Is there a formula for figuring out what percentage of the pot I need to bet on each street to get the stacks in on the river?

No because there are a finite number if decisions that are out of your control. Ie the opponent. You need to work it out on a case by case basis. If the opponent is nitty good luck getting you stack involved. If the opponent is a fish you can overbet.

Let's say on a very generic case: stacks are 100bb, you 3bet he calls we go to flop. You only need to bet just over half pot on flop turn and river to get your stack in without overbetting.

The formula you are looking for: A = B*C³

where A is the desired final pot size, B is the pot size on the flop and C is the rate at which the pot has to grow.

Example: Starting stacks 100BB, SB opens to 2,5BB, BB folds.
A = 200, B = 5, and C is what we want to find

200 = 5*C³ => 40 = C³ => 40^1/3 = C => 3,42 = C.

That means the pot has to increase by the factor of 3,42 on each street.

So we need to bet around 6 BB on the flop for a new pot size of 17 BB and around 21 BB on the turn for a new pot size of 59 BB and shove the river for 70,5 BB. That means we have to bet roughly 1,2 times the pot on each street to get all-in by the river.

In a 3bet pot SB vs. BB, the numbers could look like this:

200 = 18*C³ => 11,11 = C³ => 2,23 = C

That means we need to bet 11 BB on the flop, 24,5 BB on the turn and shove the river for 55,5 BB. So we have to bet slightly over 60% pot on all three streets to get all-in.

That was a far better explantation than mine but same principle.

In a standard pot you need to overbet. In a 3bet pot heads up you need to bet approx 60% on each street.

why would you want to know the average (smallest) amount to bet on every street? Are you assuming that this will be the path of least resistance to get stacks in by the river? I don't think this is strictly the case in all situations. Different bet sizing lines (eg Big Bet, small Bet, Big bet) might achieve more success as it looks weaker than the same sized bet on every street? Many pros will notice changes in bet sizing which you could exploit past history with. Also opponents have different tolerances (Elasticity) for bets on certain streets so a read would obviously help....

In fact If getting stacks in by the river is the goal you might try moving more of your bets to earlier streets as there might be untapped drawing equity you could make your opponent peel more with. You might also try an nicely timed overbet on the river ?

just thoughts...

Madlex equation is basically correct but that's assuming a constant growth rate, but there are different scenarios where different bet sizes might work better.

I interpreted the question as assuming you bet a constant percentage of the pot on the flop, turn and river and villain calls each time. You want the percentage to be such that on the river, you put villain all in betting the same percentage of the pot. If this interpretation is not correct the result may still be of some interest.

Assume pot = 1 on the flop and villain’s stack is S times the pot. If I did the algebra correctly, the percentage of the pot to bet each round is given by the equation,

3P+6P^2+4p^3 = S.

I’m too lazy to solve this cubic equation so I put it into Excel and used its Goal Seek function.

Here are three examples:

S = 3, P=45.7%
S= 5, P = 61.2%
S=10, P = 87.9%

Assume the pot is 10 and villains stack is 50. So, S as I defined it is 5. So you bet 6.12 on the flop (61.2% of 10). Villain’s stack is now 43.88 and the pot is now 22.24. You again bet 61.2% of the pot or 13.61 so villain’s stack is now 30.26 and the pot is 49.46. A river bet of 61.2% of the pot is 30.27, which is villain’s remaining stack

I get the same equation. Here's the wolfram input

f = 1/2 * [³√(2s+1) - 1]

where s = (stack / pot)

That's obv. true, but isn't constant growth rate the only thing that's at least somewhat interesting theory-wise?

Check, check, shove gets us all-in on the river, but I highly doubt that's what OP wanted to know.

I don't understand what principle?

if you use equatoin Madlex wrote, in 3bet pots starting pot size is bigger, so you will get smaller pot growth rate. Equation is 100% correct in all cases.

Point here is not if beting equal fraction of the pot on each street to get stacks in is right or wrong.
OP wanted to know equation for it, and as you can see, there is one.

I didn't mean to take away from your post as it was quite good.

I was merely trying to point out that betsizing should be dynamic based on the situation to maximize profits. The constant growh rate is a fantastic starting point, but it's probably not the best place to stop studying.

I don't understand why you guys are following this approach. Matlex had already given a much better approach with no need of wolfram to solve. His approach also generalizes easily to any number of streets and (for what it's worth) any number of players :

n*s + 1 = (n*b + 1)^m

s: effective stack size measured in proportion of pot
b: bet size in units of pot
n: number of players
m: number of streets

The right side is the desired final pot size and n*b + 1 is the rate of growth of the pot for a given bet size. We first want to solve for that rate of the growth, then solve for b. This gives:

b = ((n*s + 1)^1/m - 1)/n

Notice that with n = 2 and m = 3, it yields your formula.

BTW, this is not how one should be sizing their bets. Each street is very different from the others and the best bet size for each should be different as well. Besides that, computing the m-th root at the table is a pain in the ass. Not at all practical.

He didn't do that in this thread (or he kinda did but I skimmed over it).
I don't think anyone is suggesting it is. Statman was just answering OP's question. I just checked his work and solved his cubic.

Except he (nor madlex) didn't answer the OP's question. OP did not specify 3 streets

It's Madlex's approach that everyone should be paying attention to.

Is there any easy way to estimate it on the flop?

Recently, I gave an easy way in this thead. BTW, if you wanted to save yourself the hassle computing the SPR for the formulas given there, you can use these pair of formulas instead:

Bet = (Stack + Pot)/5 for 2 streets

Bet = (Stack + 7*Pot)/20 for 3 streets

These cover the most useful range of bets of about half-pot to pot size bets.

Hi, ty for your help
When I give the following formula into excel will it work?
=SUM(1/2*(2*S+1)^(1/3)-1/2)
S=SPR
3 streets 2 players
at spr 4 it gives 54% but it should be 50%
at spr 13 it gives correctly 100%

54% is correct for SPR = 4. At SPR = 3.5, b = 50%.

For example, lets assume s = 3.5. Pot is equal to 1.

Our 1st 0.5 pot bet makes our stack goes down by 0.5 to 3, but the pot goes up to 1 + 2*b, so s for the next street is s' = (s - b)/(1 + 2*b) = (3.5 - .5)/(1 + 2*0.5) = 1.5. Pot is again set to 1.

A 2nd 0.5 bet gives a new s' = (1.5 - 0.5)/(1 + 2*0.5) = 0.5. Pot is set to 1 for the last time.

Now the 3rd and last 0.5 bet puts us all in as expected.

Originally, I came up with the approximating formula b = (s + 6)/19, which is exact for s = 3.5 and 13 producing 50% and 100% bets respectively. However, who wants to to divide by 19? So I fudged it a bit to make it easier to work with. The result was satisfactory as seen in the tables I produced in the other thread.

The exact formula: b = ((2*s + 1)^1/3 - 1)/2 is correct, but verifying this can be confusing if you are not very careful. It is easy to get tripped up. Been there, done that