Hi there. Simple maths question.

Say you have 4 combos of bluffs otr and 12 combos of value.

What is the equation to come up with bet sizing that makes villain neutral to calling?

Thanks

Your missing a whole lot of I formation there to make an accurate assumption, but:
4*(pot+bet size as a %of the pot)= 12*pot
If we assume pot is 100 to negate the %:
4*(100+bet size) =12*100
1200/4 -100 =Bet size
Bet size =200

If you have the same bluff combos as value combos:
1*(100+Bet) = 1*100
100 - 100 = bet
=0.... If your perfectly balanced it makes no difference

0bluff Combos:
0*(100+bet)=1*100
100/0 -100 = infinity.....size would have to be infinitely big to call....

Hope this helps

How do we know that all our value hands beat all our opponent value hands or even miracle hands that developed by river?

The entire thing is far more complex.

Also why calling only? The opponent can raise also as a bluff or for real.

You could however design a specific situation that the bet is either folded or called because it is on no particular fold equity to raise it if the stacks are not very deep.

Just basically realize that what you are describing here as a question will not have any real value in general, only when the situation is very specific to be able to give you some very clear yes or no answers and not something more complicated that usually exists in real life.

Your bet basically should be putting opponent all in for you to be able to study more easily all cases that your value hands are no longer good always, something that is affected by the board (ie how good they tend to be).

Your value hands can definitely not be good often here something that cannot be ignored if you act first and have also no other very clear information. If your value hands are top pairs and overpairs and higher how do you know he didnt get 2p at the river or trips or set or some not easily expected straight while he was in for some flush draw etc.

So you need to severely restrict the board to be able to ask such basically simple assumption questions.

That discussion also may be better for poker theory.

i think op is asking about the proverbial river situation where he always has a polarized range and opponent always has a bluff catcher. he wants to know sizing to make opponent indifferent (point of nuetrality) with a fixed range consisting of 12 value hands and 4 bluffs. as masque pointed out, its usually not quite that simple outside of a toy game.
i posted yesterday but ended up deleting because i figured someone else could explain better than i can. but after seeing this i thought i better pop back in and clear things up, or maybe learn something myself! im not really sure whats going on here ron.
if we bet 200 into 100 on river then opponents ev of call would be (300*.25)+(-200*.75)= -75
opponent could attain a higher ev (0) by folding.
we want to find the case where opponent cant improve his ev regardless of what he does. lets try setting our equation to 0 and see what that provides.
0=(100+x)(.25)+(-x)(.75)
x=50
if we bet 50 into 100 then opponents ev is 0.
if he folds, again his ev is still 0, so its neutral.

For this I wanted to assume that only one bet could go in and that we are betting a polarised river range.

Where does .25 and .75 come from in equation?

Thanks for replies

.25 and .75 is your percentage of bluffs and value respectively.

you have 4 bluff hands and 12 value hands for a total of 16 hands.

4 bluff hands out of 16 total hands means you have 25% bluff hands
4/16=.25

12 value hands out of 16 total hands means you have 75% value hands
12/16=.75