Before I show you how to simplify that I want to make sure you're aware of the basic method of calculating expected value for any event where the outcomes are known.
EV(action) = P(outcome 1)*EV(outcome 1) + P(outcome 2)*EV(outcome 2) + P(outcome 3)*EV(outcome 3) + .... for all outcomes of the action.
P is the probability of the outcome occurring, and EV is the value from that outcome. All P values should add up to 1, or 100%.
There are three immediate outcomes to 3-betting a hand preflop:
EV(3-bet) = P(fold)*EV(fold) + P(4-bet)*EV(4-bet) + P(call)*EV(call)
If we assume he opens the button for 4bb and you 3-bet from the BB to 12bb, he folds to the 3-bet 39% and 4-bets 9.1%:
EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*EV(call)
I want to clear up that your EV of 13.5 when he folds to the 3-bet makes no sense, all you win is his raise plus the blinds, 5.5bb in my example.
As you can see, we have no value for EV(call). That value is an entirely separate EV problem that is actually quite complex, as we don't know what the flop will be, what our action will be, what his response to that action will be, and what turn and river runouts will come, not to mention the different action sequences of those runouts.
EV(call) = Sum of P(a million different things)*EV(a million different things)
...basically.
Theorists use the "R value" to simplify this problem. R stands for realization of equity. In this case I will refer to it as the percentage of the pot that you will win on the flop if your 3-bet is called. We know the pot size will be 24.5bb minus rake, if you assume a 4bb open and 12bb 3-bet. Your R value will be the percentage of that 24.5bb pot you expect to win, on average. This counts all the times you will make profitable bluffs, all of the times that you will get to check down the best hand, but also all the times you'll be able to get value from worse. It's expressed as a percentage of the 24.5bb pot, but that percentage can be over 100, and would be if for example your entire 3-bet range was AA, a hand with heavy implied odds against worse hands.
EV(call) = (24.5 - rake)*R - 11
So this expresses that you will win R% of the pot on the flop, deducting that you already lost 11 when your 3-bet was called.
The new full equation is as follows:
EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*((24.5 - rake)*R - 11)
if you assume the rake will be 1.5bb,
EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*(23*R - 11)
Now, we can set EV(3-bet) equal to zero to find out what your R will need to be when called to justify the 3-bet:
0 = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*(23*R - 11)
R = 38.2%
Reply With Quote