## Testing Barry Greenstein’s claim

In his book * Ace On The River*, Barry Greenstein gives an example of a card removal effect that he says changes the hand equities significantly (more than 2%):

“In hold’em, with no betting after the flop, all pairs are a favorite over Ace-King offsuit, by at least 52% to 48%. All pairs except Deuces with neither card in the Ace-King’s suit are favored over Ace-King suited.

If several players fold first, Ace-King suited is a favorite over most pairs.(The exceptions are Aces, Kings, and Jacks, and also Tens where one of the Tens is the same suit as the Ace-King.) Even Ace-King offsuit is now a favorite against small pairs. The reason for this is that players are more likely to play hands having an Ace or King than those containing smaller cards. Therefore,as players fold, the probability of an Ace or King coming on the board increases.”

(p. 150)(emphasis added)

From our **Rank Bias analysis** we know this last sentence to be absolutely true, but I want to test the specifics of the claimed degree of equity shifting. My guess is that Barry did this mathematically using some assumptions about typical player behavior, but didn’t test it empirically with a large hand sample. For a scenario this specific, it takes a very large sample to find enough trials meeting these criteria, and I have a very large sample.

First, let’s look at the preflop equity of the matchups described, which would be exact expectation in the case of a heads-up game (no one else can fold) where the hand got all-in preflop.

*“…all pairs are a favorite over Ace-King offsuit, by at least 52% to 48%.”*

**AhKd vs. 2h2d** – the worst case pair equity is 52.26%

**AKo vs. 22** – the pair equity is 52.65% (average regardless of suits)

**AKo vs. 88** – the pair equity is 55.16%

**AKo vs. QQ** – the pair equity is 56.76%

*“All pairs except Deuces with neither card in the Ace-King’s suit are favored over Ace-King suited.”*

**AhKh vs. 2c2s** – the pair equity is 49.92%

**AhKh vs. 2h2c** – the pair equity is 50.30%

**AhKh vs. 3c3s** – the pair equity is 50.61%

So far so good. Now we get to the part about folders changing those equities.

* *

*“If several players fold first, Ace-King suited is a favorite over most pairs. (The exceptions are Aces, Kings, and Jacks, and also Tens where one of the Tens is the same suit as the Ace-King.) “*

**AKs vs. 99** – the AK equity is 47.42%

**AKs vs. 88** – the AK equity is 47.52%

**AKs vs. 77** – the AK equity is 47.69%

**AKs vs. 66** – the AK equity is 47.69%

**AKs vs. 55** – the AK equity is 48.03%

**AKs vs. 44** – the AK equity is 48.59%

**AKs vs. 99-44** – the average AK equity is 47.83%

So the theory is that we will see **an equity shift of greater than 2.17% towards the AKs**, making that hand the favorite in these matchups. We can design a test for this.

The equities are close enough that we’ll disregard the relative frequencies of the different pairs being involved in such a matchup, and use 99-44 to get a decent sample size (and we’re going to calculate the exact equity for the actual sample anyway). 33 and 22 are very close to 50% so we’ll leave those out of the test. “Several players fold” as Barry stated, is not very specific so we’ll try different numbers of folds ahead of the first bet. And we’ll filter that after X folds before the first bet, we get a preflop all-in between AKs and 99-44.

You might say that according to Barry’s qualifier (”no betting after the flop”) we should also include hands where the players are not all-in preflop but then they check all the way to the showdown, but those hands would actually bias our test. No post-flop betting by decision is not the same as no post-flop betting being possible (in the case of preflop all-in). The equity doesn’t change, but our actual sample would be biased to only include the hands where no one chose to bet or fold post-flop, and those decisions are influenced by the cards on the board. Thus we would not have a random sample of boards, invalidating our equity calculation. So our test must be restricted to preflop all-in hands, and the result of that test can be assumed to apply to the more general case stated by Barry G.

## RESULTS

**I ran the scan on 33 million hands of full ring NLHE.** Filtering for at least (as opposed to exactly) 4 folds in front, I found 1739 hands that matched our specific criteria, or about once every 19,000 hands. So whether or not AKs is a favorite over pair 99-44 when all-in in preflop, Barry G chose an appropriate heading for that section of his book:

“SOME INTERESTING BUT NOT TOO USEFUL MATHEMATICAL FACTS:”

To be fair, we only tested all-in hands, and these matchups do come up a lot more often without resulting in an all-in situation.

Anyway, here are the results for the AKs vs. 99-44 after at least 4 folds before the first bet:

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 47.48% 0.21%47.69%1739 48.07% 0.20%48.27%0.49

**So 4 folds ahead of us is NOT enough to make our AKs a favorite over a pair 99-44**. Also notice that with equity of 47.69%, we know the average pair was around 88, which makes sense since we filtered for 99-44 and they got all-in preflop, so we’d expect this.

Let’s try at least 5 folds ahead of us now:

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 47.49% 0.21%47.70%847 48.41% 0.24%48.64%0.55

Better but no cigar. Let’s try 6 folds (meaning the all-in was either blind/blind or button/blind):

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 47.53% 0.21%47.74%385 49.87% 0.00%49.87%0.84

So close! There’s definitely a trend here. But all we have left to try is blind vs. blind at a full table, and everyone folds around to the small blind. So 7 folds in front:

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 47.54% 0.22%47.76%101 53.47% 0.00%53.47%1.15

Hmmm. Well, now I suspect we have a sample size problem with this big jump, and since the SD offset is still only 1.15. But there is definitely a trend and this test did show the AK as a favorite. But unless we can get a sample size that will show over 2SD offset it isn’t going to be very conclusive.

To get a larger sample, let’s check the second part of Barry’s statement:

“Even Ace-King offsuit is now a favorite against small pairs.”

I’m not quite sure how he defined “small pairs”, since the equity shift is going to have to be higher than the first test even if we only count 22, and he didn’t specify suit domination (giving the AK more equity). But we’ll assume he meant 22-33 to give us the best chance. Even that will need a +3% equity shift.

**AKo vs. 22** – the AK equity is 47.35%

**AKo vs. 33** – the AK equity is 46.63%

**AKo vs. 33-22** – the average AK equity is 46.99%

Based on our AKs tests, I doubt a 3% equity shift is going to happen. But let’s try it, filtering for 6 or more folds in front, thus button/blind or blind/blind get allin.

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 47.15% 0.29%47.44%192 42.19% 0.00%42.19%-1.46

Oops.

I suspect that when it comes to these low pairs, there are some other removal effects at work that cancel out the benefit to AK from folders. See this post. To see if that may be the case, let’s move up to 66-55 and see if the AK results actually get better.

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 45.44% 0.19%45.63%464 47.63% 0.43%48.06%1.05

Aha! A very significant boost for the AKo against these middle pairs. But not against 22 and 33. Again, I suspect there are other cancelling effects for the bottom pairs, and the AKo isn’t ever going to be a favorite even against those.

Ok, let’s do one last test where we check all matchups of any AK suited or not, against any pair of 99 or lower. And we’ll filter for 5 or more folds in front of the first bet. That will give us a really big sample and a good general answer about how much equity shift AK can expect after several folders. That shift can then be generally applied to all the specific cases we’ve looked at, since they all depend on folders in front tending to not hold an Ace or King.

ExpW% ExpT%/2 TotExp Hands ActW% ActT%/2 TotAct #StDev 45.71% 0.21%45.93%3432 47.70% 0.17%47.87%2.29

Now we have a clear result on the equity shift with a sufficient sample, and **the average equity boost for all AK against all 99-22 is 1.94% after 5 or more folders in front**.

Unfortunately that isn’t enough.

## CONCLUSION

We can’t say from this data that AKs ever really becomes a favorite over a medium pair if several players fold first, or even that AKo becomes a favorite over a small pair. The mean equity shift with 5 or more folds is only enough in the case of AKs vs. a baby pair, which is essentially even money already. But the two cases Barry G pointed out both appear to fall short. The equity shift just isn’t quite high enough.

I wouldn’t interpret Barry’s choice of the word “several” to mean only in the worst case of 7 folders in front at a full table, leaving only the blinds. That was the only positive result we had, and even with checking 33 million hand histories, that case is so specific that we only had 101 matches, and our SD offset was only 1.15. However, our sample size with 4 or 5 folders is enough to show that there is a trend, but that the trend isn’t going to reach a big enough equity boost to make AK the favorite.

To prove this conclusively we need to analyze more than 33 million hands of full ring NLHE. But that in itself says that the effect isn’t really something any player needs to consider or even be aware of. Nevertheless, I believe we have pretty good evidence that the claims are incorrect as to making AK the actual favorite, even though the trend is in that direction and gets close.

—–

UPDATE: an error was corrected in this post for the number of hands scanned. None of the results changed, I had simply misreported the count of scanned hands in the sample. In part 2 we’ve scanned another 119 million hands from another site.

**Click the link below or on the right of the page to go to Part 2.**