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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 |
#2
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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. |
#3
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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) |
#4
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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. |
#5
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Re: How to calculate profit
What about the times when a jack hits on the turn and the river is a queen?
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#6
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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 |
#7
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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? |
#8
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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. |
#9
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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 |
#10
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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 |
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