Hey guys,

While playing some 6 card PLO a friend offered to to a big flip and give me and edge of one more card if accepting. What is my % advantage doing a PLO-flip with 7 vs 6 cards? (and how do I do the math?)

Thanx!

You have 7c2 = 21 2-card combos
He has 6c2 = 15
You should win 21/36

Of course, diminishing returns kicks in meaning that the value of additional extra cards diminishes as the number of cards increases.

So 7v6 is probably not as good as the 21/36 calculation suggests.

What is the correct calculation then? Unfortunately it's not possible to run random 7 card vs 6 card hands in any simulator I know of

This isn't a pure "math" question so that there isn't a "formula" to calculate the edge. Kinda like there is no formula to calculate what edge QQ has vs AKo in NLHE. You simply have to run through all the possible boards and keep track of how many times QQ wins and how many times AKo wins (and ties).

Anyway, since there are way way too many possible deals of 7-card vs 6-card PLO to consider, a simulation does indeed seem to be the only way to get a "direct" answer.

Today I programmed a 2-person PLO simulator in which the two players need not get the same number of hole cards. Here are the player equities I found over 500,000* deals in each situation (the numbers in parentheses are the splits that would obtain purely by the number of 2-card combinations that each hand offers):

3-cards vs. 2-cards: 66.03% vs. 33.97% (75.00% vs. 25.00%)

4-cards vs. 3-cards: 62.31% vs. 37.69% (66.67% vs. 33.33%)

5-cards vs. 4-cards: 59.63% vs. 40.37% (62.50% vs. 37.50%)

6-cards vs. 5-cards: 57.87% vs. 42.13% (60.00% vs. 40.00%)

7-cards vs. 6-cards: 56.54% vs. 43.46% (58.33% vs. 41.67%)

You will see that the actual edge that the player receiving one additional card has is less than the hypothetical edge based purely on the combinatorics.

Let me know if you have any questions.

* Based upon a sample of 500,000 deals, the standard error of the estimate around the true equities is 0.07%. This means that we would expect the true equities to be within 0.11% of the simulation equities around 90% of the time and to be within 0.14% of the simulation equities around 95% of the time.

Bump for adding the 8-card vs. 7-card case below.

Here are the player equities I found over 500,000* deals in each situation (the numbers in parentheses are the splits that would obtain purely by the number of 2-card combinations that each hand offers):

3-cards vs. 2-cards: 66.03% vs. 33.97% (75.00% vs. 25.00%)

4-cards vs. 3-cards: 62.31% vs. 37.69% (66.67% vs. 33.33%)

5-cards vs. 4-cards: 59.63% vs. 40.37% (62.50% vs. 37.50%)

6-cards vs. 5-cards: 57.87% vs. 42.13% (60.00% vs. 40.00%)

7-cards vs. 6-cards: 56.54% vs. 43.46% (58.33% vs. 41.67%)

8-cards vs. 7-cards: 55.54% vs. 44.46% (57.14% vs. 42.86%)

You will see that the actual edge that the player receiving one additional card has is less than the hypothetical edge based purely on the combinatorics.

Let me know if you have any questions.

* Based upon a sample of 500,000 deals, the standard error of the estimate around the true equities is 0.07%. This means that we would expect the true equities to be within 0.11% of the simulation equities around 90% of the time and to be within 0.14% of the simulation equities around 95% of the time.

How did you know there are diminishing returns?

Anytime that the value of something is governed by a "maximum score" metric and the constituent elements "overlap" in creating a score, then the diminishing returns phenomenon comes into play.

What I am trying to say is that a 4-card Omaha hand, making up C(4,2) = 6 2-card combos, is not really the same thing as being dealt 6 2-card hands (12 cards which must only be used in pairs).

For if you are dealt AA72r, as an extreme example, the value of the hand comes largely from having AA. The other 5 combos (A7, A7, A2, A2, 72) contribute little in most cases. The same phenomenon is exhibited in a great many Omaha hands.

Thanks great explanation

Bump for upping the total sample size to 1 million in each case below. Again, posting these more reliable results for posterity in case they get referenced or used in the future.

Here are the player equities I found over 1,000,000* deals in each situation (the numbers in parentheses are the splits that would obtain purely by the number of 2-card combinations that each hand offers):

3-cards vs. 2-cards: 66.06% vs. 33.94% (75.00% vs. 25.00%)

4-cards vs. 3-cards: 62.30% vs. 37.70% (66.67% vs. 33.33%)

5-cards vs. 4-cards: 59.65% vs. 40.35% (62.50% vs. 37.50%)

6-cards vs. 5-cards: 57.88% vs. 42.12% (60.00% vs. 40.00%)

7-cards vs. 6-cards: 56.58% vs. 43.42% (58.33% vs. 41.67%)

8-cards vs. 7-cards: 55.61% vs. 44.39% (57.14% vs. 42.86%)

You will see that the actual edge that the player receiving one additional card has is less than the hypothetical edge based purely on the combinatorics.


* Based upon a sample of 1,000,000 deals, the standard error of the estimate around the true equities is around 0.05%. This means that we would expect the true equities to be within 0.085% of the simulation equities around 90% of the time, to be within 0.10% of the simulation equities around 95% of the time, and to be within 0.13% of the simulation equities around 99% of the time.

bro this is very informative ans useful in side bets
I figured using this information that random 8 card plo hand is 3.29 fav to 1 vs random 4 card plo hand

but I am getting crushed in side bets when i am offering these odds, can you run simulation and determine how much is 8 card plo hand is favourite against random 4 card plo hand.

Per request, here are the player equities I found over 1,000,000* deals:

8-cards vs. 4-cards: 77.03% vs. 22.97%



* Based upon a sample of 1,000,000 deals, the standard error of the estimate around the true equities is around 0.04%. This means that we would expect the true equities to be within 0.07% of the simulation equities around 90% of the time, to be within 0.08% of the simulation equities around 95% of the time, and to be within 0.11% of the simulation equities around 99% of the time.

thank you bro