The Vicious Cycle, Part IV: The Balancing Act

Having studied cycles and cycle sizes in the last few installments of the Vicious Cycle series, we now have more insight into how the cycle size is calculated and why certain machines have larger cycles. Apart from the geometric progression caused by an increasing number of reels, there is another important factor that can determine the size of the cycle. This is, of course, the number of stops per reel. However, this isn’t an arbitrary value selected by the game designers. In many cases it’s the result of the pay structure, and a large number of stops on the reels is necessitated by this pay structure.

Let’s consider another sample game where we attempt to design a compelling math model that makes players excited to play the game. We’ll assume that it’s a five-reel video slot, as these tend to have the largest cycles.

For this example, we won’t go crazy with our top award. Let’s make it a penny game because a million dollar jackpot is out of the question. Let’s pick a value that should be reasonable – say, 100,000 credits. (I picture someone in a white lab coat, imitating Dr. Evil saying, “One hundred thousand credits!”) There will only be one top award in our cycle in order to minimize the overall effect of the jackpot on the game math.

Now that we know the top jackpot, let’s determine our symbols. We will use five regular symbols, a scatter pay that triggers the bonus round, and our top award. The jackpot symbols will not be wild. For our basic symbols, we’ll use the card values 9, 10, J, Q and K. These symbols are found on many video slots on your floor, but this doesn’t imply that our machine will be configured similar to one of your existing games.

Symbol pays will start low, based on a single line, and will be multiplied by credits wagered. For a 9 and 10, each symbol will pay the same amount. Three matching symbols on a payline will pay 2 credits, four symbols will pay 10 credits, and five symbols will pay 20 credits. A Jack will pay 5 credits for three matching symbols, 40 for four matching symbols and 100 for five matching symbols. Queen and King will each pay the same, awarding 10 credits for three matching symbols, 60 for four, and 200 for five.

Our scatter symbol will pay for two symbols scattered anywhere. This won’t initiate the bonus game but will give a straight pay to the player. Two scattered symbols will double the total amount wagered in the game. Three symbols pays 10 times the wager, four pays 25 times and five pays 1,000 times. Awards of three or more scattered symbols also trigger a second screen bonus game with additional awards. Just looking at these pay amounts, we can assume that five scattered symbols will be fairly rare, as it pays 1,000 credits times the total wager, compared to five Kings paying 200 times the wager for the winning payline.

The top award is 100,000 credits for five symbols, dropping to 20,000 for four credits, 7,500 for three and 10 credits for two. The game pays for having only two matching jackpot symbols in order to increase the hit frequency of the game. This also serves to make players aware of the top-award symbols. If players are only paid when three symbols land on a payline, they will receive infrequent awards for the jackpot symbol. By paying for two of them, players see the jackpot symbols pay much more frequently and will feel they at least have some chance of hitting the top award. Should they play for 15 minutes and never get any pay for top jackpot symbols, this could create a level of dissatisfaction for the players.

Our initial configuration of the game gives us an overall payout of 273 percent with a hit frequency of 54 percent. Our cycle size is 1,779,084. We have a small number of symbols on each reel, namely: 18, 17, 19, 18 and 17. This is based upon actual game math in a simulation. I won’t show the entire math model simply because of limited space in the magazine. I will, however, show pertinent data that affects the game.

This is a 20-line game and the numbers I’ll show are based upon a single line. The exception is scatter pays, which pay anywhere. These are based upon a single credit wagered per game.

The total credit-in per cycle is 1,779,084 or $17,790.84. With the large payout percent, however, we pay out $48,632.00. A quick look at the paytable statistics shows that the majority of the overall game pay is from the scatter pay. There are 966,012 hits per game cycle. Line pays account for 53,550 games and scatter pays account for the remaining 912,462 games. Scatter pays account for 94 percent of the total hits. Removing the award for “any two scatter symbols” should reduce this, but will not likely bring the payout below 100 percent. In fact, after removing the pay for two scatter symbols, the game hits dropped to 674,808, with 53,550 line pays and 621,258 scatter hits (92 percent). This has made a very small difference.

Here the problem is the cycle size: We only have one scatter symbol per reel in order to make it occur as infrequently as possible. But remember, it counts as long as it is visible anywhere in the window. On our 5×3 game, there are 15 positions for symbols. This means that each symbol will count if it lands at the top, center or bottom of each reel. It really “expands” the symbols so that it counts in the space two before and two after the scatter symbol, yielding three positions per reel where the scatter symbol can land.  Reel 1 has 18 symbols. Our scatter symbol will hit on three of these 18 symbols, or 3/18 = 17 percent of the positions on reel 1. It is easy to see that we need to add more regular, or “line pay,” symbols in order to reduce the hit frequency of the scatter pay symbols on each reel.

But what effect will this have on the game? It will increase the hit frequency of line-hit symbols, as we’re adding more. This will reduce the frequency of scatter pays and lower the awards of the high-paying scatters, reducing the overall game payout. It will also increase the cycle size.

Adding just one more symbol per reel increases the reel counts from 18, 17, 19, 18 and 17 to 19, 18, 20, 19 and 18, and the cycle size grows from 1,779,084 to 2,339,280. The addition of one symbol on each reel causes the cycle to grow to 132 percent of the original.

In simulation, five additional line symbols were added to each reel. Three additional “9” symbols and two additional “10” symbols were placed on each reel. The game payout dropped to 158.62 percent with the hit frequency decreasing to 33.68 percent. This is much closer to a working game, but we still have a distance to go. The game cycle, however, is now 345 percent of the original, increasing to 6,144,864 games.

Adding another five symbols per reel made even further changes to the game math. The payout dropped to 117.99 percent with a 25.49 percent hit frequency. We’re closer yet, but not quite where we need to be. The game cycle has increased 776 percent from the initial game, resulting in a cycle size of 13,798,512 games.

The addition of symbols to our game has been somewhat arbitrary, adding more of the lower-paying symbols and leaving the upper-paying symbols alone. A better method would include analysis of the payout in various levels of awards, such as a breakdown of low-paying awards and high-paying awards.

By further reducing the paying amount of scatter pays, we will leave the hit frequency unchanged while decreasing the payout percent. Running the simulation with awards of 2 credits for three scatter symbols, 5 credits for four scatter symbols and 250 credits for five scatter symbols makes a considerable difference to the game payout, reducing it to exactly 85 percent.

We now have a game that is mathematically viable. We can either increase the scatter pay amounts slightly or add more winning combinations to bring the payout up to our desired level. One caveat is in order, however. Just because we are paying 95 percent, holding 5 percent, and have a hit frequency of 25 percent, does not mean this game is any good. Our payouts may be distributed in a manner that is unpleasant to the player.

There is also a major omission in our game design. Although we have made the game pay and play close to what we want, we have forgotten one important item: The scatter pay symbols will initiate a second-screen bonus game, and we have not provided any awards for this game. This will increase our payout even further. Assume that we can add our bonus game awards into the game and bring our payout from 85 percent to 95 percent by this step alone. This is more a result of good luck than careful planning.

Clearly a major factor in cycle size is the distribution of symbols in order to create a proper payout and hit frequency. When we add more reels to the game, this step becomes increasingly more difficult. The mixture of scatter symbols with line pay symbols complicates matters even more and helps to explain why some games just have large cycle sizes.

A different problem with the cycle is trying to fit large payout awards. Suppose that we want to make our top award significantly larger, perhaps $1,000,000. (Here again I picture someone dressed in a while lab coat imitating Dr. Evil while saying, “One meeellion dollars!”)

Using our last simulation, changing the upper award to $1,000,000 increases the payout to 809 percent. More than 90 percent of the overall game pay is from the jackpot. This creates a significant problem in our game design and is one that cannot be easily corrected. If we remove all payouts except for the jackpot amount, we still can’t make this game work. In order to pay the $1,000,000 jackpot, we need to take in more than $1,000,000 in credits played. On our penny game, this is 100 million credits. We could increase our cycle size to more than 100 million, but that doesn’t include any regular line pays or scatter pays and makes our hit frequency just one in 100 million games. Alternatively, we could require a significant credit wager in order to be eligible for the top award. If we set our maximum wager at 1,000 credits per game ($10 wager), we only need 100,000 games to “cover” this award. This reduces the cycle size, but makes players wager a very high amount in order to be eligible for the jackpot.

This is a balancing act that requires careful attention to exactly how the game plays and how it feels to the player. Mathematically, we can incorporate a billion dollar jackpot into a penny game. However, we may require a top wager of $100 (10,000 credits) and still have a cycle size of hundreds of millions of games. And while the top award is a draw for players, it may be only a novelty attraction in our game. Take a look at the reports of the games on your floor. Do your Thrillions and MegaBucks games receive significantly more play than the rest of your games? I would suspect that they do not. However, they do appeal to certain players and almost all of your players may be willing to try these games once or twice.

We must also consider the proportion of the games that contribute to the jackpot amount compared to those that don’t. If we have a slot machine with a million-game cycle and a single, large jackpot in that cycle, doubling the cycle reduces the overall impact of that jackpot by one-half. Now there is a large jackpot every two million games. This simple step may also make a large payout feasible. It serves to reduce the burden of the jackpot on the total game math.

There is one flaw in our logic. And this flaw can have serious repercussions for your operations. This is an excellent place to stop until next month, when we’ll wrap up our study of the Vicious Cycle. Until then, see if you can find that flaw.

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