Ask the Slot Expert: Accounting for bonus rounds in long-term payback

Question: Interesting bit on the buffets. We always enjoyed them, however, the amount of food wasted bothers us more and more. And how much food should one eat at a meal anyway?

I remember years ago the coffee shop at the Excalibur used to have a $4.99 ham and eggs breakfast special in the wee hours. The bone-in slab of round ham covered the entire plate while the 2 eggs, hash browns and 2 slices of buttered toast all sat atop the ham. And we ate it all! Makes me shutter today just thinking about it!

As far as returning to the casino now that things are getting back to a bit more normal, the habit of staying home has some roots after a year plus and are hard to break. We used to go to a casino somewhere once every few weeks but have not returned once so far. Maybe in a couple of months we will give it a try, but no buffets even if they have piles of great sausages like the one that attacked you!

I have a question. Thinking about a slot machines programmed pay back percentage, how do bonus rounds with a variable amount to win enter the into calculation?

Say a machine's hold back is 8%, the payout is calculated to be 92% over time, and there is a bonus program or two that is randomly triggered. If the bonus win gives you one of 4 or 5 possible outcomes, say 250 coins and the range is 200 coins (or maybe zero) to 2,000 coins, the programmer could account for those variations in the payback percent by designing so many bonus wins at a specific amount. Even the "mega" wins can be accounted for in programming by adjusting their frequency.

What happens to the payback percentage when players use their free will to select icons revealing wins and can keep selecting them until a “collect” or “game over” result appears? If players were to go all the way and select every possible prize every time, or the reverse and get the “game over” message with the first pick, it could impact the payback percentage. Or is the payback percent based on some average amount a player could win in such bonus round events?

Or would there even be an impact as the bonus rounds are so few and far between (or seem like it sometimes, no?) and do not have a material impact over time?

Your thoughts?

Answer: Every possible way to win money on a slot machine has to be accounted for on the PAR sheet. Even though bonus rounds are sometimes few and far between and sometimes just a waste of time when they pay a piddling amount -- or even nothing -- overall they might account for a few percentage points of a machine's long-term payback percentage.

I used to use the term "programmed payback percentage" years ago. I stopped using it after I received questions indicating that players were getting the impression that there was some component in the program running the slot machine that caused the machine to pay back a certain percentage, similar to how an ATM is programmed to check your account balance and dispense money to you.

There is no artificial intelligence in the slot machine's code to cause it to pay back a certain amount. What the program does is choose outcomes at random from a pool of outcomes. If 5% of the outcomes in the pool are mixed bars, then in the long run 5% of the spins will result in mixed bars. I switched to "long-term payback percentage" because the only thing guiding the payback percentage is Random Sampling with Replacement.

Let's look at an old, pre-bonus round slot machine. We calculate its long-term payback by looking at each winning combination and multiplying the number of ways to make it by how much it is worth. Add up all the products and divide by the total number of combinations and the amount bet to get the long-term payback percentage.

We can apply this same concept to a winning combination, in addition to the machine as a whole. We almost always know how much a winning combination will pay, but we don't for a bonus round because it's another random game.

In your first example, players are just along for the ride and wait to see how much they won in the bonus. The program in the slot machine randomly chooses the amount.

Consider the wheel on Wheel of Fortune. Even though the size of each slice is the same, the amounts are not equally likely to be selected. The program uses a number from the RNG to decide your fate. Each amount has one or more numbers assigned to it. The lower wheel amounts have more numbers mapped to them than the larger amounts. The number of numbers mapped to a slice divided by the total number of numbers gives us the probability of getting that slice.

We know the probability of hitting each amount. We can multiply the value of a slice times the probability of getting that slice and add up the products to get the Expected Value (EV) of the Wheel of Fortune bonus. And that's the number put on the PAR sheet.

It's like the old average of 3.2 children per family. No family has 3.2 children. But if you add up the number of children and divide by the number of families, you get an average of 3.2 children per family.

If may not be possible to win the exact amount of the number on the PAR sheet, but that's how much the bonus will pay, on the average.

We can go many levels deep. I've been playing a slot based on The Hobbit movies. One bonus round can trigger another bonus round. We can figure out the EV of the second bonus event, so we know how much it's worth, on the average, when it is triggered. The probability of triggering the second bonus and its EV figure into the EV of the main bonus round.

Your second example adds player interaction into the bonus. Free will. Sometimes your choices make a difference. Sometimes they don't.

There are many Chinese-themed games with a coin-picking bonus round to win one of four progressive amounts. Each coin reveals one of the progressives. Match three to win the corresponding progressive. Match three, four progressives, twelve coins on the screen -- it seems like you have a 25% chance of hitting each progressive.

Not so fast. The machine has already randomly chosen which progressive you will win. That's the only progressive that has three coins on the screen. In fact, it has six. The remaining six coins correspond to the un-hittable progressives, two coins each.

This situation is really the same as the "machine reveals bonus" situation above. The random selection process is configured such that each progressive will be chosen with a certain probability, so we can calculate the EV of the bonus.

Sometimes your choices do matter. I sometimes play a Quick Hit game with a free games bonus round. When the bonus round is triggered, you determine how many free spins you will get and your multiplier for those spins by choosing squares from a grid. Reveal three matching tiles to set your spins and multiplier.

We can calculate the EV of a spin, so we can calculate the EV of each spins/multiplier combination. We can calculate the probability that a player will choose a particular spins/multiplier combination based on how many tiles it is assigned to, so we can calculate the EV of the bonus round as a whole -- even though we don't know how many spins or what multiplier the player will get or what will be won on the free spins.

How about the pick'em game that you described? Consider a pick'em game with five tiles. One is Collect, another is Collect All, and the remaining three are worth 5, 10, and 15.

We can make a decision tree. At the top is the start of the game. We have to make a decision -- that is, choose a tile. There's a 20% chance we choose Collect and win nothing. There's also a 20% chance we choose Collect All and win 20. There's also a 20% chance we choose 5 and a 20% chance we choose 10 and a 20% chance we choose 15.

Say we choose 5. We have to make another decision. There are four tiles remaining. We now have a 25% chance of choosing Collect and winning 5. A 25% chance of choosing Collect All and winning 20. Plus a 25% chance we choose 10 and a 25% chance we choose 15.

We don't know which path a player will choose. We can work backwards once we reach a conclusion (Collect or Collect All) to figure out the EV of each path in a decision until we get to the top and know the EV when we choose 5 or 10 or 15. Given those EVs, we can now calculate the EV of the entire bonus game.

I hope this example illustrates how to use EV to deal with uncertainty. We don't know how any particular bonus round will play out, but if we know the probabilities of getting certain amounts, we can calculate the EV of the bonus round. And that's how much the bonus round will pay, on the average.

To bring this around full circle, we don't know the outcome of any individual spin. But we do know the probabilities of winning each possible amount, so we can calculate the long-term payback of a machine.

You had it exactly right. The payback percentage is based on the average amount a player can win in the bonus.

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