A Vicious Cycle, Part V: Variance

The time has come to end this series on the vicious cycle (an eerie reverberation can be heard). Unfortunately, all good things must end … with the possible exception of the cycle on the newest—or perhaps not so new—video slot games on your floor.

Last month we studied various math models of fictitious games that were based on real values. These games illustrated just how easy it is to have a game’s cycle size grow significantly and showed various methods of making the game pay large winnings while still trying to minimize the overall impact on the base game math.

Were you able to determine the flaw in our logic? There are actually two problems that we must consider. First, we based our math upon the cycle, which is theoretical. In order to achieve the target hold, we must experience each possible outcome once and only once. The second problem results from our games working with random—or pseudo-random—outcomes. We will never experience a complete cycle without any duplication of results. As a result, we experience a variance in the actual game results. Let’s examine these problems in more detail.

One Cycle is Not Enough

Last month, our game grew to a cycle with 13,798,512 possible outcomes. This isn’t unheard of, and some games do have cycles in the billions. The first problem is getting through the cycle, or at least the same number of games as the cycle, so that we have a good base for statistical analysis. Just how long would it take to complete 13 million games?

If the game is popular, we may experience 150,000 games per month (5,300 games per day, or one game every 16 seconds). Based on this level of play, it would take 92 months to complete the cycle. This requires the game to sit on your floor for more than seven and a half years. It also assumes that the game will maintain popularity and experience a linear play level during that time.

Should this game prove impossibly popular and experience 250,000 games per month, the cycle would take four and a half years to complete. However, if it is less popular, experiencing only 75,000 games per month, we’ll be waiting for more than 15 years for the same results. And to even reach 75,000 games per month, the machine requires two games per minute. With some games landing on your casino floor and leaving six months later, there is a problem in obtaining enough games to make the play close to theoretical.

The problem is even worse with larger-cycle games. A video slot with 130 million games in the cycle that experiences 150,000 game plays per month will have us waiting 72 years for a complete cycle. It might be a long shot, but I doubt you will still be working in 72 years. How many of your current games are 72 years old?

And what about games with cycle sizes in the billions? A cycle of 5 billion games will take 2,778 years to complete. Even if the game is extremely popular and experiences 500,000 games played per month (approximately five games played each second), it will still take 83 years to complete the game cycle.

This leads us to our second logic problem. Even if we experience 13 million plays on a machine with a 13 million-game cycle, it is virtually impossible that each outcome will occur once and only once. Every game played has every combination available to the player. This makes the game fair but introduces volatility. And it means that in order to reach our theoretical hold, we must experience several cycles. If we flip a coin 10 times, we aren’t likely to have tails five times and heads five times. If we flip a coin 10 million times, however, the number of heads and tails will be very close to 50 percent each. Similarly, if we have a game with the same number of games played as there is in the cycle, it does not mean that the results of those games will be the same as the theoretical results. The higher the volatility index, the more likely we are to deviate from the theoretical results.

In order to determine just how many cycles we need, we’ll turn to further simulation. This month’s simulation involves determining how many games are required in order to have every combination in the cycle selected at random at least once. In our simulation, we used a game cycle of 262,144. Each simulated game resulted in a random outcome being chosen out of every possible combination. This outcome was recorded and a new game initiated. Each outcome was tallied and the simulation stopped when every outcome was selected at least once. At the end of the run, there were 12 outcomes of 262,144 that were selected once and only once. This is only 0.00458 percent of the games. Obviously, almost all of the games were selected more than once before each individual outcome was selected once. The maximum number of times any individual outcome was selected was 32. Approximately half of the possible outcomes were chosen 13 or more times.

The problem lies within the outcomes that were selected only once as well as those that were selected 32 times. Suppose that one of these is a non-winning combination. If it occurs 32X, then other winning combinations have happened less, resulting in a lower payout (or larger hold). However, if the 32X combination happened to be a large jackpot award, then we have paid it many more times than we statistically should have. This does not, however, mean that there is anything wrong with the machine. The random combination was selected more than it should on average. This will result in a significantly small hold and a tremendously large payout. In order to compensate for this, the machine will have to be run through several cycles. Here, the law of probability will see that the overall payout and hold are brought back closer to the norm and everything will be well.

In the simulated example, the cycle is 262,144 games and we have already experienced 32 times that number of games played, or just over 8.25 million games. This process (assuming 150,000 games per month) has taken 4.7 years. Are you willing to keep this game on your floor for another 4.7 years, or 9.4 years, or even longer, just to even out the hold and payout?

This means that we cannot look at the cycle size alone and expect to reach the theoretical hold after we have experienced the same number of games as there are distinct outcomes in the cycle. If our newest super-mega-biggie jackpot game has a game cycle of 5 billion, then we may require a dozen or more times the game cycle in order to achieve close to theoretical results.

Historically, this was not as important. Consider the still-ubiquitous Blazing 7s by Bally Technologies. This game has been popular with players for several years. It still receives heavy play in casinos all across North America. What makes this game so popular?

While players enjoy high-volatility video slot games, there is still an appeal to games that are just the opposite. The Blazing 7s offers a small cycle with little volatility. Although there are numerous different versions of Blazing 7s, one of the most popular features a cycle size of only 32,768 games. With eight jackpots per cycle, a player can hit the top award every 4,096 games on average. On a busy weekend, players are sure to see at least one game hit the jackpot on a bank of Blazing 7s. The players therefore feel that they have a reasonable chance of hitting the jackpot. When they see a million dollar jackpot on a penny game, however, they realize that their chances are extremely slim. Most players likely feel that their chances of winning the million dollars is the same whether or not they even play the game.

With the Blazing 7s, the jackpot is a minor portion of the overall payout and a relatively small percentage of the wager. For the 1,000 credit top award, with 3 credits played, the player receives 333 credits per credit wagered. For a 20-line video with maximum wager of 10 credits per line, using the same percentage, the top award would be just under 67,000 credits. That is still a relatively good award, but certainly not a life-changing occurrence.

Running our simulation on Blazing 7s, we find that we still require 25–35 open cycles before each combination has been hit at least once. Even with a small cycle, there are a considerable number of games required to have every possible outcome occur. With the relatively small jackpot and the lower volatility, however, the overall effect is minimal. After one cycle’s worth of games has been completed, the variance in hold or pay will only be a couple of percentage points.

Incidentally, if the Blazing 7s game gets played once every 73 seconds on average, the cycle will be completed in only one month.

Back to Basics
Gaming guru Mickey Roemer offers the following sentiments on large-cycle games:  “I’ve been told over and over again by slot managers that cycle size, especially in high denomination, makes the game play differently. They tell me that their customers complain the games don’t hit the same. The math guys, on the other hand, will tell you that the customers are crazy and can’t really tell the difference. The math guys will also tell you that statistically there is no difference and the games can be mapped to match a smaller sample.”

What the math guys are referring to is increasing the cycle size to compensate for a large jackpot, which does not increase. If the cycle size is 100,000 games and the jackpot is 100,000 credits, then we can increase the cycle size to 200,000 games and keep only one jackpot hit or 100,000 credits. By taking every outcome in the 100,000 games and doubling it, we essentially cut the effect of the jackpot in half. If we increase the cycle by a factor of 10, then we have decreased the jackpot liability by one tenth. In essence, we now have 10 times the cycle but 10 times the hits and 10 times the nonwinning games as well. The only thing that remains unchanged is the number of jackpot hits (only one per cycle) and award amount. However, the problem occurs when the jackpot does hit, especially if it happens early in the machine’s life. Although we can map the large cycle to a smaller sample, we still have to complete the entire cycle in order to reach the specified hold and pay. When we complicate this by making the outcomes random, we need even more games. On one hand it won’t make much difference; on the other hand, it may.

Roemer remarks:  “I’m of the unscientific opinion that there is a difference. I like the smaller cycle size, especially for dollar and high-denom games. I think a cycle that repeats itself 115 times (32-stop game, for instance) is more predictable even than a game cycle that only completes one time during those 3.8 million games (156 stops). Again, totally unscientific, but I’ve been jaded by 25 years around slot machines. By the way, those 3.8 million games represent five years worth of play at 2,000 handle pulls per day. Which game would you rather play?”

Considering Roemer’s example, should the 156-stop game have only one jackpot, it will only hit (statistically) once every five years. Or at least you should hope it will hit only once every five years. If that combination comes up several times over those five years, your variance will be significant. Couple this with a progressive top award and the players will know that it hasn’t hit for five years. There may be periods of increased play when players feel that the jackpot is “due” to hit, but after several years where the jackpot has not hit, they are likely to feel that the game is rigged and that the jackpot is not possible.

For higher-denomination games, your handle is likely to be considerably less than a low-denomination game. This increases the time the machine must be on your floor.

In the case of the 32-stop game, the jackpot is going to hit several times per month. Take another look at the Blazing 7s game. The majority of these are progressive jackpots, and for good reason. The players see the amounts rising and resetting frequently. They know that the jackpot hits within a certain range, although sometimes one will go much higher than normal. However, players “feel” that the jackpot is due based on their casual observations of the normal range in which the machines pay. This may be unscientific, but such empirical knowledge does have some factual basis.

Players can get a feel for a game. They may not be able to tell the difference between a 93.2 percent payout and a 93.8 percent payout, but they do develop a sense of awareness about how a game plays and how frequently it hits. They learn what to expect in the bonus games and know that a certain bonus game pays well most of the time, while another pays very little most of the time. If asked, they likely could not explain game volatility, but they could tell you something about how the game plays. And simple economics tells them that a million dollar jackpot on a penny game is just not going to happen very frequently. Yet when they see a progressive jackpot worth $250 or $375 for a wager of 75 cents, they know that they may not win it, but they at least have a reasonable chance. And they also know that the Blazing 7s quarter game with a progressive jackpot that has climbed to $485 will be won before they come back to the casino next time.

What Does It All Mean?
What does this “vicious cycle” mean? For the players, it can be good and bad. It gives them the opportunity to win a life-changing jackpot on a low-denomination game, yet it also means that their chances of winning that jackpot are very slim. These same people play the lottery knowing they have very little chance of winning, yet they still buy tickets. After all, someone is going to win. It’s a dream that they are playing for, an opportunity to catch lightning in a bottle. Still, while they play these million dollar penny games, they want to win something—maybe enough to break even for the night, enough to buy that special something they have been wanting, or perhaps enough to pay for their vacation. Or maybe they just want to win enough to be able to play for a little while longer and have some entertainment value from the game. This can be done with a smaller cycle game. We don’t need tens of millions of games in order to accomplish this. And if we’re not giving players a hit frequency that makes them feel like they are winning, they won’t play these games for long. That makes it incredibly difficult to get enough handle to have the game math work out. Even with a popular game, it may be virtually impossible to have a complete cycle ever occur.

In order to protect yourself, and to understand your liability, you need to look closely at the size of the cycle to determine if it is vicious. Don’t forget to take into account the volatility in order to see how much variance you can expect over the long term. And definitely look at the top awards on the game. Can you afford to have the top two or three awards hit within the first week and have them all hit several times during the first six months?

Perhaps it is time to get back to basics. Players want new innovation and ways to win more. And we give them that. However, it may be too much of a good thing. In order to make games more exciting and pay more, cycle sizes increase. In order to give players even larger winnings, we increase the volatility of the machine. Yet that increases our potential risk as operators and can also affect the players’ perception of the games. Perhaps moderation is the most important factor.

We can give players large jackpots and offer them numerous bonus games with the opportunity to win thousands of credits. But we must couple this with a reasonable risk to the operator and provide games in which the players have a reasonable hope of winning larger awards.

Above all, consider how long it will take to complete one cycle, or at least the time required for the game’s handle to match the cycle size. Remember that this isn’t going to mean that the cycle is complete. You may need five, 10 or 20 times that amount of play in order to have one complete cycle. Depending on the game configuration, this may not matter. Again, this comes back to moderation in design.

As a player, you would likely prefer to play a game that runs through a cycle 115 times in a five-year period over one that requires five years to complete one cycle. As an operator, you must carefully study numerous aspects of the game in order to determine which one is best for you.

Just because a game has a cycle size of several million does not mean that it has a large liability and is, in fact, a vicious cycle. However, make sure that you are not placing several new machines on your floor that do have vicious cycles—it’s caveat emptor for the players as well as the operators.

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