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Statistically what is the best hand versus AA (other than AA)

I had read 56, 67, 78
Suited not the suit of any ace
56 being the best
I cannot figure how 56 is better
Ran on poker stove 

                          win +
    loss    win     tie   tie/2
56  1314307 391582  6415  394789.5
67  1315168 391637  5499  394386.5
78  1315602 391672  5030  394187.0

56 compared to 67 has 55 less wins but it has 916 more ties.

Q J T 9 8 is a tie. 78 has a blocker on that so I can see how it is worse.

On the bottom end 56 and 67 both have 3 cards under to make a straight on both sides.
The ace does not steel 2345.

56 has a blocker on 2345 so I can see less wins there compared to 67. But I do not get the number 55 there.

I cannot figure the difference between 56 and 67 on wins and ties.

Where are the differences between 56 and 67?

Is poker stove wrong?

Is there actually another hand that is better against AA?

paparazzo
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    Interesting question. One minor note though, it is standard to say 65, instead of 56. – Raymond Mar 31 '17 at 22:11
  • @RaymondTimmermans Are they different hands? – paparazzo Mar 31 '17 at 22:55
  • I have been over and over this. 56 picks up 2 ties but loses a win. I cannot figure why it is stronger. – paparazzo Apr 03 '17 at 21:48
  • Just read my answer. It explains it. – Raymond Apr 04 '17 at 06:02
  • @paprazzi apparently there are more than two extra ties 65 has over 1 win 87 has. I explained this in my answer. I am not a mathematician, so I dont know the exact percentages. – Raymond Apr 04 '17 at 11:16
  • @RaymondTimmermans "Apparently" is not an explanation. – paparazzo Apr 04 '17 at 11:21
  • Apparently 65 has better equity vs 87. This is a fact, because it is calculated. In my answer I explain the only way in which equities could possibly differ. This is because 65 has more ties and 87 has more wins, but APPARENTLY the equities are different. Why is this? Well it must be because 65 has more than twice the amount of extra ties compared to one extra win 87 has. I am not a mathematician, but I doubt anyone in this community will be able to answer your question better than I did. I suggest you to ask this question in mathematical communities if you want to get a better answer. – Raymond Apr 04 '17 at 14:30
  • Apparently it is better for the equity to block less straights that will tie compared to having no blockers for your winning straight. Therefore can be concluded that straights on the board occur much more frequently than 3 or 4 to a straight that can be used for 65 or 87 to win. The exact percentages aren't relevant, because you did not ask for it. I perfectly explained your question. You ask for a possible difference between 65 and 76? I explained. You asked if poker stove was wrong? No it wasn't. You ask if there is a better hand against aces other than 65? No there is not. – Raymond Apr 04 '17 at 14:40
  • @RaymondTimmermans *Apparently* is not an answer. Where are the differences means specifics hand that only ties / wins and not the other. – paparazzo Apr 04 '17 at 15:45
  • I literally answered where the differences are in specific hands in specific situations where one hand ties or wins and the other doesn't!!! – Raymond Apr 05 '17 at 09:22

4 Answers4

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I exhaustively simulated (every possibility is run) AA vs any pocket pair and AA vs any suited connector. Firstly let's cover the pocket pairs:

Obviously it is best for the under-pair to have different suits, in order to win with flushes. The best pocket pair to have is pocket eights. This is because all straight possibilities will result in a win. This is also the case with sixes and sevens (aces are blocked with pocket fives), but it is important to have a higher pocket pair, since you won't get counterfeit as often when you hit a set. For example with pocket sixes on 6777x results in a loss, while with pocket eights on 8777x results in a win. When we run the odds we get the following: 8s8c (20.20%) vs AhAd (79.80%).

Now on to the suited connectors. I didn't even bother to run suited one gappers, since they make less straights and therefore are for sure a bigger dog against aces. The suited connectors that make the most amount of straights are 56-JT. It is important that the hands' suits are different then both aces, since when the flush is made on flop or turn, the aces will not have (extra) outs. Let's make an overview of their equities against aces:

65: 23.06%
76: 23.03%
87: 23.02%
98: 22.62%
T9: 22.77%
JT: 21.72%

JT's equity is the lowest, since his straight outs are blocked by the aces and even when broadway is hit the board has to run out unpaired.

T9's equity isn't the best, since the board can run out KQJT, giving aces a better straight.

98's equity isn't best, since the board can still run out KQJT.

Now our last 3 hands are extremely close in equity. All the straight that can be made will result in a win. The equities differ however. Let me explain:

With 65, A234 and A2345 are blocked by aces. With 76, A2345 is blocked by aces. This explains the fact that 87 has more wins than 76 and 76 has more wins than 65. One would expect 87 to have the highest equity and 65 the lowest, but this isn't the case. The reason for this is the amount of ties each hand makes.

The boards A2345-89TJQ, will generally result in a tie. See this image for a nice overview of all possible 'straight boards'. The key is the A2345 straight combination. This straight will happen way less frequently compared to the other straights, because it is blocked by aces.

Essentially the goal is to have the most amount of straight ties. When 65 blocks A2345 obviously some of those ties vanish, but it's better to block A2345 compared to other straights. This is because when a straight is blocked it will occur less, relative to the amount of which it occurs. If A2345 occurs less, you will ultimately lose less absolute straight ties, compared to when another straight board is blocked. As can be seen on the picture 65 is the only hand that blocks A2345, which results in 65 having the most amount of ties.

Now let's explain the difference of the amount of ties between 87 and 76. 76 wins on A2345 and 23456. On all other boards 76 will tie. 87 wins on 23456 and 34567. On all other boards 87 will tie. Here you can see that 76 has more ties, since the six boards 76 ties on is not blocked by aces.

To conclude: I exlplained that 87 has the most wins followed by 76 and then 65. However, this difference in wins is relatively unimportant compared to the difference in amount of ties between the three hands. 65 has by far the most amount of ties, followed by 76 and then 78. Therefore 65 has the best equity, then 76 and then 87.

Raymond
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According to this odds calculator site, the reason 56s is better, is because it loses less against AA.

Here are the top 10 hands they list vs. AA:

Best Hands Against Pocket Aces in Holdem  
Rank    Hole    Common    Losses (Out of 20 Million)  
        Cards   Suits
1       AA      *0        435476  
2       65s     *0        15351376  
3       76s     *0        15362368  
4       87s     *0        15362369  
5       T9s     *0        15412798  
6       98s     *0        15445945   
7       65s     *1        15566984  
8       76s     *1        15575989  
9       87s     *1        15582347  
10      86s     *0        15607277
Herb
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  • I am more looking for specific hands to show the difference – paparazzo Apr 01 '17 at 08:01
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    How is that not answering the question? Right at the top of the chart it says, just as the topic says, "Best hands against pocket Aces." Additionally, I cover why they say that 65s is the best hand, other than pocket aces. – Herb Apr 01 '17 at 11:41
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    Not a great answer but did not deserve a down vote. +1 – paparazzo Apr 01 '17 at 15:10
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Not a complete answer
Yes 56 has more ties and less wins compared to 67
Why there are enough ties to overcome the wins is not so clear 

                          win +
    loss    win     tie   tie/2
56  1314307 391582  6415  394789.5
67  1315168 391637  5499  394386.5
78  1315602 391672  5030  394187.0

The difference between 56 and 67 is only 0.1% 

A                                               A       
K                                               K       
Q                                               Q       
J                                               J       y
T                                               T   y   y
9                                               9   y   y
8                                               8   y   y
7   X                                           7   y   y
6   X   X   y    y                              6   y   
5       X   y    y    y     y    y              5       
4           y    y    y     y    y    y     y   4       
3           y    y    y     y    y    y     y   3       
2                y          y    y    y     y   2       
A                                y    y         A       

    67      win  win  win   win  win  lose  lose    tie  tie
        56  lose tie  lose  win  win  win   win     tie  tie   
67 block +- -1   +1   +1    +1   +1   -2    0       -1   -1

On the T high straight the difference is 2 blockers versus 1
4*4*4*4*3 - 4*4*4*3*3 = 4*4*4*3 = 192
On the J high straight the difference is 1 blockers versus 0
4*4*4*4*4 - 4*4*4*4*3 = 4*4*4*4 = 256
On the 6 high tie with 2 blockers 4*4*4*3*3 = 576
Total = 1024

The simulation is 916 more ties. The difference is flush. AA can make a flush and steal some of those ties. And the 56 or 67 can make a flush and turn a tie into a win. If a suit not in either hand makes a flush it is still a tie.

If you forget about blocker 

                          win +
    loss    win     tie   tie/2
67  2       5       2     6   
56  2       4       3     5.5

To me it looks like 67 has 1 more block that would hurt it. Not getting how the blockers turn the tide but they the seem to.

paparazzo
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I’m new here but have played for a living for 40 years. The differences in the 3 hands vs AA is so minute as to not be important.But 56s > 67s > 78s vs AA. EV is the ultimate authority and the ties are what gives the slight edge to 56 & 67.

Shawn Nelsen
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  • How does 56s 67s and 78s' EV compare? The assertion isn't backed with anything and the differences are what this question is trying to find out... – Unihedron Oct 30 '22 at 00:48