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. [img]/images/graemlins/laugh.gif[/img]

[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. [img]/images/graemlins/laugh.gif[/img]

[/ 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