An EV equation

[BearHustler]

The idea was to formulate an equation that expresses the expected value of any action at the table. I don't know if this attempt is laughable, but here it goes anyway:



[img]/images/graemlins/diamond.gif[/img] There's two big parts: either our opponent(s) fold, or he/they don't. CFE * PS is our profit from the times they fold. The rest is what happens if they don't.

[img]/images/graemlins/diamond.gif[/img] 1-CFE is obvious: it's the chance that our opponents don't fold.

[img]/images/graemlins/diamond.gif[/img] There's three possible outcomes for any given hand: there's a showdown, we fold, or they fold. PE+FFE-RFE is the chance that our opponents don't fold to our current action (this doesn't have to be a bet, it could be a check or a call), but we go on to win the hand. It's when we either have the best hand at showdown (PE), or we make our opponent(s) fold (FFE) or they make us fold (RFE). FFE and FRE only count when the best hand is folded, because the rest is already accounted for by our pot equity.

[img]/images/graemlins/diamond.gif[/img] PS+CS+I(ESS) is simply the amount of money that we win, on average, if we win the hand. PS + CS is the amount of money there is to be won on this betting round. I(ESS) is what we expect to win from our opponents on future actions.

[img]/images/graemlins/diamond.gif[/img] 1-PE-FFE+RFE is of course the chance that our current action is called and we don't win the hand.

[img]/images/graemlins/diamond.gif[/img] BS + RI(ESS) is the amount of money, on average, that we lose, if we lose the hand. BS is the amount we've lost on this action. RI(ESS) is the amount of money we expect to lose on future actions.

[img]/images/graemlins/diamond.gif[/img] Of course, it's never easy to estimate our PE or FE. But the really tricky parts here are I(ESS) and RI(ESS). If we raise preflop with 44, we flop trips, and we bet out. Now we wanna calculate our EV for that bet. How should we estimate I(ESS)?

If it's K 4 8 rainbow and our opponent is very tight, I(ESS) is probably a very big number. He won't call without a good hand, and he won't let go of it easily. RI(ESS) is probably even bigger, because if we lose this hand, it will almost always be for stacks. On the other hand, if we make a Cbet on a wet flop with a missed AK, our RI(ESS) will be pretty low, because we don't plan on putting our stack in on A-high.


Examples

[img]/images/graemlins/heart.gif[/img] There's a bet and you fold.

CFE = 0
PE = 0
FFE = 0
RFE = 0
CS = 0
I(ESS) = 0
BS = 0
RI(ESS) =0

So, as we all know:

EV(fold) = 0


[img]/images/graemlins/heart.gif[/img] The pot is $100, your opponent goes all-in for $50. You have an open end straight draw, and expect to have 33% pot equity.

PE = 1/3
PS = $150
CS = 0
BS = $50
CFE = 0
FFE = 0
RFE = 0
I(ESS) = 0
RI(ESS) =0

=> EV(call) = PE*PS - (1 - PE)*BS = 1/3*150 - 2/3*(50) = +$16.6


[img]/images/graemlins/heart.gif[/img] $100NL. Effective stacks $100. UTG+1 opens for $4, 1 call, you call with 87o, rest folds. Flop is 6 5 J. UTG+1 bets $15, you push.

CFE = 33% based on his preflop raising %, and the hands you think he needs to continue vs this kind of action.
PS = 0.5 + 1 + 4 + 4 + 4 + 15 = $28.5
PE = 33%
FFE = 0
RFE = 0
ESS = $100
BS = $95
CS = $80
I(ESS) = 0
RI(ESS) = 0

=> EV = 1/3*28.5 + 2/3*1/3*(28.5+80) - 2/3*2/3*95 = -$8.6


[img]/images/graemlins/heart.gif[/img] $100NL. You open for $4 with A [img]/images/graemlins/diamond.gif[/img] K [img]/images/graemlins/diamond.gif[/img], lots of folds, BB calls.

Flop: 3 [img]/images/graemlins/heart.gif[/img] 7 [img]/images/graemlins/club.gif[/img] Q [img]/images/graemlins/club.gif[/img]
Hero bets $8

What's the EV for that bet?

We estimate villains range as: 22+,AJs+,KQs,QJs,JTs,T9s,98s,87s,76s,65s,AQo+, which is 11% of all his hands.

We think he'll call or raise or bet with: QQ+,77,33,AQs,KQs,QJs,JcTc,Tc9c,9c8c,8c7c,7c6c,6c5 c,AQo, which is 4.5% of his hands.

So our fold equity is: (11 - 4.5)/4.5 = 59%

Our pot equity vs this range is 13.6%.

CFE = 0.59
PS = $8.5
PE = 0.136
FFE = 0.02 (if we don't improve, we're not firing any more bets, and if we do improve, he won't often fold anything that beats us)
RFE = 0.04 (we are folding a whole lot of the time, but RFE only counts when we fold the best hand. Unless he starts betting out with his draws on the turn, this won't be all that often. We usually fold the best hand when he's drawing AND he doesn't hit, AND we don't improve AND he bets a missed draw on the river AND we fold)
BS = $8
CS = $8
I(ESS) = $5 (If we win, it'll probably be a showdown where we win with A-high, or if we improve on the turn or river, but he won't put in a lot of money if he can' t beat A-high or even a pair of aces)
RI(ESS) = $2 (We're pretty much done with the hand. the only times he'll make something extra is when we improve on turn or river AND no obvious draws hit, BUT he's got us beat anyway)

=> EV = 0.59*8.5 + 0.41*(0.136 + 0.02 - 0.04)*(8.5 + 8 + 5) - 0.41*(1 - 0.136 - 0.02 + 0.04)(8 + 2) = +$2.41



This is probably filled with mistakes and because some of the numbers are so hard to estimate, it might be useless as well, but lets give it a go anyway.

Toughts?

[Avicenna]

Or, please help me writing a bot. We don't have a bot forum yet? [img]/images/graemlins/grin.gif[/img]

[BearHustler]

[ QUOTE ]
Or, please help me writing a bot. We don't have a bot forum yet? [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

WTF? Any serious comments?

[willw9]

wwwtl;dr

[recallme]

What you all think about that?

[PerDoom]

What possible use could this be other than writing a bot?

[BearHustler]

[ QUOTE ]
What possible use could this be other than writing a bot?

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Yeah, one can only imagine. [censored]

[Pokey]

[ QUOTE ]
What possible use could this be other than writing a bot?

[/ QUOTE ]

I can't imagine how this information could be used to write a bot; after all, I could write a formula:

EV = P(win)*E(win) - (1-P(win))*E(loss)

Where:

P(win) is the probability that I win the hand
E(win) is the amount of money that I win when I win the hand
E(loss) is the amount of money that I lose when I lose the hand

...and such a formula would be equally accurate and equally useless for writing a bot.

No, this formula could ONLY be used to help think about the game from a mathematician's perspective. The unfortunate reality is that most of the variables in this formula are unknowable, and the art (not science) of estimating those formulas is simultaneously more important than the formula and less programmable.

Let's look at these hard-to-quantify variables one by one:

CFE: knowing the chance that our opponent will fold a hand is only weakly related to easily calculated statistics. It's related to board texture, recent history, table image, mood, and our opponent's hand, all of which we'll never know, and all of which would be prohibitively difficult to try and quantify in a bot.

PE: our pot equity is an extremely rough estimate based on our best guess for our opponent's range of possible hands. If you had 50,000 hands on an opponent you MIGHT be able to narrow their behavior to a reasonable (though typically wide) range on the flop, but by the turn or river these ranges are again art rather than science. Without knowing our opponent's possible holdings, our guess about our pot equity will remain just that: a guess.

FFE: our future folding equity is an unknown twice removed -- we don't even know our CURRENT folding equity, and trying to predict what it might be in the future is shooting in the dark. If we could see our opponent's hole cards we MIGHT be able to narrow this number down to something slightly useful, but barring that, FFE will remain a hazy estimate at best.

RFE: we can estimate this one reasonably well if we commit to a path, but that can quickly turn into a death pact. For instance, if we're up against a supermaniac and we flop top pair, we can simply say RFE = 0, since we've decided to call all the way to the river no matter what cards hit the board. Alternatively, if we call a preflop raise from a 15/2 with a small pocket pair on set value alone and miss the flop, we can set RFE roughly equal to 1. But in most instances, we're not going to be sure if our hand is best and we're not going to be sure if we'll fold in the future, and that leaves this number equally unprogrammable.

I(ESS): one of the most frequently misestimated variables in poker is the implied odds we face. Newer players routinely overestimate their implied odds. Weak-tight players routinely underestimate their implied odds. Even experts screw this one up on a regular basis. While I agree that knowing this number would be useful for the formula, I again think there's a huge difference between referencing it in a theoretical formula and applying it in any real-world setting. I doubt any computer is going to be able to come up with a reasonable estimate of this value any time soon.

RI(ESS): while this variable is related to I(ESS), it is different in that the unknowable part is not the amount of money that could go in the middle but rather the amount of money that our opponent will try to force us to place in the middle. At times we can estimate this number: against an INCREDIBLY passive opponent who is OOP, we can estimate this number at roughly zero. Against an opponent who is betting like it's going out of style we can expect it to be the effective stack sizes. We're still going to have trouble estimating the fraction of it that we expect to lose (since we don't know if we're ahead or behind most of the time). Also, more often than not our opponent's future bet sizes will be a mystery. Will he check behind on the river? Will he block from OOP? Will he CRAI? Will he pot it? Will he push? In no-limit the options are so wide-ranging that estimating even this number is going to be incredibly hard.

No, I don't see this as an attempt to write a bot; rather, I see this as a very vague, very basic attempt to list the major issues of importance in any particular poker hand. While I don't think it harms anything to talk about this formula (or similar ones), I don't think it's going to prove helpful to anybody to spend hours analyzing this structure -- in the end, there are just far too many unknowable pieces to really prove to be a useful analytical tool (in my opinion, of course).

[munkey]

Interesting equation for sure, but I agree with Pokey's
[ QUOTE ]

-- in the end, there are just far too many unknowable pieces to really prove to be a useful analytical tool (in my opinion, of course).


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An interesting analysis would be how much the individual components affect the final EV i.e. perturbation aanalysis for some simpler classic SSNL situations.

e.g. if we change the flop pot bet from 2/3 to pot how much does this affect the EV.

I did this type of thing with calling AI on the flop calculation some time ago( on paper). Potsize and FE affected the EV more than having a few more outs which surprised me a little was the conclusion.

This was a result of comments by True in that thread which I found to be true and proven by the above EV calcs.