How to calculate profit
 

In another thread there was a lot of discussion about raising AK from the BB preflop. Much of the debate centered on where profit is made, and how to calculate it.

I'd like to begin a discussion on this topic. In particular, I propose a simple method for calculating the true EV of a hand. That is to grant the pot to that hand, and have it pay out the EV of all the draws out against it.

Here is a simple application of this concept.

you: AK
opponent: KQ
flop: AT7
pot: 4SB
assumptions: you will bet the flop and KQ will call (incorrectly). you will bet the turn no matter what hits. KQ will raise a turn J and fold everything else. AK will call down a raise.
Finally, to simplify the math, we'll say a J hits the turn exactly 9% of the time.

Now, using my method, I calculate the EV for AK as follows:

The EV of the KQ draw is:

91% of the time: -1 SB
9% of the time: 11 SB (4 initially in the pot, plus 11 put in by AK post flop)

EV KQ draw = .91(-1) + .09 (11) = .08 SB (note that there is no a priori reason this number had to be positive)

Hence, EV (AK) = pot - EV (draw) = 4 - .08 = 3.92 SB

I claim that, under the assumptions for this problem, this is the EXACT EV for AK, not an estimate, and that any system of calculating profit which wants to be taken seriously must produce exactly this number.




Some argued that my method for calculating EV was totally flawed. They countered that a better method is to calculate the EV of the various options one has at their disposal (check / bet / fold) and look at the relative value gained / lost. They made some, I felt, hand waving arguments why this is superior. I challenge them to prove their case formally by producing the EV of AK, exactly 3.92, with their method. I'll even get them started, by guessing at what they mean:

EV of AK = EV (initial) + EV (flop bet) + EV (turn bet) + EV (river)

I suspect they would plugin numbers for these things something like this:

EV (initial) = 91% * 4SB <-- the pot equity of AK
EV (flop bet) = .82 SB <-- EV gained by taking a 91% advantage on a 1SB bet, compared to checking
EV (turn) = 91% * 9% * 6SB <-- EV gained by betting when KQ misses, instead of checking, that is, the equity of the draw that will fold, compared to checking +
9% * -2 <-- EV lost by betting a J with 0% pot equity, compared to checking +
9% * -2 <-- EV lost by calling the raise with 0% equity, compared to folding

EV (river) = 9% * -2 <-- EV lost by calling with 0% equity

So...

EV of AK = .91*4 + .82 + .91*.09*6 + -.09(6) = 4.46 + .4914 - .54 = 4.4114 SB

Oops! Different answer, and quite wrong I believe. What's the problem? I know where the mistake is, but I want someone else to point it out because this will lead us to, I think, an interesting discussion on why this method is so confusing to apply properly.

So, I make the following claims, both of which have been shot down as absurd in another thread:

1. my way of looking at EV is valid
2. my way is simpler
3. your way has not been proven valid (though it will be, I hope, shortly)


This question is relevant because it effects the way we fundamentally talk about the true value of a hand, and decides which framework to use when discussing questions like whether or not you should raise AK from the BB against 4 limpers. discuss.

Thanks,
Eric

Re: How to calculate profit
 

[ QUOTE ]
assumptions: you will bet the flop and KQ will call (incorrectly). you will bet the turn no matter what hits. KQ will raise a turn J and fold everything else. AK will call down a raise.

91% of the time: -1 SB
9% of the time: 11 SB (4 initially in the pot, plus 11 put in by AK post flop)

[/ QUOTE ]
I think you should revisit this section, specifically the assumption that we will make 11SB on this hand. 2SB are made on the flop (our bet and villain's call), but after that the well dries up. Villain folds if he misses his draw, and we're drawing dead if he makes it. This accounts for the massive leap in EV on this hand.

Re: How to calculate profit
 

What about the times when a jack hits on the turn and the river is a queen?

Re: How to calculate profit
 

[ QUOTE ]
91% of the time: -1 SB
9% of the time: 11 SB (4 initially in the pot, plus 11 put in by AK post flop)

EV KQ draw = .91(-1) + .09 (11) = .08 SB (note that there is no a priori reason this number had to be positive)


[/ QUOTE ]
I'll try not to run over this thread. So just a simple clarification:
Are you saying that the EV for a gutshot draw is positive in a 4sb pot? Or are you saying that the gutshot's "piece" of the pot is .08 and that AK's "piece" is 3.92? Or are you saying something else?

Re: How to calculate profit
 

Eric,

[ QUOTE ]
EV KQ draw = .91(-1) + .09 (11) = .08 SB (note that there is no a priori reason this number had to be positive)

[/ QUOTE ]
This should be .09(11) - 1 = -.01 right? He's getting a 9% chance to win 11 SB's for the cost of 1 SB.

That is the EV of the call is only slightly negative with your assumptions.

[ QUOTE ]
Hence, EV (AK) = pot - EV (draw) = 4 - .08 = 3.92 SB


[/ QUOTE ]
AK's EV if KQ calls is just 4+.01 = 4.01. If KQ is calling incorrectly then AK's EV certainly can't be less than if KQ folds right?

[ QUOTE ]
EV (flop bet) = .82 SB <-- EV gained by taking a 91% advantage on a 1SB bet, compared to checking


[/ QUOTE ]
91% of 1 SB = .91 SB's. I'm not sure where the .82 comes from or why you're comparing it to checking.

[ QUOTE ]
EV (turn) = 91% * 9% * 6SB <-- EV gained by betting when KQ misses, instead of checking, that is, the equity of the draw that will fold, compared to checking +


[/ QUOTE ]
I'm not sure what this is. Can you explain? If I were looking at is from EV I'd just look at what AK loses by giving the free card which would be .09*current pot + any river bet(s) from AK. So the EV of betting is the pot the EV of checking is the pot - whatever the above gives which will obviously be less.

[ QUOTE ]
9% * -2 <-- EV lost by betting a J with 0% pot equity, compared to checking +
9% * -2 <-- EV lost by calling the raise with 0% equity, compared to folding

EV (river) = 9% * -2 <-- EV lost by calling with 0% equity


[/ QUOTE ]
If AK is drawing dead (which as Harv points out it really isn't but that's just a minor detail overlooked) then AK is losing 100% of any bets. So I'm not sure what you're doing here either.

Basically, I'm just not making any sense of the second half of your example. But as I said in the other thread I have no issue with your method. Any method if done correctly will lead to the same EV. It shouldn't matter which perspective you look at it from.

Matt

Re: How to calculate profit
 

[ QUOTE ]
specifically the assumption that we will make 11SB on this hand. 2SB are made on the flop (our bet and villain's call), but after that the well dries up. Villain folds if he misses his draw, and we're drawing dead if he makes it. This accounts for the massive leap in EV on this hand.

[/ QUOTE ]

he's talking about how much KQ makes if it hits its Jack on the turn, the 4 sbs in the pot + 7 sbs postflop (1 sb for the flop bet, 4 sbs on the turn - bet/call, and 2 sbs on the river - check/call, for a total of 11 sbs)

Re: How to calculate profit
 

[ QUOTE ]
[ QUOTE ]
specifically the assumption that we will make 11SB on this hand. 2SB are made on the flop (our bet and villain's call), but after that the well dries up. Villain folds if he misses his draw, and we're drawing dead if he makes it. This accounts for the massive leap in EV on this hand.

[/ QUOTE ]

he's talking about how much KQ makes if it hits its Jack on the turn, the 4 sbs in the pot + 7 sbs postflop (1 sb for the flop bet, 4 sbs on the turn - bet/call, and 2 sbs on the river - check/call, for a total of 11 sbs)

[/ QUOTE ]
Sigh, I don't even know how I get dressed in the morning.

Re: How to calculate profit
 

[ QUOTE ]
you: AK
opponent: KQ
flop: AT7
pot: 4SB
assumptions: you will bet the flop and KQ will call (incorrectly). you will bet the turn no matter what hits. KQ will raise a turn J and fold everything else. AK will call down a raise.
Finally, to simplify the math, we'll say a J hits the turn exactly 9% of the time.

[/ QUOTE ]
Ok, a slight change of mind...if we're going to deal with a specific hand with specific postflop play, then it's actually possible to calculate exact EV at each spot of the hand. So I guess I'll throw out some EV numbers based on the 'traditional' way of calculating EV.

I suggest 3 numbers can help us decipher this hand: The EV for both sides immediately following the flop; The EV of the flop betting action; and the implied odds for the remainder of the hand.

First, a couple of assumptions that will make this a little easier:
1) A J will turn 10% of the time (this just makes for eaiser calculations than 9%).
2) KQ is in the SB and AK is in the BB, and there's no rake.

So...
EV immediately after the flop:
We know that KQ is going to fold the turn 90% of the time and there's 4sbs in the pot so, if there's no additional betting, AKs long-term EV is 3.6sb and KQs is .4sb.

EV of the flop action:
2sbs go into the pot and 90% of the time, AK is going to win, so the long-term EV for AK is 1.8sbs and .2sbs for KQ. Note that you can also subtract the 1sb each side put in and say that AK made .8sbs on the bet and KQ lost .8sbs on the bet, which is my preferred way of looking at it.

Implied odds:
10% of the time, KQ is going to collect 6sbs in turn and river bets for an EV of .6sbs.

So on this particular hand, with postflop action shown, we can calculate that:
AK will win, on average, 3.6+1.8-.6 sbs, or 4.8sbs. Of course, he has to put in 3sbs so his average profit, or overall EV from this hand will be 1.8sbs.
KQ will win on average, .4+.2+.6 sbs or 1.2sbs. And when you subtract his 3sbs that he put in the pot, he will lose an average of 1.8sbs.

I suggest that any alternative method of calculation must come up with these same numbers.

Re: How to calculate profit
 

[ QUOTE ]
AK will win, on average, 3.6+1.8-.6 sbs, or 4.8sbs. Of course, he has to put in 3sbs so his average profit, or overall EV from this hand will be 1.8sbs.
KQ will win on average, .4+.2+.6 sbs or 1.2sbs. And when you subtract his 3sbs that he put in the pot, he will lose an average of 1.8sbs.

[/ QUOTE ]
They're only putting in 3 SB's each after the flop the 9-10% (or actually 8.5%, but sure 10% is easier) of the time. So you still need to look at the two different possibilities...that is 90% of the time there is no river and AK wins 4SB's + 1SB and KQ loses 1 SB. 10% of the time KQ wins 11 SB's and AK loses 7 SB's. Using 10% and the assumptions and neglecting any redraw obviously KQ would be correct to call with an EV of .1*11 -1 = .1 SB.

Re: How to calculate profit
 

the 3 sbs are the 2 bets PF that make up the 4bet pot and the 1 bet each on the flop.

The turn and river bets are all encapsulated in the .6 implied odds calculation. i.e. KQ will clear 6sbs profit if he turns a J (10% of the time).

Re: How to calculate profit
 

Right but you don't just subtract 3 SB's for each from the total EV. AK's overall EV for the hand will be just under 4 and KQ's will be 4-AK's EV and just barely positive using the 10% and neglecting redraws. In actuality KQ is calling incorrectly so AK's EV must be just greater than 4 and KQ's is a little less than 0.

Re: How to calculate profit
 

Ack. You are right. I miscalculated the implied odds. I was wondering how the implied odds could be so big that they made the flop call correct (I rationalized this to myself by saying the implied odds had AK making 3 huge mistakes on the turn and river). But, in fact, on the turn KQ gives up his .6sb equity in the pot by folding 90% of the time so the implied odds needs to be 6sb*10% - .6sbs*90%, or .06sbs. i.e. KQs expected gain after the flop betting is .06sbs.

This makes the overall EV 3.6+1.8-.06 or 5.34 for AK. When you subtract the 3sbs he invested into the hand (you have to do this) his overall EV is 2.34sbs.

In other words, if I played you with this hand 100 times (and I got AK), I'd expect to profit, on average, 234sbs. If you folded to the flop bet like you should, I'd expect to make 200sbs.

Aaaand, if I just checked the flop for some reason, I'd expect to make 154sbs.

[ QUOTE ]
In actuality KQ is calling incorrectly so AK's EV must be just greater than 4 and KQ's is a little less than 0.

[/ QUOTE ]
Right. If you want to look at it from the flop onward, then AK will make 4.34 on average and KQ will lose .34sbs.

Edit: added the check=154 number.

Re: How to calculate profit
 

[ QUOTE ]
EV of AK = .91*4 + .82 + .91*.09*6 + -.09(6) = 4.46 + .4914 - .54 = 4.4114 SB

Oops! Different answer, and quite wrong I believe.


[/ QUOTE ]
Heh. Just rereading this thread. I'd guess 4.41sbs is darn close to AKs EV if the J only turns 9% (instead of 10%).

Re: How to calculate profit
 

[ QUOTE ]
What about the times when a jack hits on the turn and the river is a queen?

[/ QUOTE ]

You're correct Harv, but for simplicity sake, let's ignore this. It only complicates the math without changing any of the conclussions.

Thanks,
Eric

Re: How to calculate profit
 

[ QUOTE ]
Are you saying that the EV for a gutshot draw is positive in a 4sb pot?

[/ QUOTE ]

With the stated assumption that AK will lose 3 BB every time hit, yes. He's just barely getting odds to call at 11:1.

[ QUOTE ]
Or are you saying that the gutshot's "piece" of the pot is .08 and that AK's "piece" is 3.92?

[/ QUOTE ]

Yes, I'm also saying this. Given the assumptions stated about how these players are going to play their hands, the 4 SB in the pot get divided up 3.92 for the AK, .08 for the gutshot.

You guys are proving my point
 

SeaEagle, you are proving my point.


[ QUOTE ]
AK will win, on average... 4.8sbs

[/ QUOTE ]
[ QUOTE ]
Ack. You are right... This makes the overall EV... 5.34 for AK.

[/ QUOTE ]

You've now taken several cracks at this very simple situation and come up with two different wrong answers. Not only are your answers wrong, they are WAY wrong. Not only are the WAY wrong, they are getting WORSE as you continue to "fix" your estimate.

This just shows how much more confusing your view is. It's not easy to calculate things your way. Here's how to do it, with your request that we make a J hit 10% of the time:

When KQ misses (90%), he loses 1 SB. When he hits (10%), he gains 11 SB. Therefore:

ev (KQ) = .90 (-1) + .1 (11) = .2 SB
ev (AK) = pot - ev (kq) = 3.8 SB

This is the answer. Wasn't that simple?

Good luck.
Eric

Re: How to calculate profit
 

[ QUOTE ]
Heh. Just rereading this thread. I'd guess 4.41sbs is darn close to AKs EV if the J only turns 9% (instead of 10%).

[/ QUOTE ]

You could guess that, but you'd be wrong.

Re: How to calculate profit
 

[ QUOTE ]
This should be .09(11) - 1 = -.01 right? He's getting a 9% chance to win 11 SB's for the cost of 1 SB.


[/ QUOTE ]

No. You're ignoring the fact that when he wins, he gets his SB back, so he only loses his flop call 91% of the time.

[ QUOTE ]
AK's EV if KQ calls is just 4+.01 = 4.01. If KQ is calling incorrectly then AK's EV certainly can't be less than if KQ folds right?

[/ QUOTE ]

KQ is calling correctly. This is just a function of your missed math above though.

[ QUOTE ]
91% of 1 SB = .91 SB's. I'm not sure where the .82 comes from...

[/ QUOTE ]

No. Here's an obvious counter-argument. If you bet and are called and you have only a 50% chance of winning, you don't make any money. The thing you are ignoring here is that, while the AK wins .91 SB of his opponent's bet, he loses .09 of the bet he put in himself.

[ QUOTE ]
...or why you're comparing it to checking.

[/ QUOTE ]

I'm comparing it to checking because this is the method of calculating profit that is "typical" and was defended in the other thread.

[ QUOTE ]
I'm not sure what this is... the EV of betting is the pot...


[/ QUOTE ]

Your change increases this term, which makes the calculation even MORE wrong than before. You'll have to defend this.

[ QUOTE ]
Basically, I'm just not making any sense of the second half of your example...

[/ QUOTE ]

Yes, isn't it confusing? That's the point. Looking at things the way I did in the 2nd half is very difficult. Looking at things the first way is very easy. We should use the first way.

Good luck.
Eric

Re: You guys are proving my point
 

[ QUOTE ]
When KQ misses (90%), he loses 1 SB. When he hits (10%), he gains 11 SB. Therefore:

ev (KQ) = .90 (-1) + .1 (11) = .2 SB
ev (AK) = pot - ev (kq) = 3.8 SB

[/ QUOTE ]

Heh. I must stop drinking and posting.

You are correct. And if you look at my original post, it's correct as well (AK will average a profit of 1.8sbs on this hand). The implied odds are so huge that they overcome the "mistake" of KQ calling the flop bet.

FWIW, my "fix" introduced the same problem you had in your "wrong" answer in your original post: I double-dipped KQ losing his equity (or you would probably say that I didn't account for KQ winning the entire pot 10% of the time).

When I sober up in the morning, I'll have to think on this further.

p.s. Did you know when you made up this example that KQ was in fact correct in calling a 10-1 draw only getting 5-1? This is pretty unintuitive and I almost didn't post my original (correct) post because I couldn't believe this was true.

Re: You guys are proving my point
 

Ok, now that I'm able to think coherently about this.
Either we are doing the same thing in almost the same way, or you have something cool as hell.

First let me paraphrase your approach to make sure we're communicating on the same wavelength.
[ QUOTE ]
ev (KQ) = .90 (-1) + .1 (11) = .2 SB

[/ QUOTE ]
In English: EV is the equity you have in the final pot minus any immediate investment (I switched the order of the terms since I think it flows easier). Yes?

So the part that's different from my approach is that I calculate immediate EV but don't have a way to add in the implied odds and usually just fudge them (or screw them up as I did last night). Your approach has the implied odds baked in - if we can only figure out what the final pot will be and how often you'll win it, and if there's any further investment required to win the pot. If you have that part down, then your approach has the benefit of being far more accurate than the traditional approach. In the OP hand, your calculation correctly showed a +EV for calling while mine would normally only look at immediate odds and show a solidly -EV (-.4sbs assuming no implied odds). Obviously, the improvement in accuracy you can achieve would make a huge difference over time.

So how do you do that? Let's add just a smidgen of complexity to this hand. Let's say there's a 3rd player (P3) who acts last and has unknown cards. We can assume that KQ is still drawing to the nuts with no redraws so that he'll still win the pot 10% of the time, but now AK is sharing the other 90% with P3. So, if P3 folds to the flop bet, all the calculations are the same as above. But P3 may or may not raise, and if he does AK may or may not reraise. How do you calculate the EV if we introduce this small complication?

Re: How to calculate profit
 

[ QUOTE ]
No. You're ignoring the fact that when he wins, he gets his SB back, so he only loses his flop call 91% of the time.


[/ QUOTE ]
Oh yeah...my mistake.

[ QUOTE ]
KQ is calling correctly. This is just a function of your missed math above though.


[/ QUOTE ]
Yeah, I guess when you round the equity up to 9% and neglect the redraw it is actually.

[ QUOTE ]
No. Here's an obvious counter-argument. If you bet and are called and you have only a 50% chance of winning, you don't make any money. The thing you are ignoring here is that, while the AK wins .91 SB of his opponent's bet, he loses .09 of the bet he put in himself.


[/ QUOTE ]
OK fine. I usually keep these terms seperate when I do the EV calc and you made it sound as if this were just the one term and not both combined.

The second half is confusing because I have no idea what you're calculating or where your numbers come from. It's certainly not how I would do the calculation here so I'm not sure why you think it's any sort of standard way. Also as for comparing it to checking...I really doubt if anyone would do that for AK vs. KQ on a ATx board. I'm not sure why you think this very specific HU example relates to AK against a bunch of unknown hands. But I really doubt if anyone would feel the need to estimate the EV of checking in this example. Your first part (other than the rounding up of EV) is how I've always done it and how I've seen it done for an example like this. So I'm not sure what you feel is new here. It's a completely different situation when you're 5-6 handed. Then I'd be looking at it from an equity point of view because I don't know the opponent's hands so I pretty much have to assign ranges and run a simulation for any analysis, basing my decision off of the equity resulting from the simulation against a wide range of hands for my opponents. This is especially true with a bunch of loose limpers and the raise coming from the BB since no one had to cold-call.

AK_EV + KQ_EV = Existing_Pot is not a new formula. It's what I've always used for an example like this and what I've always seen used for an example like this. I'm not sure why you feel it's some new method.

My EV calculation
 

[ QUOTE ]
The EV of the KQ draw is:

91% of the time: -1 SB
9% of the time: 11 SB (4 initially in the pot, plus 11 put in by AK post flop)

EV KQ draw = .91(-1) + .09 (11) = .08 SB (note that there is no a priori reason this number had to be positive)

Hence, EV (AK) = pot - EV (draw) = 4 - .08 = 3.92 SB

I claim that, under the assumptions for this problem, this is the EXACT EV for AK, not an estimate, and that any system of calculating profit which wants to be taken seriously must produce exactly this number.


[/ QUOTE ]

Eric,

I might take issues with your assumptions here, but under those assumptions the math seems to be right. I calculated the same 3.92 EV under these assumptions looking at pot equity on a street-by-street basis:

AK's Pot Equity of Pot on Flop = 91% x 4 = 3.64
AK's PE of Flop Bets = (91% x 1) - (9% x 1) = .82
AK's PE of Post-Flop Bets = (91% x 0) - (9% x 6) = -0.54

Total = 3.92

Mind you, I think your method, or any other similar one (such as what I did), becomes much more difficult (if not impossible) in multiway pots, particularly if you factor in what happens when players make other non-nut hands with which they might choose to continue. I would point out that neither of our EV calculations factor in the possibility of a split pot when a Jack comes on a turn and a Quenn comes on the rvier, but for simplicity sake I think that is fine.

All the best.

Colgin

Re: How to calculate profit
 

[ QUOTE ]
EV (turn) = 91% * 9% * 6SB <-- EV gained by betting when KQ misses, instead of checking, that is, the equity of the draw that will fold, compared to checking +


[/ QUOTE ]

What is this term for? It's a duplicate term - this has already been calculated in your initial EV (the 91% win rate already counts the times your opponent folds the turn.)

Re: How to calculate profit
 

Nice post... before I post how I would do it, I think this was a mistake right?:

[ QUOTE ]
you will bet the flop and KQ will call (incorrectly).

EV KQ draw = .91(-1) + .09 (11) = .08 SB

[/ QUOTE ]


(Also, I havent read any of the replies yet, so sorry if this is a total repeat).

Anyways... here is how I would do it:
(this will make the same assumptions as you: ie. we lose 9% of the time, and ignore redraws.)

91% of the time: AK bets the flop, then the turn, and take it down. We win 5 SBs this way.

9% of the time: We bet the flop, then the turn, get raised, and call down. And lose. We lose (1+4+2) = 7 sb.

5*0.91 - 7*0.09 = 3.92


If you want to do a method of street by street, it should look like this:

Ev (initial) = 0.91 * 4sb

Ev (flop) = 1sb * 0.91 - 1sb 0.09

(91% of the time we win a SB, 9% of the time we lose)

Ev (turn) = 0.09 * -4sb

(this is where the method posted was going wrong. You never make money on the turn. If he is putting money in the pot, it is with a hand that will beat you (again, disregarding the Q on the river))

Ev (river) = -2 * .09

0.91*4 + 0.91*1 - 0.09*1 - 0.09*4 - 0.09*2 = 3.92SB

Re: How to calculate profit
 

[ QUOTE ]
Nice post... before I post how I would do it, I think this was a mistake right?:

[ QUOTE ]
you will bet the flop and KQ will call (incorrectly).

EV KQ draw = .91(-1) + .09 (11) = .08 SB

[/ QUOTE ]


[/ QUOTE ]

Yes. I wrote out the assumptions before I did the math. I just assumed calling with a gutshot in a 4SB pot was wrong.

[ QUOTE ]
If you want to do a method of street by street, it should look like this:

Ev (turn) = 0.09 * -4sb

(this is where the method posted was going wrong. You never make money on the turn. If he is putting money in the pot, it is with a hand that will beat you (again, disregarding the Q on the river))


[/ QUOTE ]

Absolutely right. I hope that this error will help illuminate why calculating EVs street by street when you want to find a true EV for a hand is confusing. Let's look more closely at the turn:

First, I hope you see that the logic of betting being profitable when your opponent misses is quite reasonable and hard to disprove. After all, if we assign a betting in this spot to be 0 EV, it implies that we have to assign a turn that goes check-check a NEGATIVE EV, which is counter-intuitive. How do we lose money if we don't put anything in the pot? Why is it that on the flop, a check-check flop should be assigned EV zero, but on the turn when he misses, it should have a negative value? This is quite hard to explain and leads to lots of mistakes when the calculations are done this way.

It seems that we first have to know what the correct action is, then we have to go back and assign the correct action to be zero, and calculate deviations from there. Of course, this assumes that we have correctly calculated the EV initial based on everyone playing perfectly for the rest of the hand. So, to do a street-by-street calculation, we have to do the following:

1. figure out what the proper action should be for the various possible cards that are dealt
2. for each situation, calculate the EV gained on whatever bets go into the pot correctly, and add these up.

I think it should be pretty clear that this method is highly error-prone for all but the most careful mathematician.


In real-world calculations, the method of estimating the value of the draw(s) and subtracting them from the pot is going to be much more feasible and simple. If you hold top pair and suspect you are up against a flush draw, you just make an educated guess at the true odds the flush faces, the EV of the call from the pot, and you have the rest. Doing it street by street from the top pair's point of view can be done, but it can easily lead to subtle mistakes that cause major errors in the answer. When you use the draw EV method, you are always going to be in the right ballpark.


So the point is, when we're arguing about the best action, PARTICULARLY when the debate is based on the idea of passing up a small edge to get a bigger edge later, we should abandon any arguments that are based on "making more on this street", and how these sum to a bigger number, since these arguments are so easily confused and misleading, and focus on arguments stating that hero's true EV is greater. This debate, I believe, will more often lead us to the correct conclussion, which is what this forum is all about.

If there are no objections to this, I'm now going to reopen a thread on the value of raising AK vs 3 limpers, with the goal of having a discussion of this, I think, interesting topic that doesn't get bogged down with the mechanics of EV calculations.


edit: by the way, there's one more reason the street-by-street arguments are misleading at times, and that is that in these calculations, it is not unusual under reasonable assumptions for the hero to lose money on the later streets. That is, the current street calculations rarely take into account the implied odds for the draw. When you look at things from the point of view of the current EV of the draw, you naturally think about implied odds and, having played these so many times, we've gotten pretty good at estimating them. You'd almost never forget to add in a couple big bets when looking at the value of a flush draw, while this is very easy to do when looking at top pair on a street-by-street calculation.

Re: How to calculate profit
 

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EV (turn) = 91% * 9% * 6SB <-- EV gained by betting when KQ misses, instead of checking, that is, the equity of the draw that will fold, compared to checking +


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What is this term for? It's a duplicate term - this has already been calculated in your initial EV (the 91% win rate already counts the times your opponent folds the turn.)

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Hi Jeff,

Good job, you found it. See my other post on this topic for a discussion.

Thanks,
Eric

Re: How to calculate profit
 

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So the point is, when we're arguing about the best action, PARTICULARLY when the debate is based on the idea of passing up a small edge to get a bigger edge later, we should abandon any arguments that are based on "making more on this street", and how these sum to a bigger number, since these arguments are so easily confused and misleading, and focus on arguments stating that hero's true EV is greater. This debate, I believe, will more often lead us to the correct conclussion, which is what this forum is all about.


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Maybe you'll cover this in the new AK thread, but I'm really interested to see how you calculate the "bigger number".

I'm pretty sure most of us calculate the immediate EV and then kinda guess at the implied EV, a method that would have had virtually all of us folding KQ on the turn in this thread. I would love to see some examples of this at work in a more complicated hand.

Re: How to calculate profit
 

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EV (turn) = 91% * 9% * 6SB <-- EV gained by betting when KQ misses, instead of checking, that is, the equity of the draw that will fold, compared to checking +

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This is the problem and if you change 6SB to zero SB, then you get the same answer. I'm not sure what your point is except that you did the 2nd one wrong.

This just shows how you make money in the early betting rounds with hands like these.