I'm working on a poker problem where I'm trying to calculate how much of the pot I want to bet (as a percentage). I'll give the nuts and bolts of my request first and then an explanation of it's usage in case that helps or matters.
If you look at this post in the poker stack exchange you will see a comment by
paparazzo where he gets the formula
s / (1 + 2s) = f. I want to reconfigure that formula to solve for "s". i.e. I know my bluff frequency "f" and I want to know what percentage of the pot "s" I should bet to make my opponent indifferent to calling. So: how do I solve for "s" in this formula; or if more appropriate, write an entirely different formula that accomplishes my goal.
Detailed explanation Often in poker you come at this problem from the perspective of "f" in the formula above. e.g. There is 1 in the pot and my opponent bet 1. How often do I have to call so that he can't profitably bluff with any two cards. In this scenario you are calling 1 to win 3 (original pot + opponent bet + your call). 1/3 = 33%.
Then you can do something like say: I have 10 value hands here and I want to bet pot. Betting pot gives my opponent odds of 2-to-1 on a call. So I need to bluff with 5 hands to make him indifferent to calling or folding. 5 bluffs making my bluff % the same as his pot-odds. If I only wanted to bet half-pot then I would be giving him 3-1 odds to call so I need to bluff 2.5 hands.
But I want to come at this problem from the other direction. Analyze a hand and when I get to the river I might end up in a situation where I just have a lot of value but not very many bluffs in my range. If I have 18 value hands and only 4 possibly bluffing candidates; then I can't go looking for other bluffs to add because they don't exist. So knowing that I'm going to be bluffing 22% of the time; what percentage of the pot do I need to bet to make my opponent indifferent to calling or folding?