Hi guys,

This is probably not in the right forum so feel free to move it, but I thought high stakes players would have more insight in this kind of theory oriented question.

Lets say we play a game where Hero always has either nuts/bluff on the river and Villain always have a bluff catcher.

What percentage of nut to bluff ratio must Hero have in order for Villain to be indifferent between calling/folding given different bet sizes? I can't seem to get the math to make sense.

Lets say Hero bet:

50% pot
100% pot
200% pot

Example:

Lets say the pot is 100 and Hero bets 100 on the river (100% pot). According to my initial thought Hero must win the pot 50 % of the time in order for his bet size to break even, risk 100 to win 100. So given this I thought I would set Hero's range to include 50% nuts and 50% bluffs.
Lets say Villan is confused about what to do and just calls 100% of the time.

Heros payoff will be: 0.5* (100) + 0.5 * (-100) = 0
Villans payoff will be: 0.5 (-100) + 0.5*(200) = 50

This is not the result we want... Villan turns a profit employing this strategy.

Whats the formula for solving the correct nut to bluff ratio, making villain indifferent to calling/folding?

If Hero has 67% nuts 33% bluff and Villain still calls all the time:

Heros payoff will be: 0.67(100) + 0.33(-100) = 34
Villans payoff will be: 0.67(-100) + 0.33(200) = 0

This result looks better and Villain seems to be indifferent between calling/folding but I came up with these numbers just by trial and error.

Am I on the right track? and how do you employ minimum defense frequency into this?

With my above calculation Villain is calling 100%, that doesn't seem right either.

According to minimum defense frequency he should be calling 1- 100(/100+100) = 50%

I would appreciate if someone could clarify some calculations/concepts for me.

Thx