Bluff-to-Value Ratio

Also known as: bluff to value ratio, value-to-bluff ratio, bluffing frequency, polarization ratio

The equilibrium proportion of bluffs to value bets in a polarized betting range, set by bet size so the caller is indifferent.

For a polarized bettor (nuts-or-air) on the river, the equilibrium ratio of bluffs to value is fixed by bet size \(s\) (as a fraction of pot), chosen so the caller is indifferent between calling and folding a bluff-catcher.

The caller risks \(s\) to win \(1 + s\) (pot plus bet), so they need equity \(\dfrac{s}{1+2s}\) to call. For them to be indifferent, the fraction of the betting range that is bluffs must equal \[\dfrac{s}{1+2s}.\] Equivalently, value:bluff \(= (1+s) : s\). Note this is the river formula: earlier streets carry more bluffs because bluffs retain equity and can improve.

• Half-pot (\(s = 0.5\)): bluffs \(= \dfrac{0.5}{2} = 0.25\) → 3 value : 1 bluff.
• Pot (\(s = 1\)): bluffs \(= \dfrac{1}{3} \approx 0.333\) → 2 value : 1 bluff.
• 2x overbet (\(s = 2\)): bluffs \(= \dfrac{2}{5} = 0.40\) → 3 value : 2 bluff.

Bigger bets justify more bluffs because the caller folds more (higher alpha). Pick bluffs by blocker value, unblock their folds, block their calls.