Variance as function of bet size and number of hands

After reading an old post on the WoO website, I think my math might be a bit off and was wondering if the Wizard himself could put me on track by spotting where the following argument breaks down.

1. The variance of single-hand Jacks or Better is 4.417542 ** 2, or about 19.5.
2. The variance of 4-play Jacks or Better is about 19.5 + (4 - 1) * 2 = 25.5. The figure for the covariance came from jazbo's old site.

3a. The variance of a $1 game of Jacks or Better, at full coin, is 19.5 x ($1 / $0.25) = 78, relative to the quarter level.
3b. Alternatively, the variance of the $1 game is 19.5 + 3 x 100% x 19.5 = 78, since a $1 game can be considered as 4 hands at the quarter level, with 100% covariance.

4. The variance of 4-play Jacks or Better at the 50-cent level is 25.5 x 2 = 51, relative to the quarter level.

5. It is less volatile, therefore, to play 4-play Jacks or Better at the 50-cent level instead the single-hand $1 game since 51 is less than 78.

There's a mistake somewhere, but where? Here's what compelled me to ask in the first place:

Quote:

Thanks for the answer about risk of ruin on power poker vs. single play. Now for a follow up... which has more volatility, $1 jacks or better, or $.50 4-play jacks or better (betting $10 per play instead of $5 per play)?

RAY FROM MAPLE GLEN, PENNSYLVANIA

From my video poker appendix 3 we can see the standard deviation for 1-play jacks or better is 4.417542. The standard deviation for 4-play jacks or better is 5.041215. Keep in mind these figures are per hand and relative to the betting unit. Adjusting for bet size and number of hands the standard deviation of $5 bet in 1-play jacks or better is 11/2*5*4.417542 = 22.08771. The standard deviation of 4 bets of $2.50 in 4-play jacks or better is 41/2*$2.50*5.041215 = 25.20608. So you are better off betting the smaller total amount in 1-play. Interestingly you can double the total amount bet in 4-play and the standard deviation only goes up by 14.12%.

Thanks for helping me get out of this knot!

Variance is measured in squared units. You can think of this as squared bets. The familiar 19.5147 for 9/6 Jacks or Better is therefore in squared bets. Since a dollar is 4 times a quarter, a "dollar squared" is 4^2 = 16 times a "quarter squared."

If this isn't intuitive, think about what would occur if you worked with standard deviations instead.

If when converting from quarters to dollars you multiply the standard deviation by 4 (i.e., 4.4175 x 4) to get the dollar standard deviation of dollars relative to quarters, I think you can see that to get the variance of dollars relative to quarters, you would have to multiply by 16.

You can't have both:

4.4175 x 4 = dollar standard deviation relative to quarters and

19.5147 x 4 = dollar variance relative to quarters, since those two expressions are unequal. This 2nd line needs to by 4.4175^2 x 4^2 to get an equivalent expression.

Quote: drrock

Variance is measured in squared units. You can think of this as squared bets. The familiar 19.5147 for 9/6 Jacks or Better is therefore in squared bets. Since a dollar is 4 times a quarter, a "dollar squared" is 4^2 = 16 times a "quarter squared."

If this isn't intuitive, think about what would occur if you worked with standard deviations instead.

If when converting from quarters to dollars you multiply the standard deviation by 4 (i.e., 4.4175 x 4) to get the dollar standard deviation of dollars relative to quarters, I think you can see that to get the variance of dollars relative to quarters, you would have to multiply by 16.

You can't have both:

4.4175 x 4 = dollar standard deviation relative to quarters and

19.5147 x 4 = dollar variance relative to quarters, since those two expressions are unequal. This 2nd line needs to by 4.4175^2 x 4^2 to get an equivalent expression.

Thanks for your reply. I knew I was off somewhere!

Interesting. Here's an exhibition of what I think I've learned (and some things that seem to follow but seem intuitively quite strange):

1. The variance of the 50-cent game above would be 19.5 x (2 x [quarter bet size])^2 = 4 x 19.5 x [quarter bet size]^2, or 4 times the variance of the quarter game.

2. For N-hand jacks or better, the variance is about 17.5 + 2N, where "2" is the covarance between hands in multihand JoB and "N" is the number of hands.

3a. If one wanted to play multihand nickels, they could play up to N hands according to the following formula:

Quote:

17.5 + 2N = (quarter variance) = 25 x (nickel variance) = 25 x 19.5

N = floor( [25 x 19.5 - 17.5] / 2) = 235 hands, or 235 x 0.25 = $58.75 per spin (if such a machine existed)

...and achieve the same level of volatility/risk as a single-line quarter game!

3a. If one wanted to play multihand quarteres, they could play up to N hands according to the following formula:

Quote:

17.5 + 2N = (dollar variance) = 16 x (quarter variance) = 16 x 19.5

N = floor( [16 x 19.5 - 17.5] / 2) = 147 hands, or 147 x 1.25 = $183.75 per spin (if such a machine existed)

...and achieve the same level of volatility/risk as a single-line dollar game!

Is this the right reasoning?

The n-play variance that you are using is only for 1 of the n hands (in a sense), so in looking for your equivalence, the left-hand side of your equations should be multiplied by n. Then you can solve and get more reasonable answers.

Below I've used numbers with a greater degree of precision to be safe, so I will set up the n-play nickels equation as:

17.5483n + 1.9664n^2 = 19.5147 x 25. Solving gets you about 11.91, so 11-play nickels is a little less variance than single line quarters and 12-play is about right but a tiny bit more. So, it's about $3 per play vs. $1.25 on a quarter single line to get nearly equivalent variance figures.

Similarly, we get 17.5483n + 1.9664n^2 = 19.5147 x 16. This solution is about 8.91, so we have 8-play quarters with less variance than single-line dollars and 9-play quarters with just a little more than single-line dollars. Or about $11.25 for 9-play quarters vs. $5.00 for single-line dollars.

Now I see. My formulas were per-hand variances, and the variance of the N-hand arrangement therefore has to be multiplied by N. It appears I was well outside my risk parameters! Thanks for the much-needed math medicine.