Hello this is duplicate, i'm trying to maximize EV (by checking or if by betting, then with what sizing?). To simplify, we take the river, we know the FE for any sizing.
I have several questions:
1) Is a value bet on the river a bet with a hand that has 50%+ equity against a call or defense range (including raises)?If the second should we take into account our reaction to this raise or just put a defensive range into the calculator and see if we have 50%+ vs it?How about value betting on the turn or flop, and depending on IP/OOP?I heard that in these cases you need more equity, how true is that?
2) separating the calculation of EV for a range and an individual hand. Is there any point at all for the first one, that is, in summing up EV of bets with value and bluffs? For this purpose, I made a whole sheet a week ago with different sizings, FE and ratio value for bluffs, and then it seemed that this was useless information, because then you can just as well sum up the total EV of bet and check, and what will it give us?According to my sheet, the total EV of bet at a certain FE becomes maximum if we bet 100% value with as large sizing as possible and with 0% bluffs, even at 90% FE, which is obviously incorrect, because we definitely should bet all bluffs having this FE. It turns out that the correct approach would be to calculate the EV of bet and compare it with the EV of the check, and do it separately for bluffs and value? And even more correct for a separate hand with a certain equity..And here the most important thing is how to apply it in a real game, because the equity against the check range and against the call range, not to mention the raise range, is different for different hands.it depends on different conditions - preflop ranges, the line of play and the type of opponent,but are there any average indicators like EV of check with A-high and second pairs> EB of bet etc. Or is it all so individual that you need to sort out a lot of hands to understand what's what?
3) the EV of bet on the river if the opponent is only call/fold:
EVbet=F*pot+C*(Eq*finalPot-bet)
right?
and if opponent adds raises, then if:
we play bet fold
EVbet-fold=F*pot+C*(EqC*finalPotС-bet) +R*(-bet)
we play bet call
EVbet-call=F*pot+C*(EqC*finalPotС-bet) +R*(EqR*finalPotR-call)
Is it correct?

1) Neither, you need to beat >(50% + bluff raises) of the defense range. When OOP you can beat a bit less. On flop and turn there are way too many variables to turn it into a succint equation imo. There's equity denial, equity, inducing...

2) Not sure what you're talking about here. Range EV is just the sum of hand EVs, you can't maximize range EV without maximizing every individual hand EV, if you find that discrepancy then your math must be wrong

3) I think your math is wrong, you're missing some EV loss from running into a better hand when you get raised (you lose your bet and the subsequent call) unless by "call" you mean that

I don't understand, can you elaborate please?

interesting,how is this explained? "a bit less" means even <50%?

So it is,i just want to say,that regardless of our sizing and FE,EVbet of the whole range always will be higher,if we bet 100% value(assuming having 100% eq) and 0% bluffs(0% eq),opp just calls or folds.
Example:we are potbet in pot 10bb having 90% FE
1)V/B=100/0
EVbet=11
2)V/B=67/33
EVbet=10
3)V/B=50/50
EVbet=9,5
4)V/B=33/67
EVbet=9
5)V/B=0/100
EVbet=8
so at first sight we want to choose 1) option, when we bet,since EV is the biggest.But these calculations do not take into account EV of check and in example above we must bet V/B=100/100.So I came to the conclusion that I need to analyze each hand separately, comparing the EV of bet and check with this hand.In short, don't pay attention, I just described the initially erroneous path to the final correct way of analyzing-comparing the EV of bet (including bet-call/fold/raise) and the EV of check(including check-call/fold/raise if OOP),ofc the situation is even more complicated,since the equity is often differs from 100% and 0% for both value hands and bluffs,plus as i mentioned eq vs call and raise range also differs.

so doubts only about the last one? Interestingly,there are no formulas nowhere for the last two cases..
i mean can we simplify and calculate EVbet-fold as EVbet(opp calls or folds) and EVfold(opp raise) separately and then sum them up and do the same for a bet-call?

Beating bluff raises doesn't count towards EV with a thin value hand in equilibrium because you lose the entire pot on average when you face a polar raise.

Right, so your first math was akin to saying "your BU RFI EV is highest if you only open AA" where you would be missing that you want to maximize your entire range at the decision point, not just the RFI one.

Your math for facing raises isn't used often because most of the time it's a bit superfluous and too difficult to use in game. The easiest simplification in equilibrium is that your thin value hands will lose the entire pot regardless of whether you call or fold facing a raise

still didn`t get it..
example:board AJ7r6r2
we bet A8s
opp calls ATs,KJs and raises 66,98s
do we have value bet with A8s?
1)if i put whole opp`s range we have 54,55% eq
2)if i put only call range we have 60% eq
which approach is correct?or if "neither",then how much equity with what range my hand needed,so i can value bet here?I mean, isn't there a strict mathematical answer?

so calling polar raise in equilibrium is not 0EV,but minus EV?i`m confused

0EV is losing the entire pot, meaning your investment Up to that point.

Yes, there is a mathematical answer. You need to beat every bluff raise plus 50% of the defense range. This is in equilibrium.
Versus an imbalanced raise you would have to calculate the EV vs raise as it wouldn't necessarily be 0

You mean calling range?
under the spoiler is an excerpt from an article by one midstakes coach, I can give a link if necessary

following his opinion, my approach "1)if i put whole opp`s range we have 54,55% eq" is correct, but you write "neither",why?

Funny, that's exactly what I'm asking, if my formula is right and if not, which is right?

I already talked about your formula, It seems almost right to me but not quite, missing EV loss from the bet investment.

No, your equity against the entire defense range is not good, i already said this. You have to calculate the EV against raises

exactly-"call" means call of the raise,i thought that was obvious.
I don’t really understand, or you deliberately give evasive answers, or you just don’t know them, but don’t want to admit (I would just write, I don’t know):
I gave a specific example with specific combos and asked a specific question - do we have a value bet with A8s? I didn’t get a specific answer(yes or now). You write that i need to calculate Ev against a raise, so show me how to do it in this example.
According to formula:
EVcall of raise=EqR(pot+raise)-call+EqR*call
where:
EqR-our equity vs raise
pot-pot before opp raises
raise-how much opp raises
call-how much we should call after opp`s raise.
let`s say we bet 5bb in pot 10bb and opp raises 15bb(3x),we need to call 10bb and we have 50% equity(3combo 98s and 3 combo 66),then:
EV call of raise=10bb
ok we got that number,what`s next?how it helps me with knowing do i have value bet with A8s or not?
let`s count EV vs opp`s calling range(FE=0):
EVbet=EqC(pot +call)-bet+EqC*bet
EqC-our equity vs call
pot-pot before we bet
call-how much opp should call
bet-how much we bet
let`s say we bet 5bb in pot 10bb,opp calls 5bb and we have 60% equity(2 combos ATs and 3 combos KJs),then:
EVbet=0,6*15-5+0,6*5=7bb
now we`ve got all numbers and?Maybe we should also multiply these Eves by the calling and raising frequencies, i.e. by 0.545 and 0.455 respectively?

If instead of asking the same question 4 times because you just want me to tell you you're right, you took the time to understand why I'm saying you're wrong, you wouldn't need such a thorough explanation.

"the EV of bet on the river if the opponent is only call/fold:
EVbet=F*pot+C*(Eq*finalPot-bet)
right?
and if opponent adds raises, then if:
we play bet fold
EVbet-fold=F*pot+C*(EqC*finalPotС-bet) +R*(-bet)
we play bet call
EVbet-call=F*pot+C*(EqC*finalPotС-bet) +R*(EqR*finalPotR-call)
Is it correct?"

EVbet=F*pot+C*(Eq*(finalPot-bet)-(1-Eq)*bet) = F*pot+C(Eq*finalPot-bet)

EVbet-fold=F*pot+C*EqC*finalPotC-bet) +R*(-bet)

EVbet-call=F*pot+C*(EqC*finalPotC-bet) +R*(EqR*finalPotR-raise)

No, you can't do any 50% rule over this, something having a mathematical answer doesn't mean it's going to be a dumbed down 3 step formula.

The only 50% rule you can have is from an equilibrium perspective, since your EV is 0 versus raises, you need to beat 50% of the defense range but considering bluff raises as hands that beat you, so you need to beat bluff raises AND still beat an additional 50% of the entire defense range.

If I'm wrong about my math I'm happy to correct it, I don't have a math background or anything close to it. If you don't understand the concepts I'm talking about, please reread before asking the same question again.

let someone else say whose formula is correct, now I’m sure that mine is (checked with a calculator). According to your formula in my example, it turns out that EVcallofRaise=5bb,in mine it`s 10bb
in any case, thanks for the time spent, in no case do I want to strain, I'm just used to the specifics.
I consider the issue of value bet is open.