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.
The request:
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.
https://poker.stackexchange.com/questions/1045/the-right-bluff-frequency
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?
You can write $$\frac s{1+2s}=f\\ s=f(1+2s)\\ s=f+2fs\\ s-2fs=f\\ s(1-2f)=f\\ s=\frac f{1-2f}$$