Casino Enterprise Management

Monday. 8:32 p.m. I was sitting at my desk once again, leaning back in my chair, relaxing. I knew she would arrive soon. Jane already had the first part of the solution to her mystery progressives mystery and seemed happy with the results. Little did she know, it was just the tip of the iceberg. I had only explained the simple part of mystery progressives: payouts guaranteed between a certain range. The machine would pick what the actual payout amount will be. A separate counter would start at the base amount and then each wager would contribute a percentage toward this target. The lucky player who happened to increase that separate counter to the point that it matched the target would win. I wonder if you had to rub the screen at the same time?  But there was more, much more. Yes, sometimes I even surprise myself. I’d worked my way to the bottom of this case and found out all the answers. Of course, Mike Mastropietro at WMS had a big part in it, but in the end, I’d found out the information she wanted.

A knock came at the door just before 9 p.m., and she walked into my office. I swear, this girl got prettier every time I saw her. I wondered if she had dressed up and done her hair because she was coming to meet me. I figured it was because she’d just come from work. “I just came from work,” she said. “I didn’t have a chance to change.” One thing was for sure, she had a knack for answering these little questions of mine.

Jane sat down, pulling the chair close to my desk. “I went over your figures today,” she said. “They look great. You really got to the bottom of this. My boss, Shannon, is very impressed. But you mentioned there was more?”

“Oh, yeah, doll. Much more. Wait till you see what I’ve come up with.”

“I can hardly wait!” she squealed.

I opened the envelope once again, spilling the contents onto the desk. Jane opened her portfolio, scanning the information and figures I’d given her the night before.

“Right, babe, here’s where we pick up,” I told her. “You see, there’s also mystery jackpots that don’t pay within a certain range. The player can be sitting at the machine, rubbing the glass when, Bam! She wins some mystery amount for some mysterious reason.”

“Right,” she replied. “It seems the same as the example you showed me yesterday, except that there’s no upper limit or guarantee of a specific range of payout.”

“That’s right. You see, these jackpots are very similar. They have a base starting point and then increase by a percentage of total coin-in. They could go for hours, days or even weeks without paying.”

“Yes, that’s right. How is it done? And more importantly, how can I calculate how much we’re contributing to these mystery pays?” she asked.

“Let’s look at the first part of your question — how they are awarded. There are two methods,” I explained. “One uses so-called ‘invisible’ symbols. These are specific symbols that generate the win. Here’s a picture of a slot game. If the player gets five of the same symbol, in this case a Trident, he wins 10,000 credits.”

“Yes,” she nodded. “The symbols have to be on a payline, from left to right, and he gets 10,000 credits multiplied by the number of credits wagered per line.”

“And if he gets four symbols, he wins 1,000 credits. Three gives him 100, and in this case, two symbols give five credits.”

“Yes. That’s just a standard payout for a video slot game.”

“And that’s exactly how the mystery payout works. The amount and the number of symbols required to pay the amount is based upon the probability of the symbols occurring. For example, suppose the mystery payout is 10,000 credits to start. You’ll probably need five symbols on a payline to award this. Or the game designers might make a mystery payout starting at 500 credits. You might need four matching symbols for this, or perhaps three. In this regard, it’s the same as a regular payout.”

“OK, I’m with you so far. I think,” she trailed off.

“But what is a ‘mystery’ is how it is awarded,” I continued. “It really isn’t much of a mystery. It is awarded exactly like a regular payout, except the player doesn’t readily recognize a symbol. The matching symbols, instead of Trident–Trident–Trident–Trident–Trident, might be Trident–Dolphin–Pearl–Bonus–Mermaid. Just as you can determine the hit frequency of other symbols occurring on a payline, based upon how many symbols exist on each reel, you can do the same with these so-called mystery symbols. You can determine how many Tridents are on the first reel, how many Dolphins are on the second, how many Pearls are on the third, etc. From this you know how frequently they occur and, consequently, the probability of this happening. This can be put into the PAR sheet just like a regular award. In fact, it is just a regular award. It’s just that it doesn’t use the matching symbols in the traditional sense. You could put the requisite symbols on the payglass and then it wouldn’t be a mystery to anyone.”

“Requisite? Isn’t that a big word for someone like you?” she asked.

Someone like me, I thought. Gee, thanks.

“I surprise myself sometimes, doll.” I said. “Just because I’m a gumshoe doesn’t mean I don’t have a good vocabulary. Anyway, back to your mystery. Selecting the payout this way is simple. It’s just that the players won’t recognize the symbols or remember them for next time. People look for, and recognize, patterns. The game designers are using a pattern, but not one that we can quickly identify.”

“So the math calculation is pretty easy,” she said. “It’s just a standard payout with an incremental progressive amount added to it.”

“Yes. But there is another way of doing this, one that makes the math a little bit more complicated. And that’s to have an entirely separate probability calculation for the award.”

“How is that done?”

“The award is assigned a probability,” I explained. “Maybe there’s a 1-in-10,000 chance of having this mystery pay awarded. In that regard, it’s like a regular payout. Perhaps four Trident symbols have a 1-in-10,000 chance of being awarded, too. But in our case, it’s separate from the game. At some point during the game, a random number is selected. It’s easy to think of it this way: We pick a number from 1 to 10,000. If the number is 1, you win the award. If it’s not, you don’t.”

“It has to be the number 1?” she asked.

“No, it could be any number, as long as we determine which number we’re looking for,” I explained. “And it can only be one of those numbers. Perhaps it has to be 10,000 or the number 8,192. It doesn’t matter. What matters is that it’s one number out of 10,000. That gives us the probability of 0.0001.”

“How did you calculate that?” she asked.

“Just divide 1 by 10,000. That gives us 0.0001 probability. Multiply it by 100 to get a percent. That means that you have a 0.01 percent chance of getting that award.”

“Oh, yeah, I see.” she said, with a puzzled look on her face. “But in that case, how do we calculate the overall payout? We know how often it happens, but how do we relate this to the overall game math?”

“Selecting this award with this method means that it’s fundamentally separate from the game math. We need a separate calculation to do this. Now, before I show you the formula, you have to realize that there are two ways this can be done.”

“Why two ways?” she asked.

“There are two ways that we can pay the mystery award. In the first case, the mystery pay is set and only changes as the progressive factor increases it. If the player wins the mystery jackpot, he is awarded the current value of the jackpot.”

“Isn’t that just common sense?”

“Yes, except for one factor,” I told her. “If I wager 200 credits each game and you wager 20 credits, should we both have the same chance of winning the jackpot? If I buy 10 lottery tickets, wouldn’t I have a greater chance of winning than you if you only buy one? The more you bet, the better chance of winning you have.”

“Oh, I get it,” she said. “How do you make it so that you have a better chance of winning ... Yeah, that complicates things, doesn’t it?”

“It just means we need to take that into account in our formula. But it does alter the math,” I said.
“Yes, it would have to. What’s the second way of winning?”

“In the second case, you don’t have any greater chance of winning the jackpot if you wager more. If you wager one credit and I wager 1,000 credits, we each have the same chance of winning.”

“That doesn’t seem fair.”

“It isn’t. But in this case, your win is a factor of your wager,” I explained. “We each have the same chance of winning, but I will win 1,000 times the amount that you will since I wagered 1,000 times more. Think of it like a three-coin multiplier. If you wager one credit and get three single bars, you win 10 credits. If I wager three coins and get three single bars, I win 30 credits. The second mystery jackpot calculation works just like that.”

“Oh, I get it,” she said. “So if we did a three-coin stepper using the theory of the first jackpot, we would each win 10 credits if we got three single bars. But because you’re wagering three credits at a time, you’re three times as likely to get the award. That evens things out. You wager more for a better chance at the same jackpot, or else you wager more at the same chance but win a larger jackpot.”

“Exactly!”

“I think my first impression of you might have been wrong,” she told me.

“Your first impression?” I asked.

“Oh, was that my out-loud voice? Um … What’s the formula for the first case?”

“In the first case, we’ve established that your chance of winning depends on the amount you wager. The probability, therefore, is a factor of your wager. This means that we have some pre-determined probability of winning the jackpot. We have to factor in the amount of credits you wagered. Here’s the formula: e.v. = [(j/bt)*p(base)*(bt/base)]+I”

“OK. What’s that in English?”

“‘j’ is the jackpot amount, the starting jackpot. Let’s do an example and say that the jackpot is $250. ‘b’ is your wager. Now, we can either express ‘b’ in credits or dollars. It doesn’t matter. But whatever we use, everything has to be the same — either all dollar amounts or all credit amounts.”

“I see. We would need to know the denomination of the machine then.”

“Correct. ‘p(base)’ is the probability of the jackpot occurring and is our base probability before we factor in the wager. Now, this is an important consideration. We could have the probability expressed as a single credit wager or the maximum credit wager. However, if we use single credit, we might have a problem. Since we factor in the wager, it is possible that we could wager enough that the probability would become greater than one. That’s not good.”

“Wouldn’t that mean that we would always win?”

“Yes, and that would be a mathematical error. It’s easier to calculate the probability based on the maximum wager, and if you bet less, the probability is reduced accordingly. Now, to factor this in, you take the amount you wagered and divide it by the wager that the probability is based upon.”

“I think you’re starting to lose me,” she said.

“Let’s work through our example, and it will become clear,” I said.

“The base jackpot is $250. Let’s convert that to credits. Assume it’s a nickel machine, so $250 multiplied by $0.05 is 5,000 credits. The jackpot starts at 5,000 credits and increases progressively.”

“Right.”

“Now, let’s say this is a 20-line game. We bet one credit per line, so our bet is 20 credits. ‘j / b’ is 5,000 / 20, which is 250. Next we need to know the base probability. Using our example earlier, our probability is 0.0001. This would be for maximum coins wagered. Let’s assume you can wager a maximum of 400 coins on this machine. We then take the wager and divide it by the maximum wager, 20 / 400, which is 0.05. Now, we multiply ‘j / b’, 250, times the probability, 0.0001. That gives us 0.025. This is multiplied by ‘wager / bet’, which is 0.05, giving us an answer of 0.00125. That’s our contribution. To show this as a percent, we multiply by 100 to give us 0.125 percent. The mystery jackpot contributes 0.125 percent of the total machine payout. Now we just add our progressive increment, say 1 percent, and the total is 1.125 percent.”

“OK. What if we had wagered 300 credits. What would our expected value be?” she asked.

“1.125 percent — the same. We pay the same amount out no matter what. We just adjust the probability. Here, look at this: 5,000 / 300 * 0.0001 * (300 / 400) = 0.00125. Multiply by 100, add 1 and voila! 1.125 percent.”

“Wow! Like magic!” she mused.

“It’s all about slot machines, math and magic, doll! No mater what we wager, the value is the same. You always win the same amount, but because you’re wagering more, you have to adjust this based upon your wager. This little formula does that.”

“You really are smart!” she quipped.

I didn’t mention Mike Mastropietro’s name. I figured it was best if she thought I was the smart one.

“OK, now you have to show me the second one,” she said. “If you win twice as much, how can the ‘e.v.’ be the same? I understand it must, because you’re paying more but having the same probability. How would you calculate this, though?”

“You use a slightly different formula that doesn’t adjust probability, but does adjust the win,” I explained. “The formula is: e.v.= ( [ j * (wager/base) ] /wager ) *p + I. We make the base amount the minimum wager, since every award is a multiple of this amount.

“Let’s step through this using the same figures we just used. Now, we don’t have to have this mystery pay work as a progressive. If it’s not a progressive, ‘I’ is 0, since we’re not adding anything.  In this case it would just be a straight mystery-pay.

“‘j’ is 5,000 credits, wager is 20 credits, maximum wager is 400 credits, and the probability is 0.0001 — ( [ 5000 * ( 20 / 20 ) ] / 20) * 0.0001 = 0.025. Now, if we wager 300 credits, our formula is: ( [5000 * ( 300 / 20 ) ] / 300 ) * 0.0001 = 0.025. In the first case, the player wins a specific amount no matter what the wager. The more he or she bets, the greater the probability of winning. In the second case, he or she has the same probability no matter what the wager. However, the more he or she bets, the more he or she wins. It all works out — it’s give and take, really. Do you want to win more or have a better chance of winning?”

“So, let’s see if I understand this,” she said. “In the first case, the mystery progressive award is the same whether you bet one credit or 1,000. However, the more you bet, the greater your chances are of being awarded the progressive jackpot. To do this we factor the probability based upon what you wager. If you have a 1-in-10,000 probability of winning the mystery progressive wagering 1,000 credits, you’d have 1,000 times less chance if you wager one credit, which would be 1 in 10,000,000. This way, the expected value is always the same. Think of it as you have a certain chance of winning if you bet one credit. If you bet 100 credits, you’re just buying 100 of those chances of winning.”

“Yes, you can think of it that way,” I replied. “It’s like buying a ticket with each coin wagered. One coin gives you one ticket. Two hundred coins gives you 200 tickets. But no matter how many tickets you have, you only win on one ticket, so you only win the specific award.”

“And in the second case, you have the same chance no matter what,” she said. “You get one ticket for each game played, if we think of the chances of winning in terms of buying a ticket. But you’re putting a certain amount of money down on the ticket. If you put down one coin, you get a certain amount of jackpot award. If you put down 10 coins, you get 10 times the prize.”

“Yes, that’s a valid analogy,” I said.

“And the jackpot amount starts at a predetermined value. If there’s a progressive, we add the amount of the progressive. Since these payouts are independent of the game, we have a separate calculation. The game can pick a random number and see if you won. If the odds of winning are one in 12,000, it picks a number from 1 to 12,000. If it’s the specific pre-determined number, you win the jackpot.”

“Exactly! You have it!” I told her.  “Any time you need a mystery solved, I’m your man.”

With that, Jane stood up and left my office. When the door closed behind her, I knew I’d never see her again. Such is the way with pretty women and me. But she left happy so, in the end, I guess that’s all that matters. Besides, I was about to start work on a new case and didn’t need any distractions. This time it wasn’t a dame who needed my help. It was a man named Jack. Last name, Daniels ...

John Wilson is the 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@icsgaming.com.