djatc Joined: Jan 15, 2013
• Posts: 4477
November 25th, 2014 at 12:52:30 AM permalink
I'm unable to figure out a formula to determine the right amount of bankroll for some out of the ordinary games (base game + promotion of some sort) and would like some guidance in doing so. I have the standard deviation, variance, and edge, just don't know where to go from there.

Also, how would I figure bankroll for games such as Ultimate X. I'm sure the variance is extremely high.
"Man Babes" #AxelFabulous
mcallister3200 Joined: Dec 29, 2013
• Posts: 3141
November 25th, 2014 at 1:05:36 AM permalink
If you have the advantage and variance I think you have all you need. % advantage divided by variance should give you a very close approximation of your optimal Kelly bet. Adjust Kelly downward to satisfactory risk levels( assumes you have choice of denom at paytable) I usually like 1/4 or 1/3 Kelly so that you don't have to immediately adjust on a downswing to be below 1/2 Kelly.
dwheatley Joined: Nov 16, 2009
• Posts: 1246
November 26th, 2014 at 1:46:52 PM permalink
Try the risk of ruin for investors formula to get a benchmark (loaded from wikipedia):

p(ruin) = ( 2 / (1+u/r) - 1 ) ^ (s/r)

where
r = sqrt(u^2 + sigma^2)
u is your edge (0.5% edge is a u=0.005)
sigma is the st.dev.

Example: u=0.005, sigma = 5 (ballparks for 1-play video poker with small edge from promo).
You bring 1000 units, formula gives
p(ruin) = 44.9%

Keep in mind this formula is a long-term risk of ruin, you just keep playing until you go broke or make all the money.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
Mooseton Joined: Sep 6, 2010
• Posts: 608
November 26th, 2014 at 7:05:40 PM permalink
I'm confused about the sigma part of the equation. I looked it up and can't seem to figure out how you figure out the sigma number. How did you get 5 for single hand vp? Is it like that for all single hand vp games? What about multi hand games?
\$1700, 18, 19, 1920, 40, 60,... :/ Thx 'Do it again'. I'll try
RS Joined: Feb 11, 2014
• Posts: 8623
November 27th, 2014 at 2:37:18 PM permalink
If it's a grind play where you have an edge on each bet, look at what the Wizard wrote on his section on Kelly criterion in WOO.

If it's a coin-in play with a necessary coin in goal....well, I was actually going to post the same question the other day but got side tracked.

If it's something else...I got no idea what you're asking. :(
djatc Joined: Jan 15, 2013
• Posts: 4477
November 27th, 2014 at 11:11:26 PM permalink
Quote: mcallister3200

If you have the advantage and variance I think you have all you need. % advantage divided by variance should give you a very close approximation of your optimal Kelly bet. Adjust Kelly downward to satisfactory risk levels( assumes you have choice of denom at paytable) I usually like 1/4 or 1/3 Kelly so that you don't have to immediately adjust on a downswing to be below 1/2 Kelly.

Ok so I'm bad at math but bear with me:

Variance: 33.9319

I got 0.001544. Do you multiply this by \$1.25 (it's a quarter machine)?
"Man Babes" #AxelFabulous
RS Joined: Feb 11, 2014
• Posts: 8623
November 28th, 2014 at 4:35:21 AM permalink
.001544 * Bankroll = how much you should bet at full Kelly.

Of course, you can't bet however much you want since there's only one bet (\$1.25) that you can make. But if you could, then that's how much you'd bet.

Note: maybe it's 0.001544/Bankroll (not 0.001544 * bankroll). I'm tired.

Edit:

Remember that the "coin in" play has different variance than the +EV promotion/bounceback/freeplay/whatever.

ie: You could play some Double STP / Ultimate X hybrid (high variance), and be given straight up cash (zero variance).

Or you could play some 0 variance game for the coin-in (ie: \$1 on every number on Roulette, straight up, where you lose \$2 every spin).....but then you have some high-variance game to redeem the free-play on.

But likely somewhere in between.

But note you can't just add +EV stuff to the -EV stuff at the same variance (EV is additive, variance is not additive....variance works in weird-a** ways).
RS Joined: Feb 11, 2014
• Posts: 8623
November 28th, 2014 at 6:14:30 AM permalink
Quote: djatc

Ok so I'm bad at math but bear with me:

Variance: 33.9319

I got 0.001544. Do you multiply this by \$1.25 (it's a quarter machine)?

Quote:

Example 2: A casino in town is offering a 5X points promotion in video poker. Normally the slot club pays 2/9 of 1% in free play. So at 5X, the slot club pays 1.11%. The best game is 9/6 Jacks or Better at a return of 99.54%. After the slot club points, the return is 99.54% + 1.11% = 100.65%, or a 0.65% advantage. The Game Comparison Guide shows the standard deviation of 9/6 Jacks or Better is 4.42, so the variance is 19.5364. The portion of bankroll to bet is 0.0065 / 19.5364 = 0.033%. By the way, this exact promotion is going on at the Wynn as I write this, for September 2 and 3, 2007.

In his example:

Variance: 19.54

Advantage / Variance = % of BR to wager
0.0065 / 19.54 = 0.00033 = 0.033%

On a \$15K BR, the optimal Kelly wager would be: 15,000 * 0.00033 = \$4.95/spin (ie: \$1 denom for \$5/spin would be slightly over kelly).

For you:

Variance: 33.9

0.0524 / 33.9 = 0.00154 = 0.154%

With your measly \$300 bankroll, you should be betting 0.00154 * 300 = \$0.462/spin. :)
djatc Joined: Jan 15, 2013
• Posts: 4477
November 30th, 2014 at 9:45:48 PM permalink
Quote: RS

For you:

Variance: 33.9

0.0524 / 33.9 = 0.00154 = 0.154%

With your measly \$300 bankroll, you should be betting 0.00154 * 300 = \$0.462/spin. :)

My math shows I need \$811. Now let me borrow \$511 lol.

That is quite interesting, seeing as that's a very low sum. Goes to show how huge edges requires minimal bankroll and very quick growth to double.

I just thought about it further, and realized that the \$811 figure is with a \$1.25 kelly bet, so as I'm losing the bankroll I would have to adjust my bets.... now how can I find out the risk of ruin? I prefer to be at 1% or close to it if possible. Thanks.
"Man Babes" #AxelFabulous
AxelWolf Joined: Oct 10, 2012