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Figuring Risk of Ruin for MultiHand Video Poker Games

Risk of Ruin (RoR) is a type of long-run bankroll calculation. Loosely defined, it tells you how much money you should have to keep from going broke, assuming you play a given game 24 hours a day forever.Bob DancerBob Dancer is one of the world's foremost video poker experts.  He is a regular columnist for Casino Player, Strictly Slots, and the Las Vegas Review-Journa land has written an autobiography and a novel about gambling.   He provides advice for tens of thousands of casino enthusiasts looking to play video poker.  Bob's website is  If the game, including slot club, returns less than 100%, the RoR is infinite, meaning you're definitely going to go broke. If you're playing a game, including the slot club, that returns more than 100%, a finite bankroll is required.

The standard formula for the RoR calculation is reasonably well known among video poker mathematicians. It uses the return on the game (including slot club) and the 5-coin standard deviation. The standard deviation is the square root of the variance, and is a measure of volatility.

Although this is a difficult calculation to perform manually, computer programs can handle it quickly. Currently, the best software available today to calculate the bankroll needed is Dunbar's Risk Analyzer for Video Poker, which is an Excel spreadsheet with a lot of macros. If you play 25¢ 9/6 Jacks or Better video poker with a 1% slot club and are willing to take a 10% chance of going broke, Dunbar says you need a bankroll of $4,550. (Dunbar rounds all of his bankroll figures to the nearest $10.)

As you may know, I've released a new video poker software program called Video Poker for Winners. VPW also has a risk of ruin calculator, and the figure it comes up with for the game in question is $4,534. The difference in the numbers is likely due both to the number of significant digits maintained in the calculation, and the amount of rounding done. I'm assuming the VPW figure is more accurate, but for practical purposes, they are identical. What if we were playing Triple Play? In that case the formula gets a lot more complicated. Each of the 2,598,960 starting hands needs to be "convolved" into its possibilities. As an example of convolving, lets look at the hand 4h 5h 6h As Kd in dollar Jacks or Better. The correct play, of course, is to hold the hearts.

Starting from '456', it's possible that we end up with nothing at all (worth $0), a high pair (worth $5), two pair (worth $10), 3-of-a-kind (worth $15), a straight (worth $20), a flush (worth $30), or a straight flush (worth $250). It's not possible to end up with a full house, 4-of-a-kind, or a royal flush. The probability of ending up with each of these hands is well known and may be found in any video poker computer trainer, including VPW (but not Dunbar, which is a specialized bankroll tool and not a complete trainer).

Each of these hands, however, may be combined with each of the others on another line. For example, it's possible in Triple Play that we end up with nothing at all on the first line, a high pair on the second line, and a straight on the third. This gives us a total return of $25. It's also possible that we end up with a 3-of-a-kind once and two pair once for the same $25. In calculating RoR, we don't care HOW we got to $25, just that we did.

It turns out that there are 37 different amounts we can end up with, starting from '456', namely:


There's only one way to reach $80, for example (two flushes and one straight), one way to reach $90 (three flushes), and no way to end up with any amount higher than $90 and lower than $250. With other starting hands, like a pair of 3s, for example, it would be possible to end up with $100 (two full houses and one two pair), so the total number of dollar amounts possible is considerably larger than the 37 shown here.

Some chances are very small, such as the chance of throwing away all five cards and ending up with two royal flushes and a full house, worth $8,045. For practical purposes, this chance is zero, but if you're doing an actual calculation, it's clearly greater than zero and must be accounted for.

Never before have multi-hand RoRs been calculated. They've been estimated by a few analysts-I've seen work by Liam W. Daily on this subject and I've heard about work by Dan Paymar (but I've not seen it), but not calculated. The typical method to make this estimate this is to use Monte Carlo techniques. In short, this means having a computer play the game hundreds of thousands of times to see what the results turn out to be.

It's a lengthy process to run Monte Carlo estimations and by necessity it can only be done on a few games. If the calculations were done on 9/6 Jacks and you were interested in 9/6 Bonus Poker Deluxe, the 9/6 Jacks figures are useless.

It's probably obvious that Five Play is more complicated to calculate than Triple Play, and Ten Play is more complicated yet. Fortunately, VPW has the capability to calculate all of these. Fifty Play and Hundred Play are considerably more complicated, and won't be available on the first release of the software.

Although I plan to write about this much more in the near future, let me give you one set of results. Assume you were willing to play NSU Deuces Wild for $25 a hand, with a 1% slot club, and a 10% chance of going broke. How does the bankroll necessary to play this game vary if you play $5 single play, $1 Five Play, or 50¢ Ten Play. These are the RoR numbers I get from VPW:

$5 Single Play $92,105
$1 Five Play $24,893
50¢ Ten Play $16,045

There's a pretty powerful lesson here: The more hands you play (for a given total bet), the lower bankroll you need-by a considerable margin. We've known this as a general principle, but up until now, we haven't been easily able to determine how much less. With VPW, that information will be at your fingertips.
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