Suppose you are at a Hold Em table with 23 players. You are in the Small Blind with 32. Everyone folds to you. You raise all in. The BB calls with AK. The board is Q J 9 6 5. What is the most accurate estimation of your equity at the point that you went all?

To answer this question, most people would open up PokerStove then input the hole cards and see 32.222%. This answer is actually incorrect. The correct answer is you have 0% equity at the point of the all in. Not because you ultimately lost the hand, but because regardless of how the cards in the remaining deck come out, you cannot possibly win. There is no combination that gives you the win.

But what if the flop is 3 3 2 2 2 you say - surely 32 must win in that scenario. Well, we know with certainty that flop is impossible. 21 players have folded, that's 42 cards dead and another 2 players have gone all in. So 46 cards are dead. Only 6 live cards remain - and we know 5 of them (the 5 cards that were dealt). No matter what card that last unknown card is, swapping a card on the board with it cannot give 32 the win because AK will always have the flush and thus always beat 32. Therefore 32 has exactly 0% equity in this situation, contrary to what conventional equity calculators tell us.

In order to calculate our equity, more information is needed than just the cards that went all in. The cards on the flop can be used in combination with the number of players that have folded to determine a more accurate estimation of our equity. The basic idea is that we can use knowledge of the cards on the board to say that those cards on the board are more likely than others to come out of the remaining live deck.

In this particular situation, to calculate the probability that the 5c is dealt on the board, we need to know two things: the probability that 5c is in the live cards, and then the probability that it is dealt. The probability that it is a live card is 1/(52 - 9). There are 52 cards in the deck, and 9 are known (the hole cards and the board cards). The probability of it then being dealt is 5/6. So the overall probability is 1/(52-9)*(5/6) = 5/258, or roughly 0.01938.

For the Q (which was dealt on the board), we already know the probability that the Q is in the live cards is 1. All that needs to be known is the probability it is dealt on the board, which is 5/6 or roughly 0.8333.

A program that correctly calculates equity would go through all permutations of the flop cards just as PokerStove does. In the case of two players all in, there are 48 C 5 = 1,712,304 boards that need to be iterated through. But for each of those boards, a probability has to be calculated for the board to run out in that way and then this number multiplies the final result. For example, the board Q J 9 6 5 has a probability of (5/6) * (4/5) * (3/4) * (2/3) * (1/2) = 0.16667. If this board is a win for AK, which it is, it would add 1 * 0.01543 to the win count for AK, unlike PokerStove which would just add 1 to the win count.

For an impossible board such as Q J 3 3 6 the calculation would be (5/6) * (4/5) * (1/(52 - 9)) * (3/4) * (0/(51 - 9)) * (2/3) * (1/2) = 0. It is 0 because the probability that the 3 is dealt given that the 3 has already been dealt is 0, since there can only be 1 card that was not in the original 5 dealt on the board - there is only 1 unknown card. The win count for AK would therefore increase by 1 * 0 for this board.

This more accurate calculation is useful for programs such as PokerTracker and Holdem Manager. It enhances the accuracy of EV calculations by essentially implicitly putting the range of cards which were folded (ie the dead cards) as a range which excludes the cards which are known. The most important thing about this modified calculation is that for any sufficiently large set of hands where there are all ins all ins, the average of this equity calculation method will give more accurate results than using the standard equity calculation. So by replacing the old method with this new one, EV calculations become more accurate in poker database programs. That is, they converge to a value closer than the equity calculation to the 'perfect' equity calculation (that is, the equity of your cards given knowledge of all live cards (by deducing them from knowledge about what cards players folded)).

If you want to know your EV of going all in with 32 it would be much more practical to put your opponent on a range of cards and forget about the issue I have discussion in this thread. That calculation would almost always be closer to your EV then my calculation, but perhaps counter intuitively, the average of my calculation would be closer than the average of a calculation that ignores known cards.

Either you, or I, am making a huge logical mistake here. Let's start here:

You go on to say that your equity is 0 at the time you went all in, because "There is no combination that gives you the win." This sounds like total nonsense to me. If the board has a 2 and no A or K, or 3 diamonds comes out, 23 wins, for example. I don't see any reason or explanation in your text for why pokerstove is wrong.

The fact that there are 23 players is a red herring unless you are taking into account playing effects, i.e. folded hands are slightly less likely to have aces in them and more likely to have low cards. The effect is very small and in most cases I doubt taking it into effect would be useful. The effect is unknown for 23-player holdem which I imagine is Quite Nitty.

Without taking this effect into account, any given folded card has the same chance to be a card you need as the card on top of the deck ready to be dealt. Equity calculators take this into account just fine.

On a 2nd reading it sounds like what you are trying to do is use information obtained after you went all in, to calculate equity at the time you went all in. I don't really think this is a method that has merit.

It's completely arbitrary. If you use your knowledge of the flop and turn to re-calculate the chance of winning on the river, then why have you stopped short of using all the post-knowledge you have, in which case the answer is always 0 or 100% (barring splits)? I don't see a cogent argument for why assuming knowledge of SOME of the cards after the fact is better than assuming knowledge of ALL or NONE of them.

But that could never happen. The point is that it can be deduced that the cards that 32o needs to beat AdKd have all been folded. 5 of the 6 remaining cards that can be dealt are diamonds, meaning that 32o will always be up against a flush and the best hand it can make is a straight.

My new method basically implies that the players who folded have folded two random cards that are not cards on the board, whereas the old method implicitly assumes that the players have folded two random cards which may have been on the board (but not in the hole cards of the players).

So would you claim that the standard 32.222% answer people give is incorrect since it uses information after the time the player went all in (it uses information about the opponents cards, in this case AdKd).

Its not arbitrary. My method uses all the available information for maximally accurate results. The information of the cards on the board is information that can be used to model the range of cards that the opponents folded - this is really the concept behind my method.

The old method to calculate equity does not use the information about the ranges of the cards which the players folded so it is less accurate.

If you ran a monte carlo simulation using a program such as CardRunners EV and set the ranges of the players who folded to be every hand except those with the cards on the board (and the hole cards of the 2 players all in), you would find that this result would be the same as my calculation, 0%.

The reason I used 23 people in my example was because otherwise the calculations would be too complicated to do by hand. If there was only 1 player who folded, I would have to trawl through (46 C 5) = 1,370,754 permutations of boards to figure out the probabilities of winning, losing and drawing. With 23 players folding, there is only 1 unknown card so it can be done by hand as there are only (6 C 5) = 6 boards. I have not yet written a program to do this automatically.

at the point of the allin the cards that would come on the board werent known yet. therefore it doesnt make sense to incorporate this knowledge into the equity calculation.

imo the only way to make equity calcs more accurate incorporating the amount of players that have folded is to estimate the folding players preflop ranges and using this info to weigh the likelyhood of certain cards that should be left in the deck.
ie if the folding players arent maniacs its more likely that the folded small cards and thus the likelyhood of big cards remaining in the deck (and in the remaining players hands) increased. i doubt that this effect would increase the accuracy of the equity calculation enough to make it worthwhile.
its definitely interesting enough from a programmers pov to give it a go.

using hands that came on the board after the fact like you suggest still doesnt make sense though.

This.

What is our equity at this point?

I think you are confused in all of this.
The fact that that 23 players have folded random cards will not influence the chance of what this last unknown card is.
If I shuffle a deck of cards and ask you to guess what the top most card is - you would clearly have a 1 in 52 chance of being right.
If I take the top card and put the other 51 face down in the muck and ask you to now guess there is still only a 1 in 52 chance of you being right although there is only one card to pick.

Much lower than the 32.64% suggested by PokerStove & co if we assume even remotely realistic folding ranges. (And obviously much, much higher than the 0% suggested by OPs method.)

Have you actually computed this with remotely realistic folding ranges? I know I did something similar, and the effect was surprisingly small.

Yeah, there was a similar thread a few months back and i did a quick simulation & was surprised how big of a difference it made. (Can't find the thread right now, but maybe i still have the script somewhere.)

Edit, thread:
http://forumserver.twoplustwo.com/sh...51&postcount=7

Hmmm... how did you perform those computations?

I don't remember the details, but I did something at a 9-handed table, and found that the equities were off by less than 0.2%.

Just shuffled a deck & dealt random hands to all players, disregard the deal if UTG...BU had a hand in their opening range. Otherwise deal a single board and track the result.

(Obviously, this approach has really poor performance & i also didn't really test the implementation in detail.)

I want you to meditate on the difference between using the knowledge of your opponents cards vs using using the knowledge of the board cards.

Then, I want you to tell me why we would use knowledge of SOME of the board cards, but not all of them. (If you used all of them, your equity would be 0% or 100% or 50%)

No player has (necessarily) folded random cards. Each player knows what their cards are and decided to fold them.

Think about all 52! ways the deck can be ordered and which ones are counted by PokerStove as a win for 32o. What you will find it is that it is counting orders of the deck as win for 32o in which a player folded a card that was on the board which we know is impossible.

The logic used in deciding to incorporate the cards on the board is exactly the same as using your opponents exact cards, but the logical inverse. Instead of looking at his cards to see what he had, you look at the cards on the board to see what cards the players who folded did not have. It is sound to use this information to produce more accurate results.

People struggle enough with normal equity calculations. It was optimistic at best (and delusional at worst) to think that people would agree with my new method to calculate equity without any evidence behind it. I will set out to prove beyond any reasonable doubt that my method is more accurate than the standard method by creating a program that will make such a calculation. Here is how it will work:

Input is the cards of all players in the hand and the cards on the board and which two players are all in. It will output three separate numbers; a standard equity calculation (the same as PokerStove), a perfect equity calculation (a calculation using information about the exact hole cards of the players who folded) and my method of equity calculation. This will be done for millions of hands and then plotted on a graph. What will be found is that my method is closer to the perfect equity calculation - the distance between the perfect equity line and my method line will be smaller than the distance between the perfect equity line and the normal equity line.

You are asking me to calculate equity without giving me all available information. It would be no different than asking me what equity I have when my opponent has A in his hand and a hidden second card. You can not model his second card as a random card since he has influence over what it is, just as you cannot model the board cards as random since the players have influence over it.

Well, perhaps you are not confused but I am.
If the other 23 players behave like terrible bots and fold all hands regardless of having AA or 23o it won't make any difference whether the unknown cards are in the rest of the deck (it's just like they hadn't played) or in the muck. The chances of hitting one card that you want is (1/ no. of unknown cards) at the decision point. You can split the calculation up into two parts and do a weighted ev of how likely it is in the remaining group and how likely it is in the unknown mucked card group. If you do this weighted sum correctly it will work out the same, ie, still 1/ (no. of unkown cards)
If you mean that you will adjust for the bias produced by the 23 not folding certain pairs of cards this is quite well known and many will have used a variety of algorithms to try to adjust for this.
I know that Barry Greenstein did write a c++ program to calculate how likely AA will be folded to if held in the BB, given his assumptions the answer is here Barry.G.Aces.cpp output

Ok, that looks legit, however I'm not sure what sample size you'd need.

The info i posted is from 1m recorded results. SD for the 1m sample should be around 0.1% fwiw, so i don't think sample size is an issue. (I found the script though, no problem to re-run with larger samples.)

Would be interesting to check what you did differently, assuming you had results close to normal Stove percentages.

Regarding the OP approach:
It should actually reduce possible redline bias slightly (but it adds additional variance compared to "normal" redline calculation). Hard to see practical applications for it.

(Imo the 23 player example and the 0% equity argument are both unfortunate choices in the OP and distract users from the main idea.)

i also remember reading somewhere that the difference when including reasonable frequencies for dead cards due to players folding was very small.

im interested in hearing more about your approach plexiq.

As mentioned, that was just a quick test script.

Really not much more to tell, i already explained earlier itt how the simulation was implemented. Maybe someone else can try to verify the results.

Feel free to suggest better opening ranges, but i think they should qualify as "remotely realistic" for our purposes.

I don't understand any of this... but feel it's stupid/wrong/pointless...

My guess is the OP doesn't have a girlfriend and I would say there is a 0% error margin in that assumption.

And if you are thinking people fold certain cards more often then your not thinking it through. You would need to know the game type... as unless it's crazy... then I would say I'm folding ATs UTG 23 handed. Ax Kx Qx will be folded in early position so that's UTG to UTG+17

Hmm.

But in answer to your question when you know what the board cards/muck cards are yes you can work out a more accurate equity - but your true equity isn't included in those cards - it's your hand against the villains hand with 5 unknown cards to come.

That's the important bit.

Because you deal this again and get same hands and same position - the board will run out different.

But hey maybe I misunderstand the whole thing - but it does seem stupid to me. But hey I failed math GCSE

Ha.

If you do this experiment you will find that your results are more accurate than the standard equity calculation.

However consider the following experiment: Take your dataset and separate out 10% of it and call it the Evaluation Set E. Call the other 90% the training set T. For every hand h1 in E where two hands went all-in at some point, find an equivalent hand h2 in T where the same two hands went all-in. Run your calculator and the standard calculator for h2, and assign the computed equity to h1. Do this for all hands in E and calculate total equity by both methods. Which one do you expect to be more accurate? Why?

Now do you see why the standard method is preferable to yours?

I am 99.9999% sure you are a douche bag and there is only a 0.0001% margin of error in that assumption.

Edit: sorry rusty