There has been much interest recently in the volatility of N-play games. An N-play game is one in which you get a single hand (flop), choose which cards to hold, and then get N draws to that hold. The trademarked name for 3-play is "Triple Play". There is also a version of 4-play (called "Lucky Draw", I believe), and versions of 5-play and 10-play are rumored to be coming. These games appear to be quite popular, but some players have complained that they are experiencing a volatility that is much higher than expected. This article will attempt to explain what is going on.
Each draw in N-play is supposed to be made independently. That is, each draw should be what you would expect if you drew N times to the same flop and hold, shuffling the remaining 47 cards (assuming a 52 card deck) between each draw. Some people have reported some unusual draws, with exactly the same drawn cards appearing on two or more lines, which could imply some non-randomness. However, it is difficult to take these anomalous observations into account, so I will assume true randomness in my analysis.
First of all, the per hand EV is unaffected. So, if you are playing a game with 0.25% advantage, then you would expect 0.25% on each hand. On a 4-play game, you would expect to win 4x0.25% of the base bet, or 1%, per draw, but since you are betting four times as much the EV is still 0.25% of the amount bet. This all implies that you use exactly the same strategy for N-play as for the same game in a standard 1-play version.
Let the variance of a given game be V. If you play four hands of the game on a standard machine, the variance of those plays is just 4V, since variance adds. But, if you play the same game in a 4-play version, then the variance will be higher. That's because the results of the four draws are not independent, but are correlated (the amount of correlation depends very strongly on the flop -- if you flop a hand where you hold all five cards, the correlation is 1.0). The statistical measure that relates the variance of the standard version of a game with its 2-play version is the covariance. I've computed the covariance values for a number of video poker games and shown them in the table below. I've chosen these to cover a wide range of games and base variances. Note that all values are given for optimal play, although very good strategies for these games will have numbers that are very close to these. If you don't recognize some of the abbrevations, they are explained at the end of the article.
It turns out the the variance of N-play has a simple linear relationship to the variance and covariance. To compute the variance of an N-play version of a game from the list, just add the base game variance to (N-1) times the covariance. For example, the variance of 9/6 Jacks in the 4-play version is 19.510 + (4-1)x1.966 = 25.408. (Note that variance is always given in units of bets squared.) The percentage column in the table gives the correlation coefficent (=100%*Covariance/Variance), which makes it easy to see that 4-play Jacks has about 30.2% more variance than the standard game per base bet. That is, for all four draws together the variance would be 4x(1.302) = 5.208 base bets squared instead of the 4 base bets squared you would get on four independent plays.
The variance values are most useful in figuring your bankroll requirements. By bankroll, I mean the *all* money that you have available for gambling, not the amount you happen to take with you to a casino. If you are employed and plan to devote a portion of your earnings to gambling, you would include the present value of those earnings in your bankroll. The result computed above means that for 4-play Jacks or Better, you need a bankroll of about 30% more to play than you would require for the standard version of the game. For 10-play (yikes!), you would need a bankroll that is 9x10.079% = 90.87% larger than for 1-play. But, you might need to take a much more than 30% or 90% extra with you to the casino to actually play these games. I will take this question up next.
People use "volatility" very loosely, in my experience, to refer to the "streakiness" of the game, by which they mean the chance of having a short-term, substantial losing streak. There isn't a crisp mathematical definition of this term, that I am aware of, and it is only loosely related to the variance of the game (which is what people too often equate to this informal term, leading to confusion). So why are people reporting higher than expected volatility for N-play games? I think it has more to do with the volume of play than the variance.
Let's consider a player that normally plays 10/7 Double Bonus for a two hour session with a speed of 600 hands/hour. That same player now plays the same coin denomination in 4-play, again for two hours. N-play machines are usually slower, so let's say the rate is 400 hands/hour. The standard deviation for 1200 hands of 1-play is sqrt(1200*28.257) = 184 bets, while the standard deviation for 4-play is sqrt(4*800*(28.257+3*3.391)) = 351. This implies, roughly, that the swings in the 4-play case will be about 90% higher than than for the same denomination in 1-play. You can use the covariances given in the table above to plug your own numbers in for you own games and speeds.
By the way, I see many people using the normal distribution as a yard stick to make statements about expected results relating to standard deviation. Remember that for the number of hands you have in a session, the distribution of results is decidedly non-normal. Instead of estimating the result as 95% confidence of being plus or minus two standard deviations (which would be true for a normal distribution), an estimate of being within (1- 1/(# std deviation)^2) is more appropriate. This says you should expect to be within +- two standard deviations 75% of the time, and within +- 3 standard deviations 88.9% (not 99.7%). Also remember that the estimate is for ending bankroll. The estimate assumes you will keep playing through any losing streak, perhaps springing back to get inside the estimated region. I guess I need to write another article on this subject sometime.
You probably recognize the abbrevations for most of these. "9/6 Jacks" is the familiar full pay version of Jacks or Better. "All Amer" is of course, All American. "10/7 DB" is 10/7 Double Bonus. The "DDYM" lines refer to the 8/6 and 7/5 versions of Double Double Your Money with Aces turned on. You can get more information about these games in the descriptions of the video poker strategy cards I sell for these games.
The next group of four games has "FPDW" = full pay Deuces Wild, and three version of Loose Deuces. I have a strategy card for Deuces Wild and will eventually have cards for the Loose Deuces variations.
The final four games use jokers. "9/5 DJ" and "9/5 DJ1K" are the full pay and super-full pay versions of Double Jokers (more description is given on my VP strategy pages, referred to earlier). The last two, "JW5K0" and "JW5K3" are for single joker progressive games found in Atlantic City (I include them mainly to examine the effect of very high variances - I doubt we'll ever seem these games in a multi-play version). The pay table for "JW5K0" is 1(TP), 2(3K), 4(St), 5(Fl), 8(FH), 16(4K), 100(SF), 1109(5K). The jackpot value is chosen to be as close to break-even as possible. "JW5K3" had a jackpot of 1438 bets, bringing the EV very close to 3%. Note that the top payout on both of these games is for five of a kind -- a royal is just another straight flush.