Casino Enterprise Management

Welcome back. Pull up a chair and make yourself comfortable as we continue our journey through the vicious cycle. (Eerie music plays.)

Last month we saw how the number of symbols on each reel, as well as the number of reels in the game, affect the cycle size. To calculate the size of the cycle, you simply take the number of stops for each reel and multiply the numbers together. If the number of stops per reel is identical, then we can calculate the cycle size using exponents, with the base being the number of stops and the exponent being the number of reels. Two reels with 32 stops per reel would be 32 squared (32²).

A sample illustration of cycle size is shown in Figure 1. Note that in this case each reel we add increases the cycle by a factor of 32. Going from three reels to five reels increases the cycle by 32 x 32 = 1,024 times.

When more symbols are added to the game, the cycle size obviously increases as well. If we have 127 symbols on each reel, the cycle size increases to 2,048,383 games for a 3-reel game and 260,144,641 games for a 4-reel game. If we have five reels, the cycle size is an incredible 33,038,369,407 (yes, that’s billion) games for a 5-reel game.

With a larger cycle size we take in more credits and can therefore pay out larger awards while still keeping the hit frequency relatively high. The larger the cycle, the more we can pay out at any one time. Before we study the downside of a large cycle, let’s take a moment and examine how the cycle size can affect bonus games. Perhaps more importantly, we’ll also study how bonus games can affect the cycle size.

Bonus Games
In the simplest sense, let’s consider a bonus game to be a regular payout. It may be available as a standard line pay or as a scatter pay. How the bonus game is initiated affects the hit frequency of the bonus game as well as the payout model. Bonus games (just like regular pays for matching symbols) pay a certain value on a payline multiplied by the number of credits wagered on that payline. Scatter pays are multiplied by the total number of credits wagered for the game since they are initiated for the game, not a payline.

In a video slot, three Bell symbols may pay 120 credits on any payline. We could change the three Bell symbols to bonus symbols and have the corresponding bonus game pay 120 credits as well. There is no difference to the math of the game—it’s all in presentation. Think of the bonus game as a type of “window dressing” where there is animation and sound to entertain the player, but really it’s just a standard payout giving the player a corresponding award.

What is different about the bonus game is that it generally has a varying payout. To make bonus rounds more exciting, payouts range from very low to very high, with enough variation so that the players have no idea how much they are going to win. In the broadest sense, we should consider a bonus game to be anything beyond a standard payout for matching symbols. That would include free games awarded to the player.

There are a number of ways we can provide multiple or varying payouts for a bonus game trigger. One way is to have the same symbols appearing on the reels but make them represent different symbols within the slot machine. In Figure 2, three matching bonus symbols initiate the bonus game. To the underlying math, however, there are two unique symbols on each reel—bonus symbol 1 and bonus symbol 2. As the two symbols create eight different combinations (2 x 2 x 2), this allows us to have eight different paying amounts. The player sees the same symbols visually and likely won’t catch on that they are different internally. However, an astute player may recognize symbols surrounding the bonus symbols and eventually figure out what pay they will receive before they enter the bonus round. Using this method, each combination of symbols is identical to a standard payout and is accounted for in the par sheet and mathematic model.

There’s nothing wrong with this configuration, and it does satisfy the requirement of providing varying pays to the player. If we want to increase a bonus round payout significantly, however, then it draws significant credits away from the base game and the standard symbol pays. This in turn decreases the hit frequency and perhaps the player satisfaction as well. The same problem was encountered with small reel sizes in the past as we discussed at the start of this series. Rather than just increase the cycle size, however, there is a method by which we can change the bonus round to provide more credits.

Making Big Big Big Bonus Round Pays
Figure 3 shows a new and improved bonus round that provides the chance of the player receiving a large windfall. The total payout from each of the eight bonus games has increased to 81,600 credits. If our base game cycle is sufficiently large, this may not constitute a significant portion of the total cycle payout. However, it may. Rather than configure the bonus round as eight separate bonus rounds, we can set it up as a single bonus round event with varying pays.

After all, we may wish to trigger the bonus round hundreds of times in the base game, so setting up individual pays for each of these would be a lengthy process.

­­In this case each bonus round symbol is treated as the same, and no matter what combination of bonus symbols it displays we enter the same bonus round. The machine then randomly determines what payout we will receive out of the eight possibilities. Each of the eight values has an equal chance of coming up. In essence, the machine rolls an 8-sided die.

Figure 4 shows how the payout relates to the overall game pay.­ In the first example, eight bonus rounds were identified in the cycle. In the second example, we only have one bonus round. You will notice, however, that the amount attributed to the bonus game is smaller, even though we have increased four of the amounts. The reason is that we are now running the bonus game as an “average,” with the amount shown in the chart as an average of each of the eight pays. This brings the payout back down to 10,200.

This slight of hand may appear to violate our earlier rule—that we can’t create new payouts, we only transfer the amount from one area to another. Yet here we’re paying 81,600 credits but it only costs us 10,200.

As the pay is an average, it means that some pays must be larger than the average, with some smaller pays to bring the average back down. Figure 5 illustrates three pays that result in an average of 400 credits. The 600 pay amount brings our average up, but it is offset by the 200 credit pay. In our eight possible awards, we add them up and divide by eight, giving us the average.

­This means that the bonus round must be initiated eight times in order to pay the average amount. If a player receives the 120-credit award, then we’ve paid out less than the average and our hold will be a little higher. If they receive the 75,000-credit award, then we have paid out much more than our average and our hold will be lower than normal. After we have had eight bonus rounds, our average is met.

This can, in effect, increase our cycle size. If there were only one bonus combination to trigger the event in the base game, then we would need eight base game cycles in order to have the eight games occur. This means that our base game cycle is really 1/8 of what it truly is. In our example, it works out evenly—eight triggers for the bonus game in the base game with eight variations of the pay. However, we could expand the possibilities of the pays to 80, meaning that we require 80 bonus games to meet the average. This relates to 10 base game cycles in order to achieve the 80 bonus game triggers.

In order to calculate how many base cycles are required to have every combination of the bonus game occur, use the formula shown in Figure 6. Simply divide the total number of bonus game possibilities by the number of bonus game triggers in the base game. The resulting answer is the number of complete base game cycles required to trigger the bonus game for the average to work out. This can be significant and you must be aware of this when selecting games for your floor.

There is one fallacy in our logic that you may have identified. Although there are eight combinations, we may need more than eight times to have every combination occur. The 120-credit pay could happen twice in a row. Recall from last time that we don’t neatly step through every game combination in the cycle. As the outcomes are determined randomly, some will occur once or more times when another may not occur at all.

If the range of pays is small, this won’t matter very much. However, if it is large, then you may experience a noticeable variance on the machine. If it pays 75,000 twice, we will notice a higher than normal payout.

Expected Value
In order to have significantly large payouts that have a smaller affect on the overall payout, there is another method used. This method uses a probability factor to determine what we expect to pay out on average, rather than having each outcome equal.

In Figure 7 we change the bonus payouts to provide a greater quantity of awards with larger amounts available. The average payout drops to 2,565 credits, even though the top award has increased to 100,000 credits.

By factoring the probability, we are in essence reducing the overall effect of the large awards. There is only a 1 percent chance of the player receiving 100,000 credits. ­­­

Although there are only 25 possible outcomes available, there are 100 mathematically. This is because the average pay is based upon a percentage, giving 100 options. Only one of these is the top award. But the lowest award, 100 credits, has a 10 percent probability, or 10 out of the 100 options. In this case, we must execute the bonus round 100 times in order to meet every outcome. Of course in a random event, it will likely not work out that nicely.

If we maintain the eight bonus game triggers, then our cycle size has actually increased by 100 / 8 = 12.5 times. Since the bonus game is only called eight times during the cycle, we must repeat the cycle again and again (almost 13 times) before we can possibly receive each of the 100 outcomes.

What Does that Mean?
The important point is that bonus games can have a major impact on your base game. Be sure to study the number of hits in the base game and compare them to the number of possibilities that exist in the bonus game. In some cases, the range of payouts may not be significant, so the overall effect on the hold and payout will be negligible. In other cases, the bonus game will be triggered more frequently than there are outcomes available. For example, the bonus game may hit 4,096 times in the base game but with only 32 possible outcomes for these games. This situation will impact the math even less, since one cycle covers multiple cycles of the base game.

Think of the bonus games in the broadest sense to include free spins and other special awards to the player. Also think of these bonus games having their own cycle size. Compare it to the number of hits in the base game to see if you are going to require more than one bonus game cycle in order to complete the bonus game cycle.

And that brings us to the big point—how does this cycle size affect the operator? If you want to pay large awards, it has to come from somewhere. A large cycle size makes this possible, but it could create serious implications for your hold. Next month we’ll clear the mud and look at another important aspect of the vicious cycle.  (Eerie music fades ... )

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.