share bluffs. Green: the bettor over-bluffs and calling prints. Red: they under-bluff and calling burns money, so fold. Bigger bets punish the caller more for guessing wrong in either direction.
| Bet | 10% bluffs | 20% bluffs | 25% bluffs | 30% bluffs | 40% bluffs | 50% bluffs |
|---|---|---|---|---|---|---|
| Quarter-pot bet | −0.10 | +0.05 | +0.13 | +0.20 | +0.35 | +0.50 |
| Third-pot bet | −0.17 | 0.00 | +0.08 | +0.17 | +0.33 | +0.50 |
| Half-pot bet | −0.30 | −0.10 | 0.00 | +0.10 | +0.30 | +0.50 |
| Two-thirds-pot bet | −0.43 | −0.20 | −0.08 | +0.03 | +0.27 | +0.50 |
| Three-quarter-pot bet | −0.50 | −0.25 | −0.13 | 0.00 | +0.25 | +0.50 |
| Pot-sized bet | −0.70 | −0.40 | −0.25 | −0.10 | +0.20 | +0.50 |
| 1.5× pot overbet | −1.10 | −0.70 | −0.50 | −0.30 | +0.10 | +0.50 |
| 2× pot overbet | −1.50 | −1.00 | −0.75 | −0.50 | 0.00 | +0.50 |
| 3× pot overbet | −2.30 | −1.60 | −1.25 | −0.90 | −0.20 | +0.50 |
Where the toy game stops
- Real rivers are not nuts-or-air. Thin value bets lose to some calls, and some bluffs have showdown value, so the right ratio moves.
- The ratio is about the river. On earlier streets bluffs have equity, which is why flop and turn betting ranges carry far more “bluffs” than this table allows.
- It counts combos, not hands. Ten value combos might be two hands. Starting hands lists how many combos each hand has.
Every bet size
- Quarter-pot bet16.7%
- Third-pot bet20.0%
- Half-pot bet25.0%
- Two-thirds-pot bet28.6%
- Three-quarter-pot bet30.0%
- Pot-sized bet33.3%
- 1.5× pot overbet37.5%
- 2× pot overbet40.0%
- 3× pot overbet42.9%
The ideas behind the numbers
- 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.