As I was getting ready to write this month's column, I got the following e-mail:

Here's a situation for you. At Fitzgerald's in Tunica a couple of weeks ago, they had a progressive for quarter VP machines at $2700. Their Jacks or Better machines were 8/5, for an expected return of 101.3%. But they also had 9/7/5 Double Bonus machines going for the same jackpot, with an expectation of 103%. I have learned from you that with optimal play, Royals come more often on a Jacks or Better machine than on a Double Bonus. So the question is, which game is the optimal game to play? The one with the highest theoretical return, or the one with the best chance of getting the Royal before someone else does?
-Steve

Dear Steve,

What a great question and the timing could not have been better, since it arrived just as I was getting ready to write this column. So, instead of just answering your question then posting this in our "GameMaster Advisor" section, I thought I'd do a full-blown article on it.

The natural reaction of anyone would be to pull a bunch of $$$ out of the pocket and get on this thing before someone else hits the Royal. But, since this opportunity has long-since passed, we can take some time to analyze the situation thoroughly so that we can take full advantage of future situations which will, I assure you, present themselves. The machine Steve saw is undoubtedly a "Multi-Play" machine which offers 4 or 5 different games, but the same progressive Royal applies to all. Not all of the games offered on these are worth playing, but Steve identified two which are worthwhile, the 8/5 Jacks and 9/7 Double Bonus. This is similar to the situation I wrote about in a column entitled "Taking a Shot", but the difference here is that we want this Royal and I assume we have some considerable $$$ to go after it. One thought I want to inject in here for all our readers: If you see a situation like this, and have friends or a spouse who also play VP, consider pooling your $$$ and time to get this Royal. If this game is at a bank of 10 machines and all 10 are being played (and all players are putting in the maximum coins), you can figure that about 5000+ hands are being played per hour and it's likely just a matter of 6 or 7 hours before the Big Banana is hit.

Probably all of the writings you've seen on Video Poker stress 'maximum return' as being the key to winning, so it would seem that our choice is simple here, Double Bonus returns more so that's the one to play. And that would be true if you played a LOT of Video Poker; you cannot let a tempting jackpot blur your vision regarding 'expected value'. But to most of you, it's a much simpler proposition: "Man, I'd like to get that $2700 Royal!" And who am I to say you're wrong? It's your $$$ and I'm here to help you as much as I can. So, with that in mind, let's get busy.

As Steve rightly points out, if you use the 'standard' playing strategy (the correct strategy for the chosen game when the Royal is at 4000 coins) for an 8/5 Jacks or Better game, a Royal Flush will show up, on average, once every 40,000 hands, but for a Double Bonus game, that figure is closer to 48,000. That alone tells you the Jacks game is a better choice. But there is a reasonable alternative. With such an inflated Royal, the playing strategy can be changed . The reason for that is, of course, that we stand to make a lot more than normal by hitting the progressive, so we'll be more inclined to break a high pair, etc. to draw to a 3-card Royal. This calls for a very cool program named "Video Poker Strategy Master" (VPSM) by TomSki which I've reviewed in a previous column. If you input the 10,800 coin Royal ($2700 on a quarter machine), and then generate a strategy table for the Jacks game, you'll see that the proper playing strategy changes quite a bit. For example, you'll break any high pair to draw to any 3-card Royal; in fact, you'll even break two-pair to draw to a 3-card Royal. The same is true for the Double Bonus game and the net effect of those strategy changes is to lower the incidence of a Royal. Here we call in another great program "Bob Dancer presents WinPoker" by Dean Zamzow. I've also reviewed this in a prior column and it's a great companion to VPSM. If you input the 10,800 coin Royal, you'll find that a Royal can be expected in the 8/5 Jacks game every 32,000 hands on average and the same is true for the Double Bonus game!! Therefore, if we change our playing strategy to exploit that big Royal, the incidence of a Royal (40,000 vs. 48,000) is no longer a factor. Are you with me on this? Steve's original question concerned which game to choose, since one has a lower return, but 'quicker' incidence of Royals (the 8/5 Jacks), yet the Double Bonus has a higher overall return, but a 'slower' Royal. The modifications we can make to the proper playing strategy reduces that difference, but what if we didn't know that at the time? Is it wrong to play a game with a high Royal by using the 'standard' playing strategy?

Well, it's not wrong, but it is inefficient. Consider this: You may be the most accurate player in the world when it comes to the 'standard' strategy, but that will actually lessen your chances for hitting the Royal whereas a person who doesn't know any better may be sitting there throwing away hands like three-of-a-kind to draw to 2-card Royals and their chances of hitting it are greatly increased by doing so! By using a playing strategy which has been calculated on Video Poker Strategy Master, you get the most efficient strategy possible. It recognizes the value of the Royal, but also recognizes that the odds of you hitting it aren't great so it helps your bankroll to survive to fight another day.

But the question remains: Jacks or Better or Double Bonus?

If you do not have the ability (or desire) to change the proper playing strategy, then the 8/5 Jacks game is the better choice, because the Royal shows up more often at this game and, because the Jacks game pays 10 for 5 on Two-pair, but the Double Bonus only pays 5 for 5, you'll likely need less $$$ to play the Jacks game. Let's run a little simulation here. I wrote about Jeff Lotspeich's 'gambler's ruin' calculator (http://members.aol.com/lotsie/GamblersRuin.html.) before and it can give us some insight here. I went to Jeff's site and ran a calculation for play at an 8/5 Jacks game and one for a Double Bonus game. Jeff's calculator assumes a 4000-coin Royal and it assumes that a player will use the 'standard' playing strategy, so it can only be used as an approximation of our situation, but it has its value. It also calculates only for the 10/7 version of Double Bonus, so the resulting figures for that game will be somewhat higher than we could expect from a 9/7 format. But anyway, there's something to be learned here. What I did was assume that we had $1000 to attack this Royal and 8 or so hours to get it. With $1000 at a quarter game, the probability of us losing all of it is pretty small and it's likely that someone will hit the Royal within the 8-hour time frame. I plugged in a $1000 bankroll and ran a simulation for 5000 hands of play at each game. For the 8/5 Jacks, the probability of losing $1000 in 5000 hands was less than 0.01% and for the Double Bonus game it was .14% (1 in 700), so 'ruin' is not a factor. Where things got interesting is when we look at the profit area of +$680 to +$1100. Now remember that Jeff's simulator treats a Royal as being worth $1000, so hitting one will most likely be reflected in the +$680 to +$1100 area. Further, it plays by using the standard strategy, which is the most efficient with a Royal of that size, but it also gives an indication of how we would do at the $2700 game if we didn't vary the basic strategy. After playing 5000 hands, the probability of ending with a profit of $680 to $1100 is 5.9% at the Jacks game and 5.8% at the Double Bonus game. Understand that neither of these results guarantee a Royal, because we could hit a bunch of quad Aces at Double Bonus or several Straight Flushes at Jacks to get there, but the most likely reason for such a result is a Royal. So, neither game is likely to tap us out and both have a similar potential for a Royal in 5000 hands of play with the Jacks having a slight edge in that regard if we don't modify the strategy.

We have some time here, so let's contrast those results with a more common situation. Instead of having $1000 to go after the Royal, let's assume we have only $300 available and run the simulation again. The primary concern with a bankroll of only $300 is, of course, the probability of going broke. Our simulation shows that for the 8/5 Jacks game, the probability of tapping out is 54% and for the Double Bonus game, it's 43%. But we have to do some interpolating here, because the simulation software offers a 10/7 Double Bonus game and the game Steve saw is a 9/7 version. This is a significant difference when it comes to a $300 bankroll because, in the course of 5000 hands, we can expect to get about 55 Full Houses and the extra 5 coin payout will give us 275 coins or $68.75; a full third of our starting bankroll. Even so, it kind of surprised me that the probability of losing the $300 was so close between the two games; I thought the Jacks game would have better 'staying' power. Further analysis of the simulations show that in the profit range of +$750 to +1050, which 'implies' a Royal, the Jacks game has a probability of 3.8% and the Double Bonus game's probability is 3.5%.

I guess what the above shows us is that in such a relatively short term, it barely matters which game you play. "Luck" is such a big factor here that it overwhelms all the mathematical factors involved. That said, there are still some conclusions we can draw here and there are some steps we can take to make our luck better. Let me set up some scenarios and you can use the one that fits best.

1. I have $300 and do not know how to vary the proper playing strategy to take into account the inflated Royal. I am a 'casual' Video Poker player.

• Recommendation: Play the Jacks game.

2. I have $1000 and do not know how to vary the proper playing strategy to take into account the inflated Royal. I am a 'casual' Video Poker player.

• Recommendation: Play the Double Bonus game.

3. I have $300 and can vary the proper playing strategy to take in account the inflated Royal. I am a 'casual' Video Poker player.

• Recommendation: Play the Double Bonus game.

4. I have $1000 and can vary the proper playing strategy to take in account the inflated Royal. I am a 'casual' Video Poker player.

• Recommendation: Play the Double Bonus game.

And, if you are a well-financed pro or very serious player, you have to go with the 'expectation' and stick with the Double Bonus game. But get a copy of "Video Poker Strategy Master" and start carrying copies of the proper playing strategy for games with higher Royals.

So you can see, Steve, that by varying the proper playing strategy, we even out our probabilities of getting the Royal by quite a bit and the big variable, luck, is out of our control so it's better to go with the Double Bonus game. But, if big bucks and a modified strategy aren't available, go with the Jacks, especially when you consider the fact that Jacks can probably be played faster than a DB game and this is, after all, a race; a race between you and the others who are stalking the Royal.

Thanks to Steve for such an interesting question and I'll see you here next time.