Poker Mathematics – Part 9 – The Bluff Formula Explained

We need to start with the standard B/E Formula.

B/E Formula for Fold Equity

The basic B/E Formula is Win Value = Lose Value or Win Value – Lose Value = 0

If you introduce b, where b = B/E Fold Equity %, the formula changes to

Win Value x b – Lose Value x ( 1 – b ) = 0

AND

Lose Value x ( 1 – b ) = Lose Value – ( Lose Value x b )

Therefore the formula Win Value x b – Lose Value x ( 1 – b) = 0 can be expressed differently and converted to the formula for both a Cold and warm Bluff.

Win Value x b – Lose Value x ( 1 – b )

equals

Win Valuex b – (Lose Value – (Lose Valuex b))

equals

Win Value x b + Lose Value x b – Lose Value

equals

b x ( Win Value + Lose Value ) – Lose Value

Therefore

b x ( Win Value + Lose Value ) = Lose Value

And

b = Lose Value / (Win Value + Lose Value) where b = B/E Fold Equity

If we consider the pot and a bet to try and win it, the win value = money in the pot and the lose value = the bet cost (cost of bluff).

We use this to build our formula for both Cold and Warm B/E Fold Equity

B/E Fold Equity = Cost of Bluff / (Money in Pot + Cost of Bluff) (This is the Cold Bluff Formula)

B/E Fold Equity = Average Loss / (Money in Pot + Average Loss) (This is the (warm) Semi Bluff Formula)

Average Loss

If we go back to basics and the standard B/E formula for the fold % in a Cold Bluff

Money in pot is Cost of Bluff = Cost of bluff / (Money in pot + Cost of bluff)

If we move to the Warm or Semi Bluff formula for the fold %, the Cost of Bluff is replaced by Average Loss (our true bet cost or EV when called), but what is Average Loss?

Average Loss is equal to our (winning) equity in the current pot and (winning) equity in villain calling cost, less the losing equity of our bet cost

(Current Pot x Equity) + (Calling Cost x Equity) – (BC x (100% – Equity))

However the (BC x (100% – Equity)) can be expressed as

BC – (BC x Equity)

Therefore the Warm or Semi-Bluff Formula can now be expressed as

(Current Pot x Equity) + (Calling Cost x Equity) – (BC – (BC x Equity))

OR

(Current Pot x Equity) + (Calling Cost x Equity) – BC + (BC x Equity)

The equity figure is a constant, amending the formula

Equity (Current Pot + Calling Cost + Bet Cost) – Bet Cost

The Current Pot + Calling Cost + Bet Cost is equal to the Final Pot and this reduces the Average Loss formula to

(Final Pot x Equity) – Bet Cost

Therefore if we replace the Cost of Bluff in a Cold Bluff with the Average Loss in a Warm or Semi-Bluff, our Ratio and Formula to determine the villain fold % is

Money in Pot: Average Loss equals Average Loss / (Money in Pot + Average Loss)

OR

Money in Pot is ((Final Pot x Equity) – Bet Cost) equals ((Final Pot x Equity) – Bet Cost) / (Money in Pot + (Final Pot x Equity) – Bet Cost)

We have hero covered after calling a squeeze bet preflop – how often does villain need to fold on the flop to make a bluff profitable?

Hero Hand 9d8d

Villain Squeeze Range 10%

Flop 2d5h6h

Effective Stacks 58.05

Current Pot 23.50

Final Pot 139.60 (23.50 + 58.05 + 58.05)

Bet Cost 58.05 (This is also the calling cost)

Equity 33%

(The flop equity is from Poker Stove and increases by 1% if you ignore the flop cards).

Average Loss is (139.60 x 33%) – 58.05 = 46.07 – 58.05 or – 11.98

Fold % is 23.50: 11.98 equals 11.98 / 35.48 or 33.76%

Therefore the opponent needs to fold more than 33.76% to make the semi bluff profitable and the true bet cost is 11.98 to win 23.50