Pricing a bluff

What you will learn

Derive a bluff’s break-even fold frequency under a stated baseline.

Prerequisites: Expected value as a weighted average

A fold wins the old pot, not your returned bet

Consider a final bet B into an existing pot P. In this toy model, the opponent folds with probability f, and you always lose if called. Compare bluffing with checking and losing for certain, whose incremental value is zero.

When the opponent folds, your bet comes back and the net gain is P. When called, the bet loses B. Thus EV(bluff)=fP−(1−f)B. This is a model of fold equity with zero showdown equity, not a general rule for every bluff.

A threshold is conditional on a behavioral estimate

Setting EV to zero gives f*=B/(P+B). A larger bet raises the required fold frequency if P is fixed. A player’s actual fold rate can depend on the bet size, so changing B while holding f fixed is a hypothetical sensitivity experiment.

Checking may have positive value if your hand can win or induce a later action. That changes the correct baseline. Likewise, a bluff that sometimes wins when called needs an extra conditional showdown term. Keep those distinctions visible before using the simple equation.

Common mistakes

• Counting the returned bet as profit after a fold.
• Using a no-showdown model when the hand sometimes wins if called.
• Forgetting that opponent behavior may change with sizing.

Keep these ideas

• Define checking’s value before comparing.
• Zero-equity bluff value uses fold and call branches.
• A price-implied threshold is not an observed fold rate.