What starting hand --> the nuts on the river most frequently?

Wow didn't get a notification that you'd posted on this thread for some reason, thanks so much for the detailed information, that's really great and interesting.

One comment though regarding why ATs is ahead of AKs - I don't think it can be due to making one card straights, because for example on a 9876 board the lone ten doesn't make the nut straight. Or am I missing something? How else could it possibly be that ATs makes the nuts more often than AKs? And why does AJs make the nuts more than AQs, and AQs more than AKs?

That's a good point. I guess PPT must be defining one-card straights as the nut high on the river..?

To test it, I can ask how often ATs is the nuts on 98762r, - it returns never. This test indicates that it does not consider one-card straights as the nuts.

A more convoluted test is to ask how often ATs will be the nuts if you only look at rivers where you have a straight, and all the Queens are dead. Doing this it still shows that you have the nut high 17.27% of the time, indicating it DOES count one-card straights as "nuts".

So I guess the answer is that PPT is just inconsistent with how it defines the nut hi. Just a bug in this old freeware.

Hmm, okay. Interesting. So is ATs really the hand that makes the nuts on the river the most often then??

I guess it depends how you classify the nuts lol

Seems to me that it would have to be the same for all those suited ace-Broadway hands.

But in practice it would be best to have the ATs because other players with flushes would be happier with their chances.

************

Actually, maybe the ATs is best because it blocks more straight flushes.

Ex. The board has QJ98s. ATs would be the effective nuts, but AKs would not be.

Ah, interesting. Maybe chillrob has cracked it! That definitely makes sense.

Can anyone say how many flops are used in the calculations above? Because we can see in the charts that ATs is the nuts 0.08% more often than AJs, which is the nuts 0.08% more often than AQs, etc. If we knew how many flops were used, then we could calculate how many flops are 0.08%, which might shed some light on the matter.

I don't think this principle can also apply with AJs vs AQs, or AQs vs AKs though, right? Remains to be seen how those hands are still 0.8% of their higher counterparts.

I think that is because AJ and AQ make more effective nut straight flushes than AK, because they can make the nuts playing one hole card.

They all make the same number of royal flushes but AQs also has the effective nuts on a board of JT98s.

AJs has the effective nuts on boards of T987 and QT98.

AKs only makes the royal.

Yeah I think you're on the right lines here. But note that AQs is not the effective nuts on JT98s because it loses to 76s, and same deal for AJs on T987 and QT98; it's not the nuts because it loses to the lower-end straight flush in both cases.

I believe what is happening that causes ATs to make the nuts more often than the higher suited aces is because there's more situations where it makes the nut flush (which is the nuts, i.e. there's no paired board) as well as simultaneously blocking straight flushes:

ATs:

KQJs
KQ9sxx
KJ9sxx
QJ9sxx
QJ8sxx
Q98sxx
J98sxx
J97sxx
J87sxx

AJs:

KQTs
KQ9sxx
KT9sxx
QT9sxx
QT8sxx

AQs:

KJTs
KJ9sxx
KT9sxx

AK:

QJTs

It seems that these few rare situations are enough to give ATs the fractional edge that it has vs the higher suited aces. However, I think there may be more to it than just the above, because in the output posted by tombos it implied that ATs is 0.8% ahead of AJs, which is 0.8% ahead of AQs:

This implies that ATs should be ahead of AJs by the same number of combos that AJs is ahead of AQs - but at least in my analysis above, that's not the case - so perhaps I'm missing something.

Ahh yeah I think you're right Telemakus, it's measuring the effective nuts

Okay, interesting, thanks! So do you think this is all there is to it then? Because these are the only situations, other than making a royal flush, where ATs would be the nuts ahead of other, higher suited aces, right? So can you shed any light on the differences in percentage values for AKs, AQs, AJs, ATs?

Huh? A higher straight flush doesn't lose to a lower straight flush.

Ah, I misunderstood, you're right of course. Interesting, you're definitely onto something here, will take a closer look in the morning.

So - including four-card straight flush boards, we can add the following boards for each respective hand:

ATs:

Nuts on:

KQJs
KQ9sxx
KJ9sxx
QJ9sxx
QJ8sxx
Q98sxx
J98sxx
J97sxx
J87sxx

Four-card straight flush boards where this hand is the nuts:

KQJ9s
QJ98s
J987s
9876s

AJs:

Nuts on:

KQTs
KQ9sxx
KT9sxx
QT9sxx
QT8sxx

Four-card straight flush boards where this hand is the nuts:

KQT9s
QT98s
T987s

AQs:

Nuts on:

KJTs
KJ9sxx
KT9sxx

Four-card straight flush boards where this hand is the nuts:

KJT9s
JT98s

AKs:

Nuts on:

QJTs

Four-card straight flush boards where this hand is the nuts:

QJT9s

Note that these "nuts" assume that the board is not paired (in which case the nut flushes detailed above would not be the nuts, as quads would be possible).

So ATs not only makes the nut flush on more boards where it blocks a straight flush from beating it as being the nuts, but it also make more straight flushes on boards where there are four cards to a straight flush.

I wonder if these two factors alone are what give ATs the ability to make the nuts more than any other hand?