I wrote a formula, which is meant to calculate the average profit, when semi-bluffing two villains at once. Contrary to the usual advice, that one should almost never semi-bluff two opponents at once, there seem to be a lot of profitable situations, where one should do so, at least if the formula is any good. The biggest issue of my attempt seems to me, that the formula gives the villains only two options to react to the bluff, which is either calling or folding. It misses out on their option to answer the bluff with a raise.
I am really quite new to poker, so it is hard for me to estimate, how much impact villain raises could have on the profitability of a semi-bluff. Would be nice, if somebody could comment on that. As it is also very likely, that I am not the first one to attempt something like this, I also would be happy if anybody could link me some material for further studying this part of poker.
So here are the variables of the formula:
As I was more interested in percentiles than in absolute values, I assumed that the original pot-size is 1.
Bet: size of the bluff, compared to the pot before the bluff
odds: probabillity to still win the hand when, bluff gets called
F1: the probabillity that Villain 1 will fold to the bluff
F2: the probabillity that Villain 2 will fold to the bluff
AAV1: average added value for Villain 1 (is explained below)
AAV2: average added value for Villain 2
Given, that our bluff gets called, then the new pot-size would be 1+(1+n)*Bet, where n is the number of villains calling the Bet. "average added value for Villain x" gives the portion of this new pot size, we expect "Villain x" to additionally add to the pot, if we still win the hand after out bluff gets called. This is strongly related to the concept of "implied-odds". In this context, I just found it easier to calculate with AAV.
Every value should be a number betwenn 1 and 0. Only "Bet" can be bigger then 1.
And the formula itself:
( F1*F2*(1+Bet)
+ F1*(1-F2)*odds*(1+2*Bet)*(1+AAV2)
+ (1-F1)*F2*odds*(1+2*Bet)*(1+AAV1)
+ (1-F1)*(1-F2)*odds*(1+3*Bet)*(1+AAV1+AAV2)
- Bet ) / Bet
Explanation:
Spoiler
The first line multiplies the probabillity, that both villains fold (F1*F2), with the money we collect in this case, which is (1 + Bet).
The second line multiplies the probabillity, that "Villain 1" folds but "Villain 2" doesn't, with the EV for this scenario, which is the odds times the sum of the pot-size after the bluff got called (1 + 2*Bet) and the value, we expect "Villain 2" to still add to it ((1 + 2*Bet)*AAV2). Above this is expressed as "(1+2*Bet)*(1+AAV2)" instead of "(1 + 2*Bet) + (1 + 2*Bet)*AAV2".
The third line is like the second, just with the inversion of "Villain 1" and "Villain 2".
The forth line multiplies the probabillity that both villains call, with the EV for this scenario.
The last line substracts the initial investment from the turnout and then divides by the investment to show the ratio of the profit relative to the investment.
As I said above, I don't know how sound this formula is, as it doesn't grant the opponents the option to raise after our bluff. Other problems are for example, that it is assumed, that the odds to win the pot are the same against one and two opponents. It also only considers unraised pots. If anybody has any suggestions or comments, I would be happy to discuss them.
Some of the findings calculated with the formula in its current shape:
Given that both opponents only would fold in 10% of the time and that they only would add an average of 10% of the value of the new pot after their call, we would still have an average profit of about 10%, to our investment, if we bluff 1/2 pot-size and our odds are 20%.
If everything stays the same, with the only difference, that one opponent folds in 90% of the cases, the average profit would decrease to about 6%. This seems counterintuitive, but is explainable with the fact, that our EV is lowest, when only one opponent stays in the game, while we are about to break even if both opponents do, because the more opponents call us, the better our pot-odds/implied-odds.
If we increase the bluff from 1/2 pot-size to 1x pot-size this would decrease the average profit to about 4% compared to the first scenario, even if it would increase both opponents probabillity to fold from 10% to 50%.