Standard Deviation for Multihand Video Poker

https://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"?

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/

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).

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

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?

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.

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.

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

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

Nice first two posts James! May your screen name RIP.

It was Rockford Files, Cannon, Kolchak-the Night Stalker, the show with Dennis Weaver as a marshall, and some other rotating hour long show every thursday in the early to mid seventies. What was the other show and the name of Dennis Weaver's?

Quote: elvis

It was Rockford Files, Cannon, Kolchak-the Night Stalker, the show with Dennis Weaver as a marshall, and some other rotating hour long show every thursday in the early to mid seventies. What was the other show and the name of Dennis Weaver's?

The Dennis Weaver show was McCloud. However if I recall it rotated with McMillan and Wife and some other show. Rockford was not part of that rotation. (Google was not consulted so I could be wrong)

I have heard of The Rockford Files. Never saw it but heard of it. But these others, Cannon, Kolchak, McCloud, McMillion and wife??? Never even heard of them. Rotating shows on a Thursday night seems a strange concept. Was that a bi-product of having two TV channels at the time? I don't know how you people survived. lol I need my 800 channels. (and usually still can't find anything worth watching).

I only know James Garner from two movies, The space cowboy movie with Tommy Lee Jones and Clint Eastwood and one of my alltime favorite movies with sally Field, Murphy's Romance. (I am sure it was a bust at box office). I think I'll look up his career, maybe there is something I am forgetting. Seems like I know him from more than just those two movies. At any rate, RIP, James Garner.

Quote: kewlj

I only know James Gardner from two movies, The space cowboy movie with Tommy Lee Jones and Clint Eastwood and one of my alltime favorite movies with sally Field, Murphy's Romance. (I am sure it was a bust at box office). I think I'll look up his career, maybe there is something I am forgetting. Seems like I know him from more than just those two movies. At any rate, RIP, James Gardner.

I really liked Maverick with James Garner, Mel Gibson, and Jodie Foster.

Quote: Wizard

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.

I would recommend replacing most of this page with two types of charts.

The first chart would tell me what happens to variance if I fix the denomination and I add hands one at a time. This could be effectively communicated in a graph as well.

The second chart would tell me, if I fix the total amount bet on all hands but change the denom, then what happens to variance. The limiting cases of what happens when the number of hands approaches infinity and one in this setup can be very instructive for people trying to learn about variance.

And of course when I say variance above, I recognize that SD is the relevant number to actually put in any tables.

Quote: JamesGarnerOK

...

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.

Oh, thanks, that makes sense.