How can I create a simple formula to calculate the bluff to value ratio for hero to make villain indifferent to calling that holds the total bluffs at 1?

For example, I would like a formula to calculate all these scenarios
Hero bets…
2x pot
1.5x pot
1 pot (1 bluff : 2 value hands)
.75 pot
.67 pot
.50 pot (1 bluff : 3 value hands)
.33 pot
.25 pot

Thanks

One of the tenets of GTO play is to bet in such a way that your opponent is indifferent to choosing one of the options available to him. This is desirable because the indifference means he cannot exploit you. Assume you have the opening bet and assume that your opponent can either call or fold.

Hero: Nuts or Air .... Villain: Bluffcatchers

We assume that hero has either value hands (100% equity) or bluffs (0% equity) and villain has only bluff catchers. Your EV in pot size units if villain calls your bet of B is

EV = V*(1+B)-(1-V)*B,

where V is the percentage of hands that are value bets.

Since a villain fold results in a zero EV for him, that implies your EV is the pot, which we have set equal to 1. We therefore set our EV to 1 to solve for V:

V+VB –B+VB = 1
V*(1+2B)= 1+B

V= (1+B)/(1+2B) (the simple formula)

Then the value to bluff ratio is V/(1-V) to 1.

The following table shows the ratio of value bets to bluffs for various bet sizes in pot size units to make villain indifferent to calling or folding:

Bet in Pot Size Units	Value to Bluff Ratio
0.2	6 to 1
0.33	4 to 1
0.5	3 to 1
0.75	2.3 to 1
1	2 to 1
1.5	1.7 to 1
2	1.5 to 1
3	1.33 to 1
4	1.25 to 1
5	1.2 to 1

Example: If you make a pot sized bet, under the model assumptions you need 2 value bets for every bluff bet to make villain indifferent to calling or folding.

Note that in practice you will always make a value bet so you base the number of bluffs on the number of value bets you see. You check hands which cannot be bet

Fantastic. Thank you!

Yes, this is super helpful! I just printed out the table to staple to the back of my hand.

Ouch!