Casino Enterprise Management

Last month we began the study of a vicious cycle designed to pique your interest in game selection, slot math and risk (to both operator and player) and to make you think about high-volatility, large-award games. We'll slow down the pace this month and look at the cycle as it relates to the slot machine's "budget." The cycle is the mathematical basis of the game and how the combinations are set up within the cycle determines how the game will play, how enjoyable it will be to your customers, and how safe it is for you to have on your floor.

Hold Budget
In a previous article, I spoke of Wilson's Law of Bonusing, which states that winnings can neither be created nor destroyed; they can only be transferred from one area to another. In other words, the slot game does have a budget of sorts. There is a limited amount of money that can be taken in, and therefore a finite amount that can be paid out. How this is arranged is up to the slot manufacturer, and it is up to operators to decide if the payout scheme is a good fit for their floor. Let's delve into this so-called budget a little more.

The budget, or mathematical configuration of the machine, is dependant upon the cycle. It dictates how many combinations are available and how much can be taken in and paid out. The number of combinations is determined by multiplying together all of the stops on each reel. For a simple three-reel mechanical, these numbers range from tens of thousands to perhaps a million. When we add more reels, the cycle size increases geometrically. Figure 1 illustrates a small cycle size. For a three-reel game with 10 stops per reel, there are only 1,000 combinations. When we add two more reels, the cycle size increases 100 times to 100,000. Using a more realistic number of slots, also shown in Figure 1, we see that our three-reel game has 373,248 combinations and our five-reel video slot has more than 600 million combinations. This video slot cycle is almost 1,700 times larger than the three-reel slot, and we begin to see how adding more reels can create a much larger game cycle. Remember that the larger the game cycle, the longer it will take to complete. In a future installment, we will examine cycle size and completion time in more detail.

As we study these cycles, we realize that we have a baseline that determines how much the machine can take in. Considering every possible combination gives us our base budget. For 373,248 games in a cycle, the machine will take in 373,248 credits. The most we can pay out, therefore, is 373,248 if we wish to break even. In order to hold 3 percent, we can pay out 97 percent, or 362,051 credits. The difference, 11,197 credits, is our hold.

This game baseline obviously doesn't allow us to offer a million-credit jackpot. Furthermore, when selecting a game that offers excitement to the player, you must consider how these winnings are broken down. This game could offer a 300,000-credit top jackpot. That would leave a mere 62,051 credits to be paid out over all of the remaining 373,247 games (we took one game away for the jackpot). Looking at this another way, we could have small five-credit pays in 62,051 / 5 = 12,410 games. That would leave 373,247-12,410 = 360,837 games where the player won nothing. Knowing the cycle size and number of winning games (hits), we can determine the game's overall hit frequency. In this case it would be (12,410 + 1) / 373,248 which is approximately 3 percent of the games. Would your player be happy winning five credits every 30 games? This is quite unrealistic and precludes a very large jackpot.

The overall mix of winnings must be perfectly balanced so that the player wins frequently. If the player wins too frequently, the winning amounts must be very small. If they win less frequently, the awards can be made larger. It's a balancing act, and the game manufacturers rely on years of experience, focus groups and considerable work and research to bring these games to your floor.

By reducing the top award, we free up more credits to be distributed among smaller awards and more frequent wins. This also allows us to offer more than one top award per cycle. This lets the players know that the jackpot can be won because they see it happen. Where the jackpot is a progressive, players can monitor the level over time and know if it has been awarded. If they see the progressive amount resetting frequently, they know the jackpot is won frequently. If it goes for several months, they realize that their chances of winning the jackpot are low and this may discourage them from playing the game.

Offering Larger Awards
Recent trends in slot games involve larger awards and play that is more volatile. In order to create these large awards, there must be a mechanism to keep the player interested with smaller, more frequent wins.

One obvious solution is to allow more credits to be wagered for each game. Instead of wagering a single credit, the player can be encouraged to wager multiple credits. A multi-line video slot that allows for 20 credits to be wagered per line can offer awards 20 times greater than if they wager a single credit per line. That takes an 800-credit payout and provides 16,000 credits. This is the trend that your customers want, but it only works if they wager multiple credits per line.

Stepper models have done this for years. Multipliers encourage extra credits to be wagered and often provide a bonus for playing maximum credits. A Double Diamond single-line game typically awards 800 credits for a single-credit win, 1,600 credits for a two-credit win and 2,500 credits for a three-credit win. The extra 100 credits have a minimal impact on the overall game math, but the player sees it as a bonus. If a quarter game with a cycle size of 373,248 were configured in this manner, the top award of $625 would account for 2,500 / ( 3 x 373,248) = 0.2 percent of the total credit-in. By adding more jackpot combinations, the hit frequency increases. Using eight jackpots per cycle, the top award pays 0.2 percent x 8 = 1.6 percent of the total credit in, but the hit frequency increases from 1 in 373,248 games to 8 in 373,248 games, or every 46,656 games.

Refer to Figure 2 to see how the game awards versus hit frequency affects the base game. On the left-hand side, we see the game cycle represented by total credits taken in. Using the 3 percent hold and 97 percent payout example, we have an 11,197-credit hold. The remaining 362,051 credits can be broken down in an almost infinite arrangement of awards. The bar shows relative awards based on one or more top jackpot awards (largest award paid), large wins, medium wins and small wins. Recall that if we pay out most of our credits in small wins, the player will win frequently, but these pays will be only a few credits. If we have a large number of large wins, the player will win infrequently but be awarded a large number of credits.

The bar on the right-hand side shows the same game cycle represented by games played. The two bars relate directly, and this shows how varying the individual awards affects the game hit frequency.

One obvious contradiction may stand out. The hold represents times when the player does not win, or wins less than their wager. These amounts are never paid back to the player but are held as the operator's income. Yet these small amounts relate to the majority of the games. Surely such a small amount will not take up the majority of the cycle.

The answer lies in the magnitude of the individual awards, which is part of the "budget" of the cycle. If one game pays 1,000 credits to the player, there must be 1,000 games that pay nothing, in order to even out the payout. For a 2-credit pay, then there must be two games in the cycle that pay nothing in order to even out this payout. Given this approach, the payout will always be 100 percent--all coins in are paid out to the player. The game design will have some adjustments in order to allow for the hold. There will be more games that pay nothing in order to account for the hold. If we have a 3 percent hold and need to "store up" 11,197 credits to make that hold, then there must be 11,197 games in the cycle that pay nothing in order to account for that hold. Yet we know that in our example, 11,197 games is only 3 percent of the total.

Awards with a large magnitude have a great affect on the game. In Figure 2's example of the bar on the right, hold does not refer to only the operator's hold, but also money that must be held in order to make up for the large payouts. A 20,000-credit payout in a single game means that 20,000 games must pay nothing to the player. This balances the budget and allows our numbers to work out. As you can see, the larger the payouts, the more losing games we require. This is the basis of the Law of Bonusing--you have a fixed budget, but how do you want to pay it?

Simple adjustments can be made to an existing game by the manufacturer in order to provide varying hold amounts. If we wish to have a game with a 4 percent hold, then certain adjustments can be made. We can reduce the overall number of awards to make up the extra 1 percent. Perhaps a number of triple-bar combinations will be removed from the game cycle. The paying amounts could also be reduced--triple-bar payouts could drop from 120 credits to 100. Either the player wins fewer credits, they win less frequently, or a combination of both. Of course these adjustments are made by the manufacturer before jurisdictional approval and sale of the game is made. Once the machine is on your floor, the cycle is fixed, the budget is set, and it's not adjusted.

This brings us to another interesting point. So far we have discussed the cycle and budget in theoretical terms. Once the cycle has been completed, we know exactly what our hold is. But on the floor, under the influence of random outcomes, we experience a variance. Our hold will fluctuate constantly. This is normal, and there are ways to determine if the fluctuations you experience are within normal limits. This, too, will be discussed in a future installment.

The Effect of Randomness on Budgets
The influence of random outcomes really throws our budget and the cycle right out the window. The cycle does not exist on your casino floor; it exists only on paper. It is a mathematical model that dictates what outcomes are available for any particular game. With random outcomes, we don't neatly step through the cycle, being careful not to select the same outcome twice. Our games are similar to the roll of a die. The die does not remember if a six was rolled last time. It won't avoid rolling a six twice in a row. And our games won't really "store up" to account for the jackpot win.

Your customers, however, are under the assumption that the games do operate on a fixed budget, under control of that sneaky operator watching from the control room. Superstitious players remove their players' card so that the casino doesn't know what they're doing. They rub the glass in hopes of instilling good chi into the machine. And they don't want to play a game that just hit the jackpot.
A man walks into a casino and sits down at a slot machine. The woman beside him turns and says, "Don't play that gameÑsomeone just won the jackpot."

"Wow, thanks for warning me," states the man as he stands up and walks away.

We know that the games don't operate this way. The jackpot, as well as every other winning and losing award in the cycle, is available in every game played. How then do we account for the jackpot winning? It may temporarily decrease our hold, perhaps significantly. If we don't actually start to catch up, we'll never recover from that payout.

The answer lies in the probability. The probability of each game is fixed, as determined from the cycle and combinations shown on your PAR sheet. Probabilities will always work out. They don't rely upon superstition, hopes, fears or the rubbing of reel glass. They do, however, take time. The player may win or lose in the short term, but the operator is there for the long haul. Over the long term, probabilities will work out and the hold will come back into range. The randomness of the machine will always cause a variance, but over time, the probabilities work out. The budget, therefore, is not based on a single cycle—at least not in the real world of your casino floor. It will take several cycles and time to have it work out.

And this, finally, brings us back to the vicious cycle. A machine with low volatility will not experience the large swings and variances in the hold and payout. There will always be a variance, but it won't be large. A volatile machine, however, can experience a large variance. That doesn't mean that it always will. Randomness can result in a high-volatility game experiencing periods of low volatility, too. But there is opportunity for the game to deviate significantly from what shows on paper. When you factor in an extremely large cycle size, perhaps in the millions or billions of games, you run the risk that there isn't enough time to allow the probabilities to work themselves out. In that case, your hold could vary significantly--higher or lower. And this, my friends, is what we'll study next month.

John Wilson is the Technology Editor for Casino Enterprise Management and Owner of ICS Gaming, providing slot consulting services and game design. He has designed several slot games in both Class II and Class III markets. He can be reached at jwilson[at]icsgaming.com.