How to calculate profit

[elindauer]

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