Best hand heads up against A, A

[Scottso]

Using the odds calculator on this site, I've come up with the best hand (besides another A,A) to be dealt when going heads-up against A,A to be 6, 5 suited (with the suit being different than either of the Aces) but I don't understand why.

According to the odds calculator, these are the probabilities of the situation above and also 7, 6 and 8, 7.

6, 5 Wins: 22.87 Ties: .37

7, 6 Wins: 22.87 Ties: .32

8, 7 Wins: 22.87 Ties: .29

Now, of course the best hand to have would be the 6,5 suited, because even though it has the same probability of beating A,A, it ties just a bit more.

But why does 6,5 win just as much as 7,6? With 6,5, there are two straight possibilities that can't be had because there are two Aces that can't be dealt to the board (for A, 2 3, 4, 5). With 7,6, Aces won't help to make a straight.

[Gunslinger]

If you have 6,5 , then a 2, 3, 4 on the board is enough to give you a straight. Any ace on the board is irrelevant in making you a straight.

Therefore, 6,5 has the same number of boards to give it a straight as 7,6 or 8,7.

[Ensano]

the suits matter because of the possibility of 4 to a flush...

[Adamm]

Ug this completely baffles me.
I ran it too, just to double-check.
So with aces vs. suited connectors of a different suit than both aces here are the deterministic results for all boards...

vs 87s: 1,315,602 wins / 391,672 losses / 5030 ties
vs 76s: 1,315,168 wins / 391,637 losses / 5499 ties
vs 65s: 1,314,307 wins / 391,582 losses / 6415 ties

So AA's win chance actually goes down versus lower suited connectors. It's just not visible because the calculator only goes to the second decimal place.
Why are there more ties with lower cards? It isn't even because of board quads or board boats (which usually explain these funky sorta stats), because AA always has a leg up on those boards vs non-ace hands.

But wtf? What sort of boards are driving up the tie%? There is no straight-making interference between the cards at all. Is the calculator busted?

The only difference I can think of is with board flushes of the connector's suit, but that should affect the lower-suited connectors negatively, right? 87s ties less in those situations than 65s simply because of the combination of five possible overcards, causing a tie (and what would be a terrible beat, I might add).

ARg.
This is the sort of thing that will drive me insane. Somebody help please. Put your math inside me.

[suitedaces84]

Equity of AA:
87s = 0.769791462
76s = 0.769674953
65s = 0.769439597

56s and 67s will win more than 87s for a couple reasons:
-56s and 67s will win on a 2345x board
-it's easier for 87s to make a straight and lose. 87s will lose on a 9TJQK board this takes away some of its straight power. If 56s or 67s make a straight they can only lose to a flush or better.

87s is better because it will chop less when there is a flush on the board.

The tough question is why will 65s win more than 76s?

[und3rstudy]

Just pointing out if you hold 56 you don't need an Ace to make the straight you only need 234 so the two aces missing doesn't affect the odds

[Adamm]

-it's easier for 87s to make a straight and lose. 87s will lose on a 9TJQK board this takes away some of its straight power. If 56s or 67s make a straight they can only lose to a flush or better.

I dunno about that. 87s won't win there, but neither will 56 and 67. Conversely, 56 and 67 will just chop on a board like 789TJ. Board overstraights only affect tying power, although there are admittedly more possible with lower cards.

I never really thought about 4-card straights and board straights and how they affect hand v hand odds. I need to ponder more. My first impression is that there is no difference, because as long as you can make a three-card straight, there needs to be no more consideration (when comparing hands like these). 56 can make a straight four different ways (234, 347, 478,789) the other two cards don't matter (for purposes of comparison here at least).

I'll continue thinking about this at red lights and during commercial breaks. More later.

[suitedaces84]

56 can make a straight four different ways (234, 347, 478,789) the other two cards don't matter (for purposes of comparison here at least).

I'll continue thinking about this at red lights and during commercial breaks. More later.

We can agree that there are an equal number of ways for 56s and 78s to make a straight. However there are more ways for 87s to lose, given that it made a straight. This is because 87s has the potential to lose to a bigger straight when it makes a straight. 56s does not have that potential.

[suitedaces84]

To add:

Both hands can make a straight 4 ways.

For 56s those are: 234xx, 347xx, 478xx, 789xx. The only way 56s can lose on these boards is to a flush or better.

For 78s those are: 456xx, 569xx, 69Txx, 9TJyy. If yy happens to be QK 78s will lose.

I see what you're saying about both hands losing on a 9TJQK board. But we do not score this as a straight board for 56s. We do score it as a straight board for 78s.

[holdemhelpem]

I was skeptical on the 6,5 so I wrote a program to verify it. I ran 1 million random simulations for each hand (1225 hands) and I counted how many times the hand lost against AdAs and then sorted the hands.

Here are the results for the top 10, the last column is how many times the hand lost against AdAs in 1 million random boards:

1: Ah Ac 21893
2: 6c 5c 767882
3: 6h 5h 768004
4: 7c 6c 768237
5: 8h 7h 768693
6: 7h 6h 768734
7: 8c 7c 768956
8: Th 9h 770165
9: Tc 9c 770312
10: 9c 8c 771809

It is hard to explain the 6,5 but it obviously has to do with flushes and straights. I will probably post the complete list on my website. Also I'll run the simulation for pocket kings to see if it provides and further insight.