Tuesday, March 13, 2012

6-Handed Hand Orderings

In a previous blog post, I explained the algorithm used to generate hand-orderings. ProPokerTools has had hand orderings for hold'em, omaha, omaha-8, big-O, and big-O8 for quite some time now. The existing orderings were generated for a full-ring game; 9 players for the big-O games, and 10 players for the others. Now I would like to share some results from my latest hand-ordering adventure.

Are you ready for a full-out nerd-out data-dump? I thought so. Let's begin.

6-Handed vs. Full Ring.

I have just finished generating hand-orderings for 6-handed versions of five flop games. The results are below. "Distance" refers to how far a particular hand is in one ordering vs. the other. The distance is given both in the number of positions and a percentage of the entire ordering.

gameaverage distancemedian distancemax distancemoved more than 5%
holdem3.98 (2.35%)3 (1.78%)16 (9.57%)22 (13.02%)
omaha540.46 (3.29%)362 (2.20%)5050 (30.73%)3730 (22.70%)
omaha-8461.35 (2.81%)319 (1.94%)3361 (20.45%)3008 (18.31%)
big-O3975.40 (2.96%)2883 (2.14%)33860 (25.18%)26436 (19.66%)
big-O83642.96 (2.71%)2409 (1.79%)36030 (26.71%)22938 (17.06%)

We can see a couple of obvious patterns. On average, most hands do not move up or down more than a couple of percentage points. However, in all cases there are at least 10% of hands that move five percent or more.

Hold'em 6-handed vs. 10-handed

Let's take a closer look at the differences for 6-handed hold'em vs. 10-handed holdem since we only have 169 hand classes to deal with. Here is a listing of all the changes between the two orderings going from 10-handed to 6-handed, where a positive number is a promotion and a negative number is a demotion. I have only listed hands with a difference of eight places or more to reduce the noise. Suited hands are in parentheses.
  • A9 10
  • A8 8
  • (98) -8
  • A5 8
  • (87) -10
  • A6 15
  • A3 10
  • A2 9
  • Q8 10
  • (54) -16
  • (64) -12
  • (T4) -10
  • Q7 9
  • J7 9
  • (53) -12
  • Q6 14
  • K2 8
  • T7 11
  • (43) -15
  • (63) -11
  • Q5 9
  • Q4 9
  • J6 11
  • T6 12
  • (42) -10
  • Q2 9
  • J3 8

Two patterns jump out; high-card strength is more important 6-handed, and suitedness and connectedness is less valuable 6-handed. Both of these patterns match most players' intuitions on hand strengths, so I'm very pleased with this result.

Run-to-Run Consistency

The algorithm used to generated hand-orderings is a stochastic process - many random trials are used to create a reasonably stable ordering. I decided to explore how much random noise is involved by generating the 6-handed hold'em ordering a total of ten times. The table below shows the difference between the first run and the other nine runs.

runaverage distancemedian distancemax distancemoved more than 5%
10.93 (0.55%)1 (0.59%)6 (3.55%)0 (0%)
20.88 (0.52%)1 (0.59%)6 (3.55%)0 (0%)
31.02 (0.60%)1 (0.59%)8 (4.73%)0 (0%)
41.03 (0.61%)1 (0.59%)9 (5.33%)1 (0.59%)
50.89 (0.53%)1 (0.59%)9 (5.33%)1 (0.59%)
60.95 (0.56%)1 (0.59%)6 (3.55%)0 (0%)
70.96 (0.57%)1 (0.59%)7 (4.14%)0 (0%)
81.04 (0.62%)1 (0.59%)8 (4.73%)0 (0%)
90.98 (0.58%)1 (0.59%)6 (3.55%)0 (0%)

As expected, there is some noise from run to run. But in all cases there was at most a single hand that moved more than 5%. This reinforces my oft-repeated claim that the hand orderings are useful as a rough guide, but shouldn't be trusted at too fine a level of detail.


The hand-ordering algorithm repeats the same process over a number of generations, with each generation becoming more 'accurate' with a doubling of the number of random trials. The idea is that after a while the ordering becomes relatively stable. Below is a listing of each generation in the 6-handed omaha ordering and how it compares to the prior generation.

generationaverage distancemedian distancemax distancemoved more than 5%
1 (first)n/an/an/an/a
21930.77 (11.75%)1439 (8.76%)13311 (81.00%)10998 (66.93%)
31417.24 (8.62%)1033 (6.29%)9779 (59.51%)9404 (57.23%)
41029.30 (6.26%)737 (4.49%)9423 (56.35%)7653 (46.57%)
5740.68 (4.51%)530 (3.23%)5704 (34.71%)5802 (35.31%)
6524.36 (3.19%)372 (2.26%)4288 (26.10%)3630 (22.09%)
7372.15 (2.26%)267 (1.62%)2795 (17.01%)1882 (11.45%)
8260.11 (1.58%)185 (1.13%)1983 (12.07%)647 (3.94%)
9185.86 (1.13%)131 (0.80%)1213 (7.38%)119 (0.72%)
10 (final)132.76 (0.81%)94 (0.57%)1193 (7.26%)8 (0.05%)

It is clear from the data that 10 generations gives a relatively high level of stability, with only a tiny fraction of hands moving more than 5%. Hold'em, omaha, and omaha-8 orderings all ran for a full 10 generations.

The 5-card omaha variants ran only for 6 generations [UPDATE 6/12/2012 - 8 generations]  due to the increased computational load (and my desire to get my computer back after the weekend), hence they are inherently less accurate than their brethren. However, this is probably mitigated by the very large number of hand classes in the 5-card variants (134,459 to be exact) which should cause some of the inaccuracies to average out (for example, if JT(987) is rated a bit high, perhaps hand classes such as JT(98)7, J(T98)7 and (JT9)87 will average it out).

How Many Hands In Common?

The table below shows the number of hands in common between the 6-handed and full-ring orderings for several hand ranges.

gametop 10%11%-20%21%-30%31%-50%51%-100%

The data shows that premium hands in one ordering are almost always premium hands in the other. The biggest differences appear in the medium-strength hands, those in approximately the top 15%-35%.

Some All-in Equity Results

The data below shows hand race results for each game. Each entry lists the full-ring equity and then the 6-handed equity. Equities are for the top 10% hand.

game10% vs. 11%-20%10% vs. 21%-35%10% vs. 11%-20% vs. 21%-35%

We can see that in all cases but one, the equity for the top 10% hand increases under the 6-handed ordering, although the improvement is never very significant.

Ordering Files

Below are the data files for the new 6-handed hand orderings:

Data Overload

I think that's enough data for now. If there are some more data points you would like me to collect, feel free to comment here and I'll see what I can do.