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.