Why Two Dimes Data Is Wrong (Continued...)

[Mendacious]

I played this hand recently on Prima, which really irritated me and I ran it on 2 dimes and the percentages only aggravated me further when I analyzed it as a 3 way hand.

** Game ID 579278870 starting - 2005-11-15 19:06:04
** Brave Starr [Omaha H/L] (0.50|1.00 Pot Limit - Cash Game) Real Money

- sliwezia sitting in seat 1 with $114.53
- hazan-stepha sitting in seat 2 with $147.06
- Aifix sitting in seat 3 with $58.31
- lagiro sitting in seat 4 with $94.01
- TOLDYOUSO sitting in seat 5 with $66.99 [Sitting out]
- seminole2005 sitting in seat 6 with $98.97
- carnesio sitting in seat 7 with $19.00
- Mendacious sitting in seat 8 with $105.45 [Dealer]
- open_cobra sitting in seat 9 with $62.65
- udomyex sitting in seat 10 with $96.00

open_cobra posted the small blind - $0.50
udomyex posted the big blind - $1.00
carnesio posted to play - $1.00
** Dealing card to Mendacious: 3[img]/images/graemlins/club.gif[/img] A[img]/images/graemlins/club.gif[/img] 8[img]/images/graemlins/heart.gif[/img] 4[img]/images/graemlins/spade.gif[/img]
sliwezia folded
hazan-stepha called - $1.00
Aifix folded
lagiro called - $1.00
seminole2005 folded
carnesio checked
Mendacious called - $1.00
open_cobra called - $1.00
udomyex bet - $4.00
hazan-stepha called - $4.00
lagiro folded
carnesio called - $4.00
Mendacious called - $4.00
open_cobra called - $4.00

** Dealing the flop: Q[img]/images/graemlins/club.gif[/img] 2[img]/images/graemlins/club.gif[/img] 6[img]/images/graemlins/heart.gif[/img]
open_cobra checked
udomyex bet - $21.00
hazan-stepha folded
carnesio went all-in - $16.00
Mendacious raised - $100.00
open_cobra folded
udomyex went all-in - $72.00
carnesio shows: 2[img]/images/graemlins/heart.gif[/img] 5[img]/images/graemlins/heart.gif[/img] 6[img]/images/graemlins/spade.gif[/img] 6[img]/images/graemlins/diamond.gif[/img]
udomyex shows: A[img]/images/graemlins/heart.gif[/img] Q[img]/images/graemlins/spade.gif[/img] 2[img]/images/graemlins/spade.gif[/img] 8[img]/images/graemlins/diamond.gif[/img]

** Dealing the turn: 9[img]/images/graemlins/club.gif[/img]

** Dealing the river: 2[img]/images/graemlins/diamond.gif[/img]
carnesio wins $66.00 from the main pot
udomyex wins $154.00 from side pot 1

UDOMYEX called a huge raise all in on a hand where I came over him with was over 57% on the flop against 2 opponents, AND UDO's PE was a paltry .083.

pokenum -o8 ac 3c 8h 4s - 2h 5h 6s 6d - ah qs 2s 8d -- qc 2c 6h
Omaha Hi/Low 8-or-better: 666 enumerated boards containing Qc 2c 6h
cards scoop HIwin HIlos HItie LOwin LOlos LOtie EV
4[img]/images/graemlins/spade.gif[/img] A[img]/images/graemlins/club.gif[/img] 3[img]/images/graemlins/club.gif[/img] 8[img]/images/graemlins/heart.gif[/img] 251 254 412 0 467 0 9 0.576
6[img]/images/graemlins/spade.gif[/img] 6[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/heart.gif[/img] 2[img]/images/graemlins/heart.gif[/img] 113 341 325 0 0 77 0 0.341
Q[img]/images/graemlins/spade.gif[/img] 2[img]/images/graemlins/spade.gif[/img] 8[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/heart.gif[/img] 35 71 595 0 0 381 9 0.083


HOWEVER,

If you view the hand as a heads up between UDO and I (which my raise really accomplished because of the short stack, UDO may be a 62/38 dog to me, which isn't nearly so bad, AND his call gets him some equity in a decent sized main pot.

cards scoop HIwin HIlos HItie LOwin LOlos LOtie EV
4[img]/images/graemlins/spade.gif[/img] A[img]/images/graemlins/club.gif[/img] 3[img]/images/graemlins/club.gif[/img] 8[img]/images/graemlins/heart.gif[/img] 366 369 451 0 558 0 9 0.617
Q[img]/images/graemlins/spade.gif[/img] 2[img]/images/graemlins/spade.gif[/img] 8[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/heart.gif[/img] 173 451 369 0 0 460 9 0.383


So how you do the 2 dimes analysis is obviously very relevant to assessing the true odds the players had relative to various portions of the pot, not just as a whole.

The EV and odds are the same. The only difference is that the low draw hand will have lower variance. Write out the calculation for mean and standard deviation if you can't see this.

BTW Buzz since you [censored] this one up so badly, here's another gem from the archives. Is this the same Buzz getting pwned by Ray Zee?

http://www.twoplustwo.com/digests/ot...9_msg.html#532
[ QUOTE ]
Buzz (to Ray Zee): "However, I think you're missing the boat here. When there is an eight or better for low qualifier, I don't think low hands are better than high hands."

[/ QUOTE ]

P.S. Your posts are always great, don't often get a chance to make fun of you [img]/images/graemlins/smile.gif[/img]

[ QUOTE ]
[ QUOTE ]
FatBallz (Wintermute?)

Consider running this example 100 times. In either case, you risk $2500. In either case, you end up with $5,000 on average. The scenarios are equivalent in equity and risk.


[/ QUOTE ]

They may have the same EV, but the variance (risk) will be different.

[/ QUOTE ]

Variance and risk are definitely not synonyms. The risk is the same in both cases: $2500. The variance is not.

[Mendacious]

Are you agreeing that the variance differs for each situation? If so, can you explain why if the EV is the same and the risk is the same that the variance is different?

[ QUOTE ]
Are you agreeing that the variance differs for each situation? If so, can you explain why if the EV is the same and the risk is the same that the variance is different?

[/ QUOTE ]

The risk and EV are clearly the same, as my example shows (risk = $2500, average return = $5000).

The variances do differ. To get an idea of why this is, start by considering the distributions of outcomes in each scenario.

In the draw-to-a-low scenario, if you hit your low all 100 times, you will end up with $10k. If you hit your low zero times, you will end up with $0. The distribution is symmetric around the $5k mark.

In the draw-to-a-scoop scenario, if you win every time, you will come away with $20k; if you lose every time, you come away with $0. The distribution is asymmetric, with a fatter edge near the lower end in exchange for a long skinny tail to the upper end. It's intuitive to see that these two distributions will have different variances... it may not be intuitive, but the variance of the scoop distribution is actually higher.

To see this mathematically, model each scenario with the binomial distribution (which captures the sum of n random draws of probability p). The variance, by definition, is n*p*(1-p). So, for the low-only situation, the variance is:

1/2*100*0.5*0.5 = 12.5 pots

where we throw in the extra factor of 1/2 because we only get half of the pot with a success, which is our unit of measure.

For the scoop situation, the variance is:

100*.25*.75 = 18.75 pots

So, drawing to a low here is the better option for the risk-averse. On the other hand, if you want to win $20k one in 4^100 times, you should prefer drawing to the scoop.

[beset7]

[ QUOTE ]
The EV and odds are the same. The only difference is that the low draw hand will have lower variance. Write out the calculation for mean and standard deviation if you can't see this.

BTW Buzz since you [censored] this one up so badly, here's another gem from the archives. Is this the same Buzz getting pwned by Ray Zee?

http://www.twoplustwo.com/digests/ot...9_msg.html#532
[ QUOTE ]
Buzz (to Ray Zee): "However, I think you're missing the boat here. When there is an eight or better for low qualifier, I don't think low hands are better than high hands."

[/ QUOTE ]

P.S. Your posts are always great, don't often get a chance to make fun of you [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

That's a pretty good thread thanks for digging that up.

[Buzz]

O.K. let’s revisit this topic (which is honestly not very well settled in my own mind). Something seems not quite right about equating winning half a pot twice to scooping once.

I agree that winning half a pot twice is equivalent to scooping once and losing once. But.... well, let me try to explain with an example from a passive $4/$8 game.

Let’s say your opponents will contribute a total of $80 to the pot and it will cost you $24 to play a hand all the way to the showdown.
• When you scoop, you will be awarded all $104 in the pot of which $24 is your own investment.
• When you split the pot, you will be awarded $52, of which $24 is your own investment.

Each time you split such a pot, you only actually win $28. When you win half such a pot twice, what you actually win is $56 (and you get your own investment back). If you go home after splitting two such pots and losing none, you’ll be $56 richer.

On the other hand, when you scoop once, you actually win all $80 your opponents have contributed to the pot (and you get your own investment back). If you go home after scooping one such pot and losing none, you’ll be $80 richer.

You have to scoop one pot and also lose another to end up with the same number of dollars as splitting two pots and losing none.

If instead, you scoop one pot and then get out of the next (instead of contributing and losing $24), in that case, scooping the one pot is better than splitting two pots.
*
I realize that winning half the (same sized) pot twice is the same as scooping once and losing once. (You get your own $24 back plus winning a total of $80 invested by your opponents).

But if you scoop once and don’t lose at all, you’ll end up with more money than if you win half the pot twice and don’t lose at all.
*
Let's consider it from the standpoint of odds after the turn.

When you’re looking at your hand and the board after the turn,
• if it will cost you $16 more to continue to see the showdown if you like the river card,
• if there will be $104 in the pot at the showdown if you continue (including the $8 you’ve already invested plus the $16 it will hencefore cost you to see the showdown), and
• if seeing the river will only cost you $8 if you miss (instead of $16 because you will fold if you miss) then at that point in the hand,
• then you are getting 88 to 8 implied pot odds to win the whole pot.

(The pot at that point contains $8 of what was formerly your own money plus at the showdown it will contain $80 of your opponent’s money).

If you only win half the pot, you’ll be awarded $52, of which $16 is what you will henceforth contribute if you call the turn bet.

When you’re only playing for half the pot, if you continue, you’ll be playing to win $36. You’ll be investing $8 to possibly win $36. You’re getting 36 to 8 implied odds for half the pot.

So when you’re playing to win this whole pot, on the third betting round your implied pot odds are 11 to 1, and when you’re playing to win half this pot,
on the third betting round your implied pot odds are 4.5 to 1.

I'm not done thinking this out yet, and I'm not sure I ever will be.

But although I agree that scooping (the same sized pot) 2000 times in 10000 plays to the showdown is the same as winning half the (same sized) pot 4000 times in 10000 plays, there still seems something amiss in equating two half wins to one scoop.

If there is a "blatant" mistake here, I'm oblivious to it (although I'm sure I'm very capable of making a mistake).

It certainly is not my intention to lead anyone astray. I am, as always, looking for the truth. Alas, sometimes I make mistakes. I will continue to correct them if I see them. (In this regard, the record speaks for itself).

Buzz

Buzz, whenever I said that your mistakes in this thread were blatant, I mispoke. Truth be told, it's really easy for anyone to confuse themself by drawing up complicated examples. But really the best way to go about this problem is to express it as simply & generally as possible and then see that this covers all possibilities. Running example after example will just allow one to make an error in logic somewhere that will lead you to believe you've found a counterexample, when in fact one has honestly just made an error somewhere.

An anecdote--when I first got into gambling (period) about 5 years ago, I read up on the Martingale system, wrote my own Matlab blackjack simulator, and convinced myself that I could change the EV of blackjack by modifying my bet size, essentially doubling bets when I win repeatedly (in theory to accentuate winning streaks and downplay losing streaks). I even went to lengths to justify this falsehood in my mind by repeating something I heard somewhere about how the players' ability to change their bet size is how they can beat blackjack (not realizing at the time that this referred to card-counting). I mentioned my theory to my brother, and in 2 seconds he told me I was full of it. He explained that no matter what I did with my bet size, I was still going to have the same EV in the game, just the amount wagered would vary.

Well, I got pissed and spent about 3 hours writing down series after series of possible outcomes, trying to demonstrate to myself that my system would yield a winning result in a -EV game. In the end, I realized that I had indeed just made a mistake somewhere, and that when I thought about it my brother's direct, simply way, I saw that this clearly captured all cases, and that I was wrong. (It turns out that modifying bet size affects variance--you can change the shape of the distribution of outcomes, putting greater weight in small wins at the expense of risking an unlikely catastrophe... martingale in a nutshell.)


Now the thing is, when you draw up a very complicated example, it's very difficult to pinpoint each error in logic. I suspect that in your example above, the problem may lie in the statement:

[ QUOTE ]
But if you scoop once and don’t lose at all, you’ll end up with more money than if you win half the pot twice and don’t lose at all.

[/ QUOTE ]

Somehow I get the feeling that this statement implies that you are changing the odds... but I have to admit, I haven't thought about this carefully enough to be sure this is where things have gone awry here. I just know that something in this line of reasoning is out of whack, because a simple, general analysis shows that these two draws have identical EV & risk, just as my brother *knew* with certainty that the EV of blackjack is axiomatically fixed (barring card-counting, etc).


Finally, I apologize for dragging this thing out in the other thread. It gave me satisfaction to point out your mistake then and before because I have been on the other end of it, which is immature. I promise I won't bring this thing up again.

[M.B.E.]

[ QUOTE ]
Let’s say your opponents will contribute a total of $80 to the pot and it will cost you $24 to play a hand all the way to the showdown.
• When you scoop, you will be awarded all $104 in the pot of which $24 is your own investment.
• When you split the pot, you will be awarded $52, of which $24 is your own investment.

Each time you split such a pot, you only actually win $28. When you win half such a pot twice, what you actually win is $56 (and you get your own investment back). If you go home after splitting two such pots and losing none, you’ll be $56 richer.

On the other hand, when you scoop once, you actually win all $80 your opponents have contributed to the pot (and you get your own investment back). If you go home after scooping one such pot and losing none, you’ll be $80 richer.

[/ QUOTE ]

All you've proved in this example is that it's better to have a 1/2 probability of scooping than a 2/3 probability of splitting.

Someone has to put you out of your misery, might as well be me [img]/images/graemlins/laugh.gif[/img]

As you know, poker is about the long run, and you agree the two are equal in the long run. But in the short run, you think the hi draw is somehow better. So let me ask you this:

You have $100 to gamble. Which of the following do you prefer to do, and why?

- A 1 in 10,000 shot at a million
- A 1 in 2 shot at $200

Is either superior? If the $100 was your food money for the week, which would you take? If you were a millionaire, which would you take?

But you get what I'm saying. So let's look at what you're saying:

[ QUOTE ]
You have to scoop one pot and also lose another to end up with the same number of dollars as splitting two pots and losing none.

[/ QUOTE ]
Correct, so the E.V. is equal in this case.

[ QUOTE ]
If instead, you scoop one pot and then get out of the next (instead of contributing and losing $24), in that case, scooping the one pot is better than splitting two pots.

[/ QUOTE ]
Aha! But that will only happen 1/4 of the time. The other 3/4 of the time, you lose your $24 and there's nothing you can do about it. Your intuition is overlooking this fact. I better write this out to make it clearer:


L = low draw (50%) - L makes $28 or loses $24
H = high draw (25%) - H makes $80 or loses $24

Pretend two concurrent games are running. In one a person has a high draw, in another, a low draw. Pretend only one hand is played. These are the ways it can turn out:

L wins, H wins : H +80, L +28, H nets $52 more (this will happen 1/8 of the time)
L loses, H wins: H +80, L -24, H nets $104 more (this will happen 1/8 of the time)
L wins, H loses: H -24, L +28, L nets $52 more (this will happen 3/8 of the time)
L loses, H loses: H -24, L -24, they both lose $24. (this will happen 3/8 of the time)

So, only 1/8 of the time H will net twice as much as L, but this is balanced out by the fact that L wins $52 more an extra time than H. I think this is where your intuition failed. When H loses and L wins, L actually wins $52 more, not $28 more.

Your next point is about implied odds:

[ QUOTE ]
if seeing the river will only cost you $8 if you miss (instead of $16 because you will fold if you miss) then at that point in the hand, you are getting 88 to 8 implied pot odds to win the whole pot. then you are getting 88 to 8 implied pot odds to win the whole pot.

[/ QUOTE ]

If you still have money to bet on the river, then of course the high draw is favorable and has better EV. No one is debating this (I hope). However, the original question was about being all in and having no more money to bet or make on the river.