Hello,here is situation:
We are on flop in heads up and we didnt put any money before flop.
We have a hand,that has a 33%eq against opp`s range and it wont change on a turn and river(so 2 hands in opp`s range have 100%eq and 1 hand has 0%eq).Villain potbets 1bb in pot 1bb,we call.Same on a turn,he bets 3bb in pot 3bb.And river 9bb in pot 9bb.
What is the formula,that shows that our EVcallFlop,EVcallTurn and overall EVhand/line<0??Obviously EVcallRiver=0
EVhand/line=EVcallFlop+EVcallTurn+EVcallRiver right?

p.s afaik EVcall=Eq(pot+call)-bet+Eq*bet

I’ve been doing some research on early street EV analysis with future betting and your post addresses that issue. You have villain betting a consistent fraction of the current pot, in this case, f = 1.0. You can show that the showdown equity you need is the following:

eq = (Y^n – 1)/(2*Y^n),

where Y = (1+2*f) and n = number of future bets to be made (e.g, for turn, n=2)

For the posted example, f = 1.0 and n = 3, so Y = 3. Therefore, the needed equity is

eq >= (3^3 – 1)/(2 * 3^3) = 26/54 = 48.2%

To see this another way, if the flop pot is 1, the bets at each street for both players are 2, 6 and 18 for a total of 26 and the total pot is then 27. You put in half of the player bets or 13. Therefore you contributed 13/27 = 48.2%, so that is your “fair” share.

A similar analysis can be done if hero is the bettor with showdown equity of eq. Then the constant fraction of the pot he should bet is given by the following:

f = 0.5*((1/X)^(1/n)-1),

where f = 1-2eq.

Naturally, adjustments should be made as events unfold.

EDIT. For the last "where", it should read

where X = 1-2*eq

Thanks for reply.
Additional question:if we call flop in our example,that means if villain bets on turn/river even 1bb in pot 1000bb we should fold?))
In practice,if we think that we have 33% sd eq vs cbet on a dry flop(wa/wb),but also know,that opp is aggro player and often barrels,we should immediately fold our hand,if he potbets flop?
Will calling with hands,that have +5-15%eq to potodds be acceptable decision?
And most important-it turns out that he can exploit us, forcing us to throw out our equity and we cant do anything about it (if we play without raises)?

So formula can be used when we are both callers and betters?
And same for ?
I calculated for a second formula-if we have 33%,then f=48%,we can bet or call almost half of pot.But if i use f=1/2 in first formula it shows,that i need 44%eq and not 33%.
Wrong calculations?

The first formula is for when hero is the caller as it determines what equity he needs for breakeven if villain always bets f*Pot.

The second equation is for when hero is the bettor and has an estimated showdown equity. It determines what fraction of the pot he should bet on each street to assure break-even. Note- although I didn’t mention it, the equation applies only for equity < 50%. If the bettor has an estimated equity >= 50%, one strategy is to bet the largest amount he thinks villain will call if that results in EVbet > Pot, the EV if villain folds.

In both cases, things might and probably will change as betting continues. For example, if hero is the caller and villain changes his bet amount fraction, then hero’s required equity will change.

If hero is the bettor, as board cards are dealt, his estimated equity may change so he may have to adjust his betting strategy.

If both players are continually adjusting, I guess we can say they approach an equilibrium??? The anomaly will likely be that the equity estimates of both players will not sum to 100%.

Thanks man,great formulas!
Still it seems,that both formulas works,no matter whether we make call or bet.Just my manual calculations were wrong.I used online calc. and here are results:
f=1,eq=0,48
f=0,5,eq=0,44
f=1/3,eq=0,39
But in practice, when we make a bet, we often have FE, so it’s better to use it all,when we are callers.
And again what do you think about that?

The equations I posted make the strong assumption that villain continues to bet a constant fraction of the pot if you are the caller or that your equity stays the same if you are the bettor. In a following post I noted that one should naturally adjust if the assumptions are not true and the model can accommodate adjustments – i.e. you now have a new decision point.

It was also assumed that no raises or folds occur. As with most math analyses of this type, you have to adjust as necessary – go with the flow. If villain bets 1bb into a 100bb pot, I don’t need math to tell me I have to call or raise even if I have an 8 high – maybe he has 7 high.

I meant all other circumstances remain unchanged,except sizing.You still have your 33%sd eq on a turn,which wont change on the river.So your point is,that if V bets any size<potbet on a turn,we should call(or better example is,when we check-check turn and he bets on the river)?Because despite the fact,that we spent our "credit" on a flop already,if we fold to sizing<potbet on the river,we make mistake.
I just try to extrapolate that to real situation.If V bets pot on the flop and we know,that he always barrels,then best decision is fold.And my question is why he can so easily exploit us with his sizings?We cant even raise lol,bc he`ll just fold his 0%eq hand and calls with another two.

Here is one way at looking at what the model is doing.

It’s the turn. Villain tells you he will bet the pot and river no matter what cards come if you will call both times. You decide – ok. Then each put in 1 on the turn making the river pot 3 and each put in 3 on the river so the total pot after all bets is 1+ 2 +6 = 9. You put in a total of 4 so your equity at the decision point has to be at least 4/9= 44%. Equivalently, suppose villain went all in on the turn with a bet of 4. With a call, you would have the same pot odds (1.25 to 1) requiring equity of 44%

It isn’t that complicated. The OP asked about future betting and this is one approach. How useful it is can be questioned.

Doesn’t it seem strange to you,that you realize 0% of hand's equity instead of “legitimate” 33%,simply because of villain's sizing?
Anyway thx for your help,mb someone else will post their thoughts..