self-balanced hand rankings

[eastbay]

Recently there have been some threads about ranking hands vs. random hands, or ranking hands vs. selected hands for their win pcts. If you're jumping into this thread fresh, you might want to search for and read those first to get the background on this. The basic context is "at the end, when blinds are huge, HU, what hands are best for going all-in?"

I put some of those results up here: http://rwa.homelinux.net/poker/hand-rankings.html

The question then came up - what happens if you keep repeating the process, re-ranking hands as they do against the top hands of the list that was just generated?

What I did was start with the hand rankings associated with how hands do against a randomly chosen hand:

http://www.jazbo.com/poker/huholdem.html

I then took the top half of those hands, and re-ranked according to how each hand fares against this new list of select hands. I then took the top half of that list and ranked all the hands against that. And so on.

The process does converge quite quickly to a list that does not change upon re-ranking. I think it took 3 iterations.

The list on the left below is the ranking and win pct against a random hand. The list on the right is the new "self-balanced" list which is the result of the re-ranking iteration I described, using the top half of each subsequent ranking (the top 84 hands, to be precise).

vs. random | self-balanced (50%)

AAo: 0.852037 AAo: 0.852774
KKo: 0.823957 KKo: 0.786445
QQo: 0.799252 QQo: 0.732733
JJo: 0.774695 JJo: 0.699019
TTo: 0.750118 TTo: 0.669922
99o: 0.720573 AKs: 0.669236
88o: 0.69163 AKo: 0.652736
AKs: 0.670446 AQs: 0.641415
77o: 0.66236 99o: 0.633504
AQs: 0.662089 AJs: 0.622761
AJs: 0.653927 AQo: 0.622686
AKo: 0.65318 ATs: 0.607714
ATs: 0.646024 AJo: 0.602546
AQo: 0.644298 88o: 0.6019
AJo: 0.635612 ATo: 0.586468
KQs: 0.634004 A9s: 0.579613
66o: 0.632847 77o: 0.575587
A9s: 0.627812 KQs: 0.567876
ATo: 0.627196 A8s: 0.560263
KJs: 0.625673 A9o: 0.556357
A8s: 0.619438 66o: 0.553482
KTs: 0.617886 KJs: 0.550502
KQo: 0.614538 KQo: 0.544779
A7s: 0.60984 A7s: 0.543824
A9o: 0.607708 KTs: 0.535916
KJo: 0.605667 A8o: 0.535642
55o: 0.603249 55o: 0.533648
QJs: 0.602592 A6s: 0.527961
K9s: 0.599885 A5s: 0.526257
A5s: 0.599229 KJo: 0.525986
A6s: 0.599058 A7o: 0.518098
A8o: 0.598706 A4s: 0.517687
KTo: 0.597369 44o: 0.511433
QTs: 0.594676 A3s: 0.511124
A4s: 0.590336 KTo: 0.510402
A7o: 0.588392 K9s: 0.507929
K8s: 0.583123 A2s: 0.504211
A3s: 0.582203 A6o: 0.500889
QJo: 0.581327 A5o: 0.499103
K9o: 0.578099 QJs: 0.49305
A5o: 0.576945 A4o: 0.489862
A6o: 0.576804 33o: 0.489583
Q9s: 0.576643 A3o: 0.482721
K7s: 0.575377 K8s: 0.480828
JTs: 0.575279 K9o: 0.480425
A2s: 0.573789 QTs: 0.480021
QTo: 0.572887 A2o: 0.475245
44o: 0.570228 22o: 0.470103
A4o: 0.567276 K7s: 0.466971
K6s: 0.566407 QJo: 0.465063
K8o: 0.560181 JTs: 0.45915
Q8s: 0.560177 K6s: 0.454267
A3o: 0.558426 Q9s: 0.45305
K5s: 0.557929 K8o: 0.451482
J9s: 0.556625 QTo: 0.451136
Q9o: 0.553584 K5s: 0.443539
JTo: 0.552457 K7o: 0.436688
K7o: 0.551853 K4s: 0.435184
A2o: 0.549265 J9s: 0.432513
K4s: 0.548846 JTo: 0.429237
Q7s: 0.543023 Q8s: 0.428724
K6o: 0.542212 K3s: 0.428569
K3s: 0.54055 T9s: 0.426697
T9s: 0.540275 K2s: 0.423425

J8s: 0.540156 K6o: 0.422876
33o: 0.536931 Q9o: 0.422256
Q6s: 0.536126 J8s: 0.412752
Q8o: 0.535978 K5o: 0.411397
K5o: 0.533119 Q7s: 0.410816
J9o: 0.532492 T8s: 0.409699
K2s: 0.532117 98s: 0.409047
Q5s: 0.527694 Q6s: 0.406624
T8s: 0.523344 K4o: 0.402398
K4o: 0.523254 87s: 0.401403
J7s: 0.523248 J9o: 0.400719
Q4s: 0.518553 Q5s: 0.400122
Q7o: 0.517636 J7s: 0.398888
T9o: 0.515296 97s: 0.396527
J8o: 0.514881 Q8o: 0.396259
K3o: 0.514236 T7s: 0.39591
Q6o: 0.51022 76s: 0.395908
Q3s: 0.510192 K3o: 0.395209
98s: 0.508008 T9o: 0.394834
T7s: 0.50639 Q4s: 0.393548
J6s: 0.506059 86s: 0.389837
K2o: 0.505067 K2o: 0.389606
22o: 0.50334 65s: 0.388659
Q2s: 0.50169 Q3s: 0.387367
Q5o: 0.50118 J6s: 0.384417
J5s: 0.499868 Q2s: 0.383829
T8o: 0.497192 96s: 0.38376
J7o: 0.496799 T6s: 0.382255
Q4o: 0.491256 J5s: 0.381525
97s: 0.491177 54s: 0.381423
J4s: 0.490705 75s: 0.381279
T6s: 0.489407 J8o: 0.379583
J3s: 0.482316 Q7o: 0.377172
Q3o: 0.482174 T8o: 0.376602
98o: 0.48095 98o: 0.375978
87s: 0.479363 J4s: 0.375293
T7o: 0.479061 85s: 0.374928
J6o: 0.478422 64s: 0.372926
96s: 0.474283 Q6o: 0.372488
J2s: 0.473782 J3s: 0.369675
Q2o: 0.472934 95s: 0.368701
T5s: 0.472163 87o: 0.367893
J5o: 0.471789 T5s: 0.367251
T4s: 0.465305 53s: 0.366116
97o: 0.462958 J2s: 0.366104
86s: 0.462433 Q5o: 0.36554
J4o: 0.461843 74s: 0.365054
T6o: 0.4609 J7o: 0.364793
95s: 0.457219 T4s: 0.36355
T3s: 0.456925 97o: 0.362635
76s: 0.453718 76o: 0.362227
J3o: 0.452735 T7o: 0.361889
87o: 0.450488 43s: 0.35944
T2s: 0.448395 84s: 0.358449
85s: 0.44545 Q4o: 0.358441
96o: 0.444893 T3s: 0.357946
J2o: 0.443464 63s: 0.357131
T5o: 0.442489 86o: 0.355704
94s: 0.43862 65o: 0.354818
75s: 0.436755 T2s: 0.354352
T4o: 0.435021 94s: 0.352282
93s: 0.432643 52s: 0.352081
86o: 0.432389 Q3o: 0.351749
65s: 0.431334 93s: 0.349265
84s: 0.427016 J6o: 0.349202
95o: 0.426671 73s: 0.34914
T3o: 0.425925 96o: 0.349042
92s: 0.424152 Q2o: 0.347823
76o: 0.423207 T6o: 0.347252
74s: 0.418493 54o: 0.347227
T2o: 0.416663 75o: 0.346689
54s: 0.414534 J5o: 0.346141
85o: 0.414255 92s: 0.345695
64s: 0.413333 42s: 0.345694
83s: 0.408735 62s: 0.343039
94o: 0.40669 83s: 0.342429
75o: 0.405099 82s: 0.341623
82s: 0.402716 85o: 0.339807
73s: 0.400359 J4o: 0.339419
93o: 0.400175 32s: 0.339395
65o: 0.399423 64o: 0.337991
53s: 0.39693 72s: 0.335287
63s: 0.395336 J3o: 0.333384
84o: 0.394448 95o: 0.33298
92o: 0.390959 T5o: 0.331259
43s: 0.386419 53o: 0.33088
74o: 0.385478 J2o: 0.329425
72s: 0.381559 74o: 0.329317
54o: 0.381532 T4o: 0.327264
64o: 0.380084 43o: 0.323787
52s: 0.378493 84o: 0.322163
62s: 0.37669 T3o: 0.321247
83o: 0.374818 63o: 0.321114
42s: 0.36829 T2o: 0.317262
82o: 0.368256 52o: 0.315778
73o: 0.366002 94o: 0.315401
53o: 0.362627 73o: 0.3123
63o: 0.360756 93o: 0.312173
32s: 0.359844 42o: 0.309001
43o: 0.351438 92o: 0.308215
72o: 0.345816 62o: 0.305946
52o: 0.342826 83o: 0.305037
62o: 0.340731 82o: 0.304059
42o: 0.331977 32o: 0.302362
32o: 0.323012 72o: 0.297382

Discuss.

The most obvious thing that I see is the increased rank of AK and AQ, which makes intuitive sense - if you're up against more aces, your big aces are going to do even better due to the "domination" effect.

Also, it might be interesting to compare this against the Karlson/Sklansky rankings.

Another interesting exercise would be to repeat with a "tighter" criterion for selecting the top hands. Say, the top third.

eastbay

[Che]

AKo ranks above AQs and ATo is ahead of A9s

...but...

AQo is below AJs and AJo is below ATs


The differences are miniscule for the most part, but it still seems odd at first glance. I wonder why it worked out that way...

Che

BTW- Thanks for sharing all this data eastbay. /images/graemlins/laugh.gif

[dana33]

Great job, eastbay. Fascinating stuff.

[ QUOTE ]
The most obvious thing that I see is the increased rank of AK and AQ, which makes intuitive sense - if you're up against more aces, your big aces are going to do even better due to the "domination" effect.


[/ QUOTE ]

I also noticed some interesting patterns with the pairs. The medium pairs (99 down to 66) drop significantly in rank, while TT and above stay even. 44 and below increase significantly; 55 stays even.

[Bozeman]

"using the top half of each subsequent ranking (the top 84 hands, to be precise). "

To use the top 50% you need to weight these by frequency; as it is, you are probably selecting significantly less that 50% of hands since pp's and suited hands are more likely to be in the top.

Craig

[eastbay]

[ QUOTE ]
"using the top half of each subsequent ranking (the top 84 hands, to be precise). "

To use the top 50% you need to weight these by frequency; as it is, you are probably selecting significantly less that 50% of hands since pp's and suited hands are more likely to be in the top.

Craig

[/ QUOTE ]

What I mean by top 50% is 50% of the "hand types" and nothing more. It doesn't mean the top of 50% of hands you would see; it would be more or less depending on what kinds of hands those are. In any case this is a bit of an arbitrary parameter. It might be interesting to vary it. But for the time I spend on this stuff I think there's more interesting things to look at.

When it comes to computing the win percentage of each hand against the set, I not only account for the relative frequency of each hand, but also the fact that you're holding two of the cards in the deck already, which changes the frequency of some opposing hands.

eastbay

[eastbay]

[ QUOTE ]
AKo ranks above AQs and ATo is ahead of A9s

...but...

AQo is below AJs and AJo is below ATs


The differences are miniscule for the most part, but it still seems odd at first glance. I wonder why it worked out that way...

Che

BTW- Thanks for sharing all this data eastbay. /images/graemlins/laugh.gif

[/ QUOTE ]

Yeah I noticed some weird stuff, too. Little aces ranking higher? Does that make any sense? I don't see how.

eastaby

[Bozeman]

Well, what fraction of hands is this?

[eastbay]

[ QUOTE ]
Well, what fraction of hands is this?

[/ QUOTE ]

13*6 pairs = 78
42*4 suited = 168
29*12 offsuit = 348

594/1326 = 45%, counted as a fraction of all possible hands you could be dealt.

eastbay

[ZeeJustin]

[ QUOTE ]
The most obvious thing that I see is the increased rank of AK and AQ, which makes intuitive sense - if you're up against more aces, your big aces are going to do even better due to the "domination" effect.

[/ QUOTE ]

Oddly enough, A2o goes UP in value. I would have assumed teh opposite since it is more likely to be dominated. Can anyone explain this?

Thank you for compiling this btw. It seems like it could be very helpful.

[eastbay]

[ QUOTE ]
[ QUOTE ]
The most obvious thing that I see is the increased rank of AK and AQ, which makes intuitive sense - if you're up against more aces, your big aces are going to do even better due to the "domination" effect.

[/ QUOTE ]

Oddly enough, A2o goes UP in value. I would have assumed teh opposite since it is more likely to be dominated. Can anyone explain this?

Thank you for compiling this btw. It seems like it could be very helpful.

[/ QUOTE ]

That one bothers me as well. It bothers me enough to think I need to double-check the answers.

eastbay

[Losing all]

I would guess ace/little goes up due to the fact that no self respecting player will fold their BB to an allin by an aggeressive player if they hold kj or the like.

Since were on the subject, I was wondering last night about how big a favorite top card/bottom card was to two middle cards. assume no flush or straight draws, something like A6o vs k7o?

[Nottom]

[ QUOTE ]
Since were on the subject, I was wondering last night about how big a favorite top card/bottom card was to two middle cards. assume no flush or straight draws, something like A6o vs k7o?

[/ QUOTE ]

These are normally in the 60-40 range

From twodimes:
pokenum -h ac 6d - ks 7h
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
Ac 6d 1026088 59.92 680824 39.76 5392 0.31 0.601
Ks 7h 680824 39.76 1026088 59.92 5392 0.31 0.399

[Dave Holdsworth]

Since you have taken the top 50% of hands (or 45% actually) then that group must include a fair number of large cards lower than A, such as Kx, Qxs (higher in proportion than the top 15% or in the top 100%). Given that, I would expect A2o to rise in value which seems to be the case.

I guess if you took the top 15% self-corrected the A2o would go back down again.

[Losing all]

Thanks. I was thinking it was a little less (like 58/42) good to know.

[dana33]

You expressed some doubt about the correctness of your results, but I have done my own calculations which exactly match yours. I also did the self-balanced calculation for the top 25% of hands (top 42 to be precise), which took a few more iterations to converge than the 50% case. My results are as follows for the top 84 hands in each case:

vs random vs top 84 vs top 42
AA 0.8520 AA 0.8528 AA 0.8501
KK 0.8240 KK 0.7864 KK 0.7406
QQ 0.7993 QQ 0.7327 QQ 0.6966
JJ 0.7747 JJ 0.6990 JJ 0.6580
TT 0.7501 TT 0.6699 AKs 0.6243
99 0.7206 AKs 0.6692 TT 0.6210
88 0.6916 AKo 0.6528 AKo 0.6059
AKs 0.6704 AQs 0.6414 AQs 0.5886
77 0.6624 99 0.6335 99 0.5824
AQs 0.6621 AJs 0.6228 AQo 0.5675
AJs 0.6539 AQo 0.6227 AJs 0.5567
AKo 0.6532 ATs 0.6077 88 0.5534
ATs 0.6460 AJo 0.6026 AJo 0.5335
AQo 0.6443 88 0.6019 77 0.5266
AJo 0.6356 ATo 0.5865 ATs 0.5237
KQs 0.6340 A9s 0.5796 66 0.5056
66 0.6328 77 0.5756 ATo 0.4984
A9s 0.6278 KQs 0.5679 55 0.4858
ATo 0.6272 A8s 0.5603 A9s 0.4825
KJs 0.6257 A9o 0.5564 44 0.4635
A8s 0.6194 66 0.5535 KQs 0.4631
KTs 0.6179 KJs 0.5505 A8s 0.4551
KQo 0.6146 KQo 0.5448 A9o 0.4540
A7s 0.6098 A7s 0.5438 33 0.4411
A9o 0.6077 KTs 0.5359 KJs 0.4358
KJo 0.6057 A8o 0.5357 KQo 0.4340
55 0.6032 55 0.5336 A7s 0.4332
QJs 0.6026 A6s 0.5280 A8o 0.4245
K9s 0.5999 A5s 0.5263 A5s 0.4233
A5s 0.5992 KJo 0.5260 QJs 0.4201
A6s 0.5991 A7o 0.5181 A6s 0.4193
A8o 0.5987 A4s 0.5177 KTs 0.4192
KTo 0.5974 44 0.5114 22 0.4186
QTs 0.5947 A3s 0.5111 A4s 0.4154
A4s 0.5903 KTo 0.5104 QTs 0.4144
A7o 0.5884 K9s 0.5079 JTs 0.4125
K8s 0.5831 A2s 0.5042 A3s 0.4085
A3s 0.5822 A6o 0.5009 KJo 0.4049
QJo 0.5813 A5o 0.4991 A2s 0.4013
K9o 0.5781 QJs 0.4930 A7o 0.4010
A5o 0.5770 A4o 0.4899 T9s 0.3972
A6o 0.5768 33 0.4896 K9s 0.3947
Q9s 0.5766 A3o 0.4828 J9s 0.3923
K7s 0.5754 K8s 0.4808 Q9s 0.3922
JTs 0.5753 K9o 0.4805 A5o 0.3901
A2s 0.5738 QTs 0.4800 QJo 0.3881
QTo 0.5729 A2o 0.4753 KTo 0.3868
44 0.5702 22 0.4701 98s 0.3859
A4o 0.5673 K7s 0.4670 A6o 0.3856
K6s 0.5664 QJo 0.4651 T8s 0.3823
K8o 0.5602 JTs 0.4592 QTo 0.3820
Q8s 0.5602 K6s 0.4543 A4o 0.3815
A3o 0.5584 Q9s 0.4531 JTo 0.3802
K5s 0.5579 K8o 0.4515 87s 0.3787
J9s 0.5566 QTo 0.4512 J8s 0.3783
Q9o 0.5536 K5s 0.4435 Q8s 0.3781
JTo 0.5525 K7o 0.4367 K8s 0.3779
K7o 0.5519 K4s 0.4352 K6s 0.3757
A2o 0.5493 J9s 0.4325 76s 0.3753
K4s 0.5488 JTo 0.4293 K7s 0.3746
Q7s 0.5430 Q8s 0.4287 A3o 0.3741
K6o 0.5422 K3s 0.4286 97s 0.3724
K3s 0.5405 T9s 0.4267 65s 0.3707
T9s 0.5403 K2s 0.4234 86s 0.3706
J8s 0.5402 K6o 0.4229 K5s 0.3703
33 0.5369 Q9o 0.4223 T7s 0.3674
Q6s 0.5361 J8s 0.4128 Q6s 0.3670
Q8o 0.5360 K5o 0.4114 A2o 0.3662
K5o 0.5331 Q7s 0.4108 T9o 0.3641
J9o 0.5325 T8s 0.4097 K4s 0.3637
K2s 0.5321 98s 0.4090 96s 0.3637
Q5s 0.5277 Q6s 0.4066 J7s 0.3637
T8s 0.5233 K4o 0.4024 Q7s 0.3634
K4o 0.5233 87s 0.4014 54s 0.3622
J7s 0.5232 J9o 0.4008 Q5s 0.3618
Q4s 0.5186 Q5s 0.4001 75s 0.3605
Q7o 0.5177 J7s 0.3989 K9o 0.3603
T9o 0.5153 97s 0.3965 J9o 0.3591
J8o 0.5149 Q8o 0.3963 T6s 0.3586
K3o 0.5143 T7s 0.3959 Q9o 0.3586
Q6o 0.5102 76s 0.3959 K3s 0.3570
Q3s 0.5102 K3o 0.3953 85s 0.3555
98s 0.5080 T9o 0.3949 Q4s 0.3553

Note that A2o drops back down in relative value when you're up against the top 25% as opposed to the top 50%. All pairs below 99 also move up in (relative) value.

P.S. Sorry this is so ugly, but the posting software is ignoring the tabs in the data.

[dana33]

A very interesting thing happened when I tried to compute the self-balanced ranking against the top 10% of hands (top 17). The process does not converge to a single ordered set, but cycles through the following 4 ordered sets. After reaching the ordering in the last column, it cycles back to the first:

AA AA AA AA
KK KK KK KK
QQ QQ QQ QQ
JJ JJ JJ JJ
AKs TT AKs AKs
AKo AKs TT AKo
TT AKo AKo TT
AQs 99 99 99
99 AQs AQs AQs
AQo 88 88 AQo
88 AQo AQo 88
AJs 77 AJs AJs
77 AJs KQs KQs
KQs 66 77 ATs
66 KQs KJs KJs
55 KTs ATs 77
AJo QTs KTs AJo
44 JTs QJs 66
ATs KJs AJo 55
33 ATs A9s A9s
22 QJs JTs KQo
KJs T9s KQo 44
KTs AJo QTs QJs
QTs 55 A8s KTs
KQo K9s KJo JTs
JTs KQo 66 33
QJs Q9s ATo A5s
T9s J9s K9s ATo
A9s A9s A5s A8s

(I include only the top 30 or so hands above, and have omitted the probability figures.)

[eastbay]

[ QUOTE ]
A very interesting thing happened when I tried to compute the self-balanced ranking against the top 10% of hands (top 17). The process does not converge to a single ordered set, but cycles through the following 4 ordered sets. After reaching the ordering in the last column, it cycles back to the first:

AA AA AA AA
KK KK KK KK
QQ QQ QQ QQ
JJ JJ JJ JJ
AKs TT AKs AKs
AKo AKs TT AKo
TT AKo AKo TT
AQs 99 99 99
99 AQs AQs AQs
AQo 88 88 AQo
88 AQo AQo 88
AJs 77 AJs AJs
77 AJs KQs KQs
KQs 66 77 ATs
66 KQs KJs KJs
55 KTs ATs 77
AJo QTs KTs AJo
44 JTs QJs 66
ATs KJs AJo 55
33 ATs A9s A9s
22 QJs JTs KQo
KJs T9s KQo 44
KTs AJo QTs QJs
QTs 55 A8s KTs
KQo K9s KJo JTs
JTs KQo 66 33
QJs Q9s ATo A5s
T9s J9s K9s ATo
A9s A9s A5s A8s

(I include only the top 30 or so hands above, and have omitted the probability figures.)

[/ QUOTE ]

Neato. I thought that might happen at some % (actually I thought it might happen at any % until I verified otherwise for 50%).

This is cool, but, I think we're into the realm of academic uselessness now. I've probably been there awhile. /images/graemlins/smile.gif

eastbay