Hello,

a small thread about starting hands in PLO

Literature and online information is rather scarce.
However, there is a list of starting hands from 2006 available on the ProPokerTools website accessible here: http://www.propokertools.com/orderings/ohordering.txt
You will find a link explaining the method of calculation to achieve this ranking.

This list contains 16432 types of starting hands in Omaha according to the "Hold'em" sort (which has 169 types of hands).
There was an article on Wikipedia explaining the classification of PLO hands : https://web.archive.org/web/20151004...bility_(Omaha)

Here is a proposal for ranking starting hands in PLO not based on their relative values but on their "absolute value".
Personal definition of absolute value: "number of boards for which the hand is the nuts".

For those who are not familiar with PLO, the fact of taking 2 private cards in the final combination limits the pots splitted.
The other aspect also concerns a specificity of the PLO where the action takes place with big hands, hence the importance of having the nuts.

For those who want to leave it at that, you can simply take these 2 lists sorted by board count

• a list of all PLO hands (270725 possible hands)
• a list of 16432 combinations

For those interested in theory, here are the different possibilities of nuts on any board:

Straight Flush

For all hands, as soon as a straight flush was possible, the board was taken into account even if a higher straight flush was possible.

Example: on a board 8 7 6 3 2, all the hands with 5 4 were counted even if a hand with T 9 is max.

Quads

As soon as a hand makes quads, the board was counted (even if a straight flush or higher quads was also possible).

Full

Only the full max were taken into account. Note that if quads or straight flush were possible, they were not removed from the calculation.

Example: A A 7 6
- Board with an Ace and a pair AKTT2 - full max AAATT
- Board starting with 776 to make a full max 77766 - a board with one or two higer cards can give a full max (KJ776 can give a better full with KK, JJ, K7 or J7)
- Board starting with a three or four of a kind - a board with 88852 gives the full max 888AA (the possible quads has not been taken into account)

Other example: 7 6 5 4
- Possible Boards: 776xx - 775xx - 774xx - 665xx - 664xx - 554xx

Flush

All boards giving a nut flush have been calculated. Attention, it is madatory that on the board, there is no pair or trips or quads to avoid a full.
It is possible to have a max flush Q high if on the board there was AK suited.
And so there is only one board where the nut flush is 9 high : on the board A K Q J T

Straight

Same principle, all the boards giving a straight max have been calculated. This included all boards without any pair or flush possible.
Be careful, only the nuts straigh are taken into account.
For example, a QT952 board has only a combination of max quint KJ and not J8.

Set

This concerns the boards with no pair, no flush or no straight possible.
This is the last possible combination with a hand like AAxx, KKxx or QQxx.
Example of boards: AK982 or KQ872 or QJ762
A board starting with J has at least one possible straight...

It is therefore impossible to have the nuts with 2 pairs, one pair or a high card.
Last but not least, the straight flushes have been subtracted from the squares, full and flush to avoid doubles.

The theory over, for those of you who like numbers, here are a few of them.

There are 270,725 starting hands in PLO (1326 in NLHE)
There are 2,598,960 possible boards (5 cards from 52)
For a PLO hand, there are only 1,712,304 possible boards left (5 cards from 48).

Best hand : A:club A K K 432,674 boards giving the max game
Number of Straight flush: 1,980
Number of quads: 30,312
Number of full max: 227,878
Nnumber of nut flush: 134,424
Number of nut straight: 21,732
Number of nut set: 18,348

Board average: 80,096

Median hand: T 9 7 4 - represented by (T7)94 with 69490 boards max.
Straight flush number: 1,936
Number of quads: 3,958
Number of full max: 7548
Number of nut flush: 32
Number of nut straight: 56,016
Number of nut set: 0

Max of board
that can make straight flush: 7,656 (294 hands involved, all combinations of 2 combo JTs to 54s without a pair)
that can make quads: 30,316 (1428 hands - all hands with 2 pairs except those with straight flush possible)
that can make full max: 227,920 (6 hands only: AAKK offsuit because we removed the possible straight flushes with AAKK suited)
that can make nut flush: 136,164 (72 hands, all AAxx double suited but with xx between 9 and 6 because the straights flush are removed for other xx)
Max of board that can make a nut straight: 149,218 (24 hands KQJT offsuit)
Max of board that can make top set: 18,852 (the same 6 hands AAKK off suit as for full)

Limitations of this calculation

• I started from the idea that you have to have nuts. It's debatable, but a selection criteria is necessary to rank the hands.
  If some people have suggestions for criteria, I'll take them.
  
• It is also possible to question the fact of having a full max on a board containing a trips with a pair of Aces since a quads is possible.
  The number of boards starting with a trips (not a square) is around 12000. The impact is negligible on big AAxx hands but more important on weaker AAxx hands like AAAx.
  Don't hesitate to give me your opinion to redo this ranking.
  
• Of course, it remains just a theoretical calculation because if it is necessary to compare 2 hands, the result will be completely different. For example, if we take the worst hands of this ranking, 3 3 3 3 and 2 2 2 2, the 1st hand wins 100% of the time against the 2nd. Also, the number of boards has been calculated on all possible boards without taking into account an opponent's hand. If you take 2 AAxx hands, you can remove all the quads, full max...

If you need to compare 2 hands, I suggest you to go to http://www.propokertools.com/ which allows you to compare PLO hands.

Thanks for your feedback and comments!
Ronanko

There is a small mistake here

Better formulation: "A board J High has at least one possible straight..."