The easiest way to solve these types of problems is just to keep track of contributions to the pot.
For example, consider a triple barrel bluff, pot-pot-pot sized bets with a starting pot of 1.
Villain always calls flop and turn. How often do we need them to fold the river?
- Starting pot = 1
- Villain's contribution so far = 1 + 3 = 4
- Hero contribution if we bluff river = 1 + 3 + 9 = 13
So on the river we are
risking 13 to win 5 (villain's contribution so far plus the starting pot)
alpha = risk / (risk + reward) = 13 / (13 + 5) =
72.2%
So if they always call flop and turn, we need them to fold at least 13/18 = 72.2% on the river for the triple barrel bluff to be profitable.
Let's try another example where villain folds 60% on the river:
We can calculate that triple barreling is not the most profitable line, losing 2.2 pots total. You'd be better off just giving up on the flop. However, you must pull the trigger on the third barrel if you have already double-barreled.
The intuition here is that your river bet is profitable; they're overfolding 10% on the river.
But it's not profitable enough to recuperate your flop and turn bet.