A bluff-catch toy game

Derive response thresholds from a small river model without claiming a complete poker equilibrium.

13 minutes · advanced · Original Poker4Quant lesson

What you will learn

Calculate the defender’s indifferent bluff fraction and the zero-equity attacker’s indifferent call rate.

Prerequisites: Indifference · Pricing a bluff · Ties and shared pots

The defender sees a mixture of value and bluff bets

Take a heads-up river decision with old pot P before the attacker’s bet B. The defender faces P+B and must call B. Assume their hand beats every bluff, loses to every value bet, never ties, and there is no rake. Let z be the fraction of bluffs among observed bets.

Relative to folding, call EV is z(P+B)−(1−z)B. Indifference requires z=B/(P+2B). This is a bluff fraction among bets, not the fraction of all weak hands that should bluff. The attacker’s hand distribution and value-betting choices connect those quantities.

The attacker’s zero-equity bluff faces a call rate

Let c be the defender’s call probability. A pure bluff wins P on a fold and loses B on a call, relative to checking and losing. Bluff EV=(1−c)P−cB, zero when c=P/(P+B).

These two indifference conditions describe specific margins. They do not alone prove an equilibrium of full poker or even of an unspecified toy game. To complete a game solution, define available hand types, prior weights, all actions and off-path behavior, then check every incentive.

Worked example

P=100 and B=50. Find the bluff fraction making the defender indifferent, and the call rate making a zero-equity bluff indifferent.

The defender faces a 150 pot and a 50 call. Indifference requires z=50/(100+100)=1/4.
At z=1/4, call EV is 0.25×150−0.75×50=0.
The attacker’s bluff is indifferent at c=100/(100+50)=2/3.
Then (1−2/3)×100−(2/3)×50=0.

The two frequencies are 25% bluffs among bets and a 2/3 call rate at the specified margins. They condition on different populations.

Common mistakes

Using P before the bet as the pot the defender can win.
Confusing bluff fraction among bets with bluff rate among weak hands.
Presenting two threshold equations as a solved full poker strategy.

Keep these ideas

Define the pot before and after the bet.
State the population behind each frequency.
An indifference margin needs a complete game for an equilibrium claim.