Disclamer: this is rather a long post as I wanted to include all my thoughts on the subject so people responding would be able to be specific if they were to correct me, also, I do appreciate anyone willing to read through this and give feedback



BTN: 101.4 BB
Hero (SB): 173.2 BB
BB: 217 BB
UTG: 119.4 BB
CO: 161.4 BB

Hero posts SB 0.4 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has A T

fold, fold, fold, Hero raises to 3 BB, BB calls 2 BB

Flop: (6 BB, 2 players) 4 5 Q
Hero bets 3.2 BB

How can we calculate EV for this flop bet?

preflop BB calls with 307 combos

on the flop villain has 278 cmbs, versus hero's flop bet villain will continue with 197 combos and this range has 52% equity vs Hero's holding
197/278=0.71 thus villain continues 71% of the time and fold 29% of his starting range on the flop

Now, betting 3.2bb into 6bb pot hero needs 35% folds for a play to be break even
risk/risk+reward
3.2/6+3.2=0.35
villain doesn't fold enough so bet itself is -EV

We can calculate EV of this bet:

times villain folds * pot - times villain continues * Hero's bet

0.29*6-0.71*3.2=-0.53bb

but this calculation is only true for pure bluffs as we assume that when villain continues Hero will always lose, in the example above hero has 48% equity versus a range villain would continue with

Yes, villain can deny us realizing our equity through raiseing and/or betting on other streets, his further actions can effect (or are effecting) his range, turn and river cards will change our equity, but do we care about all that when we are calculating EV of a singular action on a given street? all we need is villain's continuationg range and percentage of times he'd fold, and from that we get the EV of a play, realizing our equity after the play does not depend on a play we are calculating for

But then how do we calculate the EV of that bet?

times villain folds * pot + times villain calls * (hero's equity * pot) - times villain calls * (villain's equity * hero's bet)

0.29*6bb+0.71*(0.48*6bb)-0.71*(0.52*3.2bb)=+5.69bb

as in case we had 0% equity (a pure bluff) the formula would look like this:

0.29*6bb+0.71*(0*6bb)-0.71*(1*3.2bb)=-0.53bb
same as 0.29*6-0.71*3.2=-0.53bb


========================


MP: 115.6 BB
CO: 73.2 BB
BTN: 100 BB
SB: 117 BB
Hero (BB): 100 BB
UTG: 136 BB

SB posts SB 0.4 BB, Hero posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has 5 K

fold, fold, CO raises to 2 BB, fold, fold, Hero calls 1 BB

Flop: (4.4 BB, 2 players) 6 9 5
Hero checks, CO bets 2.2 BB, Hero calls 2.2 BB

How do we calculate EV for Hero's call on the flop?

villain opens CO with 333 combos
on the flop he has 307 combos
he bets 2.2bb into 4.4bb pot
his betting range has 170 combos and has 60% equity versus hero's holding

Hero would be calling 2.2bb to win 6.6bb
risk/risk+reward
2.2/2.2+6.6=2.2/8.8=1/4=25%
so we need 25% equity to have 0 EV
0.25*6.6bb-0.75*2.2bb=+0bb
but we have 40% equity versus villain's betting range, therefore:
0.4*6.6bb-0.6*2.2bb=+1.32bb

so is it fair to say that this flop call brings us +1.32bb?

also, if hero would raise here to 6.6bb and villain would continue with 40% of his betting range having 84% equity when he continues we'd have the next scenario:
hero is risking 6.6bb to win 6.6bb
risk/risk+reward=6.6/6.6+6.6=1/2=50%
thus this raise is +EV
how much +EV?

times villain folds * pot + times villain calls * (hero's equity * pot) - times villain calls * (villain's equity * hero's raise)

0.6*6.6bb+0.4*(0*6.6bb)-0.4*(1*6.6bb)=+1.32bb
so if hero had 0% equity when villain continues hero'd make 1.32bb on this raise, but he has 16% equity, thus:

0.6*6.6bb+0.4*(0.16*6.6bb)-0.4*(0.84*6.6bb)=+2.16bb

Therefore, if hero's estimations are correct raise to 6.6bb in this spot will bring 0.84bb more then a call, thus if we'd have only these two options raise is an optimal play

Also, all that should mean that when we are bluffing with a hand that has some equity versus villain's continuation range we need less folds to break even on a play



If all EV calculations above are correct, I can safely assume that I can solve any spot accurately: call, bet, raise, and I guess check would just realize EV you already have considering your holding and villain's range and fold is always 0 EV

Also, in all the examples above I was useing Equilab hand vs range calculations, I assume if I'd use range vs range calculations nothing (except the percentages) would change

What you are suggesting is that you can make an early street decision by doing a standard EV analysis if you can accurately estimate villain’s range so that various win/lose probabilities can be estimated. Also, such analyses require having to make assumptions on how villain will respond to various hero actions.

You recognize that future actions can lead to different results but then state “do we care about that?” Well, there are many who do care about that. I have often done early street analyses similar to what you have done but then most always have stated that the results only offer a first-cut look and should be adjusted for factors not explicitly considered.

For example, two different hands may result in approximately equal EV’s, say 33 and JTs, when using showdown equities. But the former has poor realized equity having only two outs for a set while the suited connector has many possibilities for improvement.

I should also note that some of those who do care suggest that most early street analyses are a waste of time and effort.

With regard to your second example, you made an error in the win amount for a hero raise (it’s not 6.6) and then in the EV calculation you have villain’s equity with a call at 100% when you previously stated he has 84%. Putting in the “correct” numbers, I find that hero’s EV with a raise is 2.45. I put the word correct in quotes because we actually do not know if that is so, but in doing most EV calculations, one must make assumptions on various input values.

exactly

I assume he has only two responses, he either folds or continues

you are ripping this out of the context, I stated "but do we care about all that when we are calculating EV of a singular action on a given street", our future actions or future actions of a villain do not change the variables we need to solve the EV formula for the spot

this might be true, but if I'm gonna do it anyway I might as well make sure I'm doing it right (or, in this case, accurately)

first-cut look sounds good enough for me
also, what are these factors and how we can properly adjust?

right, but here I'm interested in EV presented as +/- x-amount of bbs after a certain action, so that I can not only see is a play + or - EV but how much bbs a given play would generate
also, playability of a hand can not be accounted for as far as I know, so I as well might just ignore it as the variable we need we already have

I didn't get that, did I **** up again somewhere?

We have a pot of 4.4bb and villain's bet 2.2bb
if we are raiseing and villain folds we'd get 6.6bb

I do have "correct" numbers in:
0.6*6.6bb+0.4*(0.16*6.6bb)-0.4*(0.84*6.6bb)=+2.16bb

I see that I referred to the wrong equation in looking at the second example. But, there is still the win amount error.

This is wrong: 0.6*6.6bb+0.4*(0.16*6.6bb)-0.4*(0.84*6.6bb)=+2.16bb

It should be: 0.6* 6.6 + 0.4*(0.16*(6.6 + 4.4) – 0.4*0.84*6.6 = 2.45

I feel like you could estimate "playability" from the flop onwards.

Something like pick a certain equity threshhold say 1/N where N is the number of players currently involved and as you calculate equity collect the number of cards over turn and river that give you equity greater than that.

The more cards that break that threshhold for you relative to the number of cards remaining to be dealt seems like it would be an ok metric for "playability".

Seems like it might lead to some odd results in multiway pots where hands with extremely low equity result in "playable" hands but maybe factoring in the relative number of those vs the number of remaining cards would be enough to counter balance that effect.

yes I did **** up again

that is: times villain folds * pot + times villain calls * (hero's equity * (pot + amount villain needs to call)) - times villain calls * (villain's equity * hero's raise)

so then we have this:
this also is true for the first example

that is: times villain folds * pot + times villain calls * (hero's equity * (pot + amount villain need to call)) - times villain calls * (villain's equity * hero's bet)

so then we have this:
0.29*6bb+0.71*(0.48*(6bb+3.2bb))-0.71*(0.52*3.2bb)=+6.05bb



that's not the poin

Sorry didn't mean to hijack your thread..just found that idea of modeling playability more interesting than the EV calculations you were presenting.

One thing you might be able to do to simplify is instead of nesting big long formula's you could break the EV equations up and do sub cases and then add all the sub case EV's together. It does make it a little trickier to appropriately assign probabilities and make sure you have all of amounts correct for each subcase though.