Omaha sets vs. over-sets; simulation

[Phat Mack]

<pre><font class="small">code:</font><hr>
_____Board____ _____Hand_____ _______________Opponents_______________
___________ FD SD CC ___________ BL 1 2 3 4 5 6 7 8
Kd9h6s2sAc 0 0 0 KsKhJdJc 0 98066 96119 94191 92342 90379 88417 86485 84644
KdTh5s2s9c 0 1 0 KsKhJdJc 2 95215 90590 85950 81316 76938 72894 68592 64480
KdTh6s2sAc 0 1 0 KsKhJdJc 2 93253 86766 80572 75105 69260 64133 59288 54641
--------- - - - Total AA's- - 1962 3824 5378 6211 7678 8761 9304 9839
AAA wins by - - - Opponent 1 - 1902 1856 1706 1574 1530 1419 1310 1267
AAA wins by - - - Opponent 2 - - 1827 1740 1616 1551 1412 1294 1203
AAA wins by - - - Opponent 3 - - - 1675 1626 1552 1371 1322 1220
AAA wins by - - - Opponent 4 - - - - 1621 1560 1398 1314 1217
AAA wins by - - - Opponent 5 - - - - - 1512 1416 1248 1221
AAA wins by - - - Opponent 6 - - - - - - 1423 1314 1201
AAA wins by - - - Opponent 7 - - - - - - - 1366 1240
AAA wins by - - - Opponent 8 - - - - - - - - 1277

Total----- - - - -------------- - 1902 3683 5121 6437 7705 8439 9168 9846
</pre><hr>

The above chart logs Poker Probe simulations to assess the value of an unimproved set of Kings when an Ace is on the board.

All simulations were of 100,000 deals.

In the first simulation there is a board of Kd9h6s2sAc. No straights or flushes are possible. Our hero has KsKhJdJc and has flopped a set of Kings. Hero has the second nut hand, the only hand that beats him is AAxx which makes a set of kings. Against one opponent, hero loses slightly less than 2000 (2%) of the hands. As the number of opponents increases, the number of losses to sets of aces increases in a linear fashion; slighly less than 2% for each additional opponent. Against eight opponents he loses 15.4% of the time.

The second and third simulations were inspired by comments made by Buzz. Unfortunately, however, we can't blame Buzz for any of this. [img]/forums/images/icons/smile.gif[/img] The idea is to run a sim where a set of kings can only be beaten by a straight that requires QJxx in an opponents's hand. Then run a sim where our hero's same hand and same set of kings can only be beaten by a straight requiring an opponent to have QJxx OR a set of aces requiring opponent to have AAxx. The increase in the number of beats meted out to hero in the later sim will reflect the marginal utility, or in this case the increase in danger, of a possible over-set.

The second simulation employed a board of KhTd5s2s9c. Hero played KsKhJdJc. Hero loses to QJxx. The numbers to the right represent hero's wins against from 1 to 8 opponents.

The third simulation employed a board of KdTh6s2sAc. Hero again plays KsKhJdJc. Hero loses to QJxx or AAxx.

The line beneath the third sim is the difference between the two numbers, which should represent the number of losses to AAxx. Beneath that is the actual number of wins by each opponent during the third sim. Against one opponent, hero beat a straight 95,215 times; beat a straight AND a set of A's 93,253 times, thereby losing to a Set of A's 1962 times. During the actual 3rd simulation, Opponent 1 beat hero with a set 1902 times. (This is a difference of 60 hands. Simulations were of 100,000 deals. If the number of deals were increased to approach infinity, I assume that this difference would approach 0.)

These sets of A's held by opponents appear to decrease in number as the number of opponents increases. This is because these are the number of ace sets that win hands. The number of ace sets should be constant, but the number of winners shoud decrease as the increasing number of opponents make straights more likely.

Different web browsers will display the above matrix differently. I apologise in advance for any incoherent jumbling. I used one of the 2+2 large type options to post this.

[Buzz]

Mack - Very nicely done!

[Against one opponent, hero beat a straight 95,215 times; beat a straight AND a set of A's 93,253 times, thereby losing to a Set of A's 1962 times.

When that one opponent was holding AAQJ, the win would be by virtue of the straight, rather than the higher set. When the board is AK962 and Hero’s hand is KKJJ, that one opponent could have AAXX, where XX is anything, including another ace, 2340 possible ways. Of these, 24 could be AAQJ. Thus about 1% (24/2340) of the losses you attribute to AA are actually losses to the straight when your opponent holds AAQJ. I like the way you kept this factor as small as possible by using a pair of jacks in Hero’s hand.

That doesn’t change (or explain) the discrepancy between 1962 and 1902, which seems to be simply a matter of experimental uncertainty (the luck factor).

Lots of food for thought here.

Thanks.

Buzz

[Phat Mack]

</font><blockquote><font class="small">In reply to:</font><hr />
That doesn't change (or explain) the discrepancy between 1962 and 1902, which seems to be simply a matter of experimental uncertainty (the luck factor).

[/ QUOTE ]

I divided 1902/1962 = .9694, a ratio between the predicted number and the actual from the ace-containing board.

Then I re-ran the same sims for 1,000,000 deals and got: 952,098 - 935672 = 19426. There were 19,149 actual winning Ace sets for the run with the latter board. 19,149 / 19,426 = .9857.

Then I re-ran the sims for 2,500,000 deals (the max allowed by Poker Probe) and got: 2,379,766 - 2,331,931 = 47,835. There were 47,690 winning ace sets during the latter run. 47,690/47,835 = .9970.

100,000 deals .9694
1,000,000 .9857
2,500,000 .9970

I think the discrepancy is an indicator of the accuracy of the sample size.

Interestingly, in a heads-up situation there is a relatively small number of outcomes, 48 choose 4. Unfortunately, Poker Probe doesn't allow the option of playing all possible outcomes here, although it does in some other situations.

[Buzz]

“I think the discrepancy is an indicator of the accuracy of the sample size.”

Mack - I agree the discrepancy is an indicator of the sample size (number of
deals).

My background is in science, thus the "experimental uncertainty"
terminology. Other things being the same in a scientific experiment, the
larger the sample size, the less the experimental uncertainty.

What I meant about the luck is that someone can play garbage and draw out
on you, and maybe do that for a whole evening, but over the long haul the
luck factor will tend to even out. The longer the haul, the more the luck
factor evens out.

It's especially interesting that after 100000 trials (deals), a number that seems like a very large number of deals, there was still a 3% discrcepancy
(which I'm thinking of as related to "luck.") That 3% discrcepancy
somehow seems like a lot, because 100000 deals seems like a lot. Yet maybe 100000 shouldn't seem like such a large nunber when we're
thinking in terms of sample size.

Interesting data. Thanks for gathering it.

Buzz

[DPCondit]

Good job Mack.

I don't think we need to run any scenarios here, or get into any essay contests. From what I can see, our hand/s is/are extremely powerful all the way. Not only that, but both an ace and a deuce, tends to counterfeit the low draws, lots of scoops, here, nice hand/s.

Thanks for sharing your research,
Don