new sims A3QQ vs AA3Q - nut high stacked board

[DPCondit]

Omaha Hold'em hi/lo 8/b, 100000 combinations tested.

Hand 1:
AcQdQs3c


Board: 6c7c8s Kc Jh

Hand | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
------+------+------+------+------+------+------+------+------+------+
High | 67777| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Draw | 20193| 4891 | 4670 | 4755 | 4739 | 4793 | 4640 | 4787 | 4734 |
Lose | 0 | 90128| 90409| 90242| 90175| 90267| 90360| 90144| 90347|
Scoop | 32223| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Low | 0 | 4981 | 4921 | 5003 | 5086 | 4940 | 5000 | 5069 | 4919 |
------+------+------+------+------+------+------+------+------+------+
Win% |70.98%| 3.66%| 3.58%| 3.63%| 3.67%| 3.61%| 3.61%| 3.67%| 3.58%|
------+------+------+------+------+------+------+------+------+------+

Omaha Hold'em hi/lo 8/b, 100000 combinations tested.

Hand 1:
AsAcQd3c


Board: 6c7c8s Kc Qh

Hand | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
------+------+------+------+------+------+------+------+------+------+
High | 53764| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Draw | 18061| 3102 | 3080 | 3188 | 3203 | 3090 | 3075 | 3091 | 3199 |
Lose | 0 | 92729| 92782| 92726| 92722| 92908| 92776| 92934| 92650|
Scoop | 46236| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Low | 0 | 4169 | 4138 | 4086 | 4075 | 4002 | 4149 | 3975 | 4151 |
------+------+------+------+------+------+------+------+------+------+
Win% |77.55%| 2.84%| 2.82%| 2.82%| 2.82%| 2.75%| 2.82%| 2.74%| 2.85%|
------+------+------+------+------+------+------+------+------+------+

Ok, both hands are stacked so that they win high every time. Where it says "win", this is ev %. Because all hands have 50% of the equity (high) locked up as a given, and there is ALWAYS a made low, we simply subtract 50% from the win ev%, then multiply by 2, this gives the low equity. Because all low wins register as a scoop (with the high locked up), and highs only register as a scoop if and only if the low side wins without splitting, all scoops represent low wins. All draws (split pots) represent low splits, because the high, nut flush, cannot split.

Does this make sense so far?

Therefore, by this reasoning, hand 1 wins low (no split)about 32% of the time, and hand 2 wins low (no split) about 46% of the time. Hand 1 has about 42% of the low equity, and hand 2 has about 55% of the low equity.

Don

[DPCondit]

To verify these results, I am running the same hands as a showdown simulation on Turbo O/8, just going to look at the low hand distribution to see if it matches.

Ok, simulations done, here are the results:
50,000 hands each

A3QQ won low 31.8%, tied low 20%
A2 won low 48.3% of the time

AA3Q won low 46.3%, tied low 18%
A2 won low 35.6% of the time

Looks pretty close, I'd say these are all pretty good results.

Don

[Phat Mack]

So another advantage of AA in this game is that it deprives an opponent of having the ace to make low, and the advantage is substantial. I wonder if hands with A22, when A2 makes nut low, are stronger than I have imagined them to be, and by the same margin.

[DPCondit]

I prefer AA2 for obvious reasons. However, having an extra Ace or Deuce both give you protection (to a degree) against splits, and also a certain amount of counterfeit protection (less cards in the deck to counterfeit you). In actual play(not showdown), the AA2 deuce would give you more protection in the sense that there are a lot more hands that your opponents will go forward with containing an ace, as opposed to hands containing a deuce. But in straight showdown, AA2 should have the same LOW equity as A22.

Good luck,
Don

[Buzz]

Hi Don - The way you set up those simulations is very clever! I’m impressed!

"Does this make sense so far?"

Here’s how I read your simulation data in terms of deducing how often you run into at least one opponent holding at least one ace and at least one deuce:

32223 - AcQdQs3c won for high and won for low
20193 - AcQdQs3c won for high and tied for low (a quarter or a sixth)
47584 - AcQdQs3c won for high and lost to A2 for low

46236 - AsAcQd3c won for high and won for low
18061 - AsAcQd3c won for high and tied for low (a quarter or a sixth)
35703 - AsAcQd3c won for high and lost to A2 for low

Therefore, according to your simulation data, when the board is 6c7c8sKcJh,
an opponent has A2XX 47.584% of the time when you hold AcQdQs3c, while
an opponent has A2XX 35.703% of the time when you hold AsAcQd3c.

Beautiful work, Don.

I had previously calculated that
an opponent has A2XX 40.6% of the time when you hold AcQdQs3c, and
an opponent has A2XX 29.3% of the time when you hold AsAcQd3c.

The discrepancy between values based on your simulations and values based on my calcluations, I suspect, is because you took five specific cards out of the deck to make a board - and none of these cards was an ace or a deuce. Therefore the pack used to make opponent’s hands for your simulations had a higher ratio of aces and deuces to the other cards than the pack I visualized when doing the (tedious) calculations. Therefore your simulated percentages for encountering acey-deucy should have been higher than those I calculated. And they were.

In other words, your simulated values for encountering at least one opponent with at least one ace and at least one deuce are river values, values after you know where nine cards are. My calculated values are pre-flop values, values when you only know where four cards are. The ratio you’ve generated in your simulations is actually 47584/35703 or 2.67/2, rather than 3/2. (For comparison, the ratio I calculated was actually 2.77/2, rather than 3/2).

It’s no big deal. In either event, you should clearly see that the likelihood of running into A2XX when there are three aces available is not exactly 3/2 times the likelihood of running into A2XX when there are only two aces available. (I think that’s how we got started on this).

Thank you again for the simulations. Very helpful.

Buzz

[DPCondit]

Yes, obviously my sims are river values, but there was no way around that to properly separate the high and low values in this particular simulation. One thing this does not account for is the number of times you win low when a deuce hits, or the reduced number of counterfeits by an ace when you hold AA3 vs A3, so there will be some improvement there that is not measured simply by looking at the number of opponents likely to hold A2. I may work on that problem later.

Don

[DPCondit]

Oh wait, you did a calculation for no ace on board. However, about 20% prob. for no ace with AA3Q and 10% for A3QQ, these numbers are way too low. You are not going to get counterfeited by an ace 80% of the time with the first hand, and 90% for the second hand, so these numbers cannot be correct.

How about for AA3Q 1-(46/48*45/47*44/46*43/45*42/44), or about 20% prob. of being counterfeited by an ace, or
1-(43/48*42/47*41/46*40/45*39/44) or about a 44% chance of being counterfeited by an ace or three.

For A3QQ 1-(45/48*44/47*43/46*42/45*41/44), about a 29% prob. of being counterfeit by an ace, and 1-(42/48*41/47*40/46*39/45*38/44) or about a 51% chance of being counterfeited by an ace or three.

A lot of rounding there, so numbers may be slightly off.

Those probs of being counterfeited still seem eerily high, but are actually just preflop odds, once you flop a low draw (or a made low), your odds of being counterfeited are much much smaller.

Probability of being counterfeit after the flop, and after the turn:
AA3Q 1-(40/45*39/44), about a 21% chance after the flop, and 1-(39/44) or about an 11% chance after the turn.

A3QQ 1-(39/45*38/44), about a 25% chance after the flop, and about a 14% chance after the turn 1-(38/44)

So the AA3 improves even more over the A3 including counterfeit protection, although not by a huge amount.

Don

[Buzz]

“Oh wait, you did a calculation for no ace on board.”

Right. I just figured the probability of encountering at least one opponent in a nine handed game holding at least one ace and at least one deuce when you have A3QQ and also when you have A3QQ. The board has nothing to do with what I calculated.

“However, about 20% prob. for no ace with AA3Q and 10% for A3QQ, these numbers are way too low. You are not going to get counterfeited by an ace 80% of the time with the first hand, and 90% for the second hand, so these numbers cannot be correct.”

Not sure what you mean here. I don’t think we’re on the same wavelength.

“How about for AA3Q 1-(46/48*45/47*44/46*43/45*42/44), or about 20% prob. of being counterfeited by an ace, or
1-(43/48*42/47*41/46*40/45*39/44) or about a 44% chance of being counterfeited by an ace or three.

For A3QQ 1-(45/48*44/47*43/46*42/45*41/44), about a 29% prob. of being counterfeit by an ace, and 1-(42/48*41/47*40/46*39/45*38/44) or about a 51% chance of being counterfeited by an ace or three.”

O.K. the set ups look right. You can still win or get a share of the pot with these hands, as when the board on the river is A-2-3-4-5.

“A lot of rounding there, so numbers may be slightly off.”

No matter.

“Those probs of being counterfeited still seem eerily high, but are actually just preflop odds, once you flop a low draw (or a made low), your odds of being counterfeited are much much smaller.”

Yes.

“Probability of being counterfeit after the flop, and after the turn:
AA3Q 1-(40/45*39/44), about a 21% chance after the flop, and 1-(39/44) or about an 11% chance after the turn.

A3QQ 1-(39/45*38/44), about a 25% chance after the flop, and about a 14% chance after the turn 1-(38/44)”

the set-up looks O.K.

“So the AA3 improves even more over the A3 including counterfeit protection, although not by a huge amount.”

Looks right to me.

regards,

Buzz

[DPCondit]

Oh, sorry, I must have misread your post from the other thread.

Don

[Phat Mack]

One thing this does not account for is the number of times you win low when a deuce hits, or the reduced number of counterfeits by an ace when you hold AA3 vs A3, so there will be some improvement there that is not measured simply by looking at the number of opponents likely to hold A2. I may work on that problem later.

I've got a copy of Mike Caro's Poker Probe on an old 386 box. I remembered that it gives more information about hi-lo hands than Poker Calculator, so I thought I would check it out. I set up AAQ3r and AQQ3r against eight random opponents. I set two flop cards to 5s4h, thinking that if the hands made a 5-high, that would indicate a 2 hit. A 6-, 7- or 8- high low would indicate that a 2 didn't hit the board. A win with a 6-, 7- or 8- high would indicate that the hands were not against an A2, a loss would indicate that they were. The increase in win% when going from one to two held aces seems correct. Both AAQ3 and AQQ3 seem to have the same number of ties for low when making 6-, 7- and 8-high lows. Strange. I would expect the AAQ3 to have fewer ties.

<font class="small">Code:</font><hr /><pre>
Results from POKER PROBE version 1.00b . . .


----------------------------------------
| ok-8 to qualify | random |
| ^ ^ | ^ ---------------------------------------
| GAME: hi-lo Omaha | DEALS: 100,000 analyzed: 100,000 |
| Board: 5s 4h | update: 10,000 per hour: 151,369 |
|----------------------------------------------------------------------------|
| Hand | Sweeps |Hi Shares|Lo Shares| Pct | Odds-1 |QPR+|
|------------------------|---------|---------|---------|-----|----------|----|
|#1: As Qh Qd 3c | 4,483| 7,499| 35,087|21.29| 3.70| 214|
|#2: | 1,774| 11,645| 8,073| 9.86| 9.14| 88|
|#3: | 1,807| 11,642| 8,067| 9.85| 9.15| 86|
|#4: | 1,753| 11,633| 8,055| 9.84| 9.16| 87|
|#5: | 1,846| 11,573| 8,186| 9.88| 9.12| 87|
|#6: | 1,749| 11,418| 8,032| 9.72| 9.28| 85|
|#7: | 1,846| 11,538| 8,209| 9.87| 9.13| 87|
|#8: | 1,762| 11,527| 8,144| 9.84| 9.17| 86|
|#9: | 1,801| 11,526| 7,982| 9.75| 9.25| 87|
| | | | | | | |
| Average statistics &gt;| 2,091| 9,763| 7,838|11.11| 8.00-1| 100|
|-----------------------------------------------------------------------|----|
| Discards: |Log+|
------------------------------------------------------------------------------
- ^ ^ ^ P O K E R P R O B E (C) M I K E C A R O 1 9 9 O ^ ^ ^ -
- - - - - - - - - - - - - - - - - - - - - - - - - -

GAME: hi-lo Omaha (ok-8 to qualify) ITEMIZED chart
Board: 5s 4h

Hand | Sweeps |Hi Shares|Lo Shares| Pct | Odds-1 |QPR+
------------------------|---------|---------|---------|-----|----------|----
#1: As Qh Qd 3c | 4,483| 7,499| 35,087|21.29| 3.70| 214
----------------------------------------------------------------------------
|HIGH | Wins | Losses | Ties |WL%|| LOW | Wins | Losses | Ties |WL%|
|-----|--------|--------|--------|---||-----|--------|--------|--------|---|
|5 x A| | | | || 5-hi| 12145| 0| 12354|100|
|St/Fl| 0| 0| 0| --|| 6-hi| 4905| 8657| 4161| 36|
| 4 x| 310| 0| 0|100|| 7-hi| 3626| 8246| 2310| 31|
| Full| 3876| 2011| 14| 66|| 8-hi| 2645| 6898| 1584| 28|
|Flush| 0| 0| 0| --|| 9-hi| | | | |
| Strt| 1678| 18389| 1612| 8||1O-hi| | | | |
| 3 x| 778| 6321| 9| 11|| J-hi| | | | |
| 2 pr| 106| 31562| 13| 0|| Q-hi| | | | |
| AA| 0| 5652| 0| 0|| K-hi| | | | |
| KK| 0| 0| 0| --|| A-hi| | | | |
| QQ| 0| 17180| 0| 0|| Pair| | | | |
| JJ| 0| 0| 0| --|| 2 pr| | | | |
|1O-2s| 0| 0| 0| --|| 3 x| | | | |
|No pr| 0| 0| 0| --||Worst| | | | |
----------------------------------------------------------------------------
Discards:
</pre><hr />
<font class="small">Code:</font><hr /><pre> Results from POKER PROBE version 1.00b . . .


________________________________________
_ ok-8 to qualify _ random _
_ ^ ^ _ ^ _______________________________________
_ GAME: hi-lo Omaha _ DEALS: 100,000 analyzed: 100,000 _
_ Board: 5s 4h _ update: 10,000 per hour: 147,650 _
__________________________________________________ ____________________________
_ Hand _ Sweeps _Hi Shares_Lo Shares_ Pct _ Odds-1 _QPR?_
__________________________________________________ ____________________________
_#1: As Ah Qd 3c _ 2,809_ 8,331_ 41,960_25.15_ 2.98_ 296_
_#2: _ 2,115_ 11,604_ 7,252_ 9.43_ 9.61_ 80_
_#3: _ 2,095_ 11,504_ 7,116_ 9.31_ 9.74_ 78_
_#4: _ 2,169_ 11,532_ 7,388_ 9.46_ 9.57_ 79_
_#5: _ 2,017_ 11,317_ 7,253_ 9.29_ 9.77_ 78_
_#6: _ 2,119_ 11,432_ 7,188_ 9.31_ 9.74_ 78_
_#7: _ 2,044_ 11,347_ 7,256_ 9.30_ 9.75_ 78_
_#8: _ 2,020_ 11,346_ 7,177_ 9.26_ 9.80_ 78_
_#9: _ 2,165_ 11,586_ 7,258_ 9.42_ 9.61_ 79_
_ _ _ _ _ _ _ _
_ Average statistics &gt;_ 2,173_ 9,818_ 8,539_11.11_ 8.00-1_ 100_
__________________________________________________ ____________________________
_ Discards: _Log?_
__________________________________________________ ____________________________
_ ^ ^ ^ P O K E R P R O B E (C) M I K E C A R O 1 9 9 O ^ ^ ^ _
- - - - - - - - - - - - - - - - - - - - - - - - - -

GAME: hi-lo Omaha (ok-8 to qualify) ITEMIZED chart
Board: 5s 4h

Hand _ Sweeps _Hi Shares_Lo Shares_ Pct _ Odds-1 _QPR?
__________________________________________________ __________________________
#1: As Ah Qd 3c _ 2,809_ 8,331_ 41,960_25.15_ 2.98_ 296
__________________________________________________ __________________________
_HIGH _ Wins _ Losses _ Ties _WL%__ LOW _ Wins _ Losses _ Ties _WL%_
__________________________________________________ __________________________
_5 x A_ _ _ _ __ 5-hi_ 15113_ 0_ 9294_100_
_St/Fl_ 0_ 0_ 0_ --__ 6-hi_ 7600_ 6893_ 4109_ 52_
_ 4 x_ 288_ 0_ 0_100__ 7-hi_ 5696_ 7044_ 2477_ 45_
_ Full_ 4003_ 1718_ 32_ 70__ 8-hi_ 4106_ 5945_ 1593_ 41_
_Flush_ 0_ 0_ 0_ --__ 9-hi_ _ _ _ _
_ Strt_ 2114_ 18329_ 1173_ 10__1O-hi_ _ _ _ _
_ 3 x_ 1074_ 5845_ 14_ 16__ J-hi_ _ _ _ _
_ 2 pr_ 267_ 31648_ 17_ 1__ Q-hi_ _ _ _ _
_ AA_ 2_ 23077_ 0_ 0__ K-hi_ _ _ _ _
_ KK_ 0_ 0_ 0_ --__ A-hi_ _ _ _ _
_ QQ_ 0_ 0_ 0_ --__ Pair_ _ _ _ _
_ JJ_ 0_ 0_ 0_ --__ 2 pr_ _ _ _ _
_1O-2s_ 0_ 0_ 0_ --__ 3 x_ _ _ _ _
_No pr_ 0_ 0_ 0_ --__Worst_ _ _ _ _
__________________________________________________ __________________________
Discards:
</pre><hr />