AxiomOfChoice
AxiomOfChoice
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April 21st, 2014 at 12:24:16 PM permalink
http://wizardofodds.com/games/video-poker/appendix/3/

I've been reading through this and I don't quite understand it.

Starting right at the beginning -- the first chart, which represents single-play. What is meant by "variance on deal" and "variance on draw"? Variance of what?

The only thing that I could think of, is, if you have two random variables X and Y, where X is the EV of your hand after the deal, and Y is the difference between the actual payout (post-draw) and X, then "variance on deal" could be Var(X) and "variance on draw" could be Var(Y).

Then X+Y would represent the complete hand -- E(X+Y) would be your expectation for the game and Var(X+Y) would be your variance. The problem here is that X and Y are not uncorrelated, so Var(X+Y) is not equal to Var(X) + Var(Y) -- there is covariance between the two random variables which must be considered.

So, then, is something else meant by "variance on deal" and "variance on draw"?
DRich
DRich
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April 21st, 2014 at 1:17:38 PM permalink
Jazbo did a write up about 15 years ago about multi-play variance and volatility. This is where I first read and understood it.

http://jazbo.com/
Order from chaos
AxiomOfChoice
AxiomOfChoice
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April 21st, 2014 at 1:24:04 PM permalink
Quote: DRich

Jazbo did a write up about 15 years ago about multi-play variance and volatility. This is where I first read and understood it.

http://jazbo.com/



Thanks.

I understand this. I just don't understand what the numbers in the Wizard's columns "variance on deal" and "variance on draw" represent. In the first chart (1-play) he seems to be breaking the variance of the game up into two components, and adding them up. But they can only be added if they are variances of uncorrelated random variables, and I can't figure out what those two things could be (with the requirement that they are uncorrelated).
tringlomane
tringlomane
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April 21st, 2014 at 10:04:22 PM permalink
Yes!!! You saved me from starting another thread!!! hahahaha

I was going to ask this tomorrow, because I was asked by someone else about it, and I can't really say with certainty anything that's not written in App. 3, or jazbo.com

I have always wondered if Var_Deal = (EV_dealt hand - EV_game)^2 for all dealt hands, but I really don't know.

We may have to PM Wiz or JB to bring this thread to their attention and hopefully get the answers I think we both want.

Edit: I am too tired to read this tonight, but here is an example of in calculating multi hand SD for blackjack written by Donald Catlin. I would assume you'd need to do VP the same way? Bleh.

http://catlin.casinocitytimes.com/article/blackjack-variance-37494
RS
RS
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April 22nd, 2014 at 4:21:46 AM permalink
On the topic of covariance, how would one figure out the draw & deal variance? On appendix 3 (link in OP) it has 3 or 4 of them for common games. But how do you figure it out, for say, NSUD or 9/5 JoB?
Wizard
Administrator
Wizard
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April 23rd, 2014 at 11:19:20 AM permalink
You're not to be the first to be confused by that page. It is on my project list to rewrite that page, using variance and covariance terminology instead of the variance on the deal and draw. For now, I don't have time to explain it here.
It's not whether you win or lose; it's whether or not you had a good bet.
JamesGarnerOK
JamesGarnerOK
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July 20th, 2014 at 6:41:22 PM permalink
Let X be the player's total revenue from a single deal in multi-hand video poker.

Let H be the initial hand dealt. (H is a discrete random object from a set of 2,598,960 hands).

Each individual starting hand has attached to it a conditional variance in the player's revenue; this is Var(X | H). It is simply the variance in the revenue received from the probability distribution of ending hands attached to that initial hand, assuming it is played optimally. The "variance on draw" is E[ Var(X | H) ]. This is the average of all of the conditional variances, summed up over all possible initial hands.

Also attached to each starting hand is a conditional expected revenue, E(X | H). This is the mean of the probability distribution of revenues of ending hands attached to this particular starting hand. These conditional EVs vary from initial hand to initial hand. So one can calculate the variance in the list of 2.6 million (or 134459) conditional means. This is the "variance on deal". Mathematically, it is Var[ E(X|H) ].

It is a standard theorem in probability that the unconditional variance in total revenue is the straight sum of these two parts. That is,
Var(X) = E[ Var(X|H) ] + Var[ E(X|H) ].

By splitting up the total variance like this, one can get a feel for how much volatility is coming from differences in EVs among the initial hands dealt, and how much volatility is coming from getting lucky on draw.
Buzzard
Buzzard
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July 20th, 2014 at 7:34:13 PM permalink
Jim Rockford once told Angel that being dead was definitely not OK !
Shed not for her the bitter tear Nor give the heart to vain regret Tis but the casket that lies here, The gem that filled it Sparkles yet
JamesGarnerOK
JamesGarnerOK
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July 20th, 2014 at 9:28:19 PM permalink
Quote: Buzzard

Jim Rockford once told Angel that being dead was definitely not OK !



Yeah, but if you read my obit, you'll see that I grew up in the area around Norman, Oklahoma, and the "OK" refers to that rather than the admittedly poor state of my health.

However, I must say that death does wonders to free up one's spirit to find places such as this. Hell, I'm happier than Maverick right now.

JG
tringlomane
tringlomane
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July 21st, 2014 at 12:40:19 AM permalink
Nice first two posts James! May your screen name RIP.

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