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

7 July 2021

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?

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.

Here are the latest figures from https://www.cdc.gov/covid-data-tracker/#cases_totalcases.

Totals Weekly Increases
US NV US NV
Date Cases  Deaths  Cases  Deaths  Cases  Deaths  Cases  Deaths
07/06   33,545,316   603,181   334,763   5,697   75,104   1,373   2,234   27
06/29   33,470,212   601,808   332,529   5,670   87,507   2,057   3,020   24
06/22   33,382,705   599,751   329,509   5,646   75,420   2,157   1,930   22
06/15   33,303,285   597,594   327,579   5,624   95,797   2,293   1,560   17
06/08   33,207,488   595,301   326,019   5,607   114,250   3,762   2,271   21
06/01   33,039,238   591,539   323,748   5,586   123,333   3,709   991   27
05/25   32,969,905   587,830   322,757   5,559   174,125   4,234   1,676   26
05/18   32,795,780   583,596   321,081   5,533   223,966   4,230   2,301   27
05/11   32,571,814   579,366   318,780   5,506   303,856   4,687   2,541   33
05/04   32,267,958   574,679   316,239   5,473   343,348   4,908   2,559   40
04/27   31,924,610   569,771   313,680   5,433   383,163   4,958   2,747   65
04/20   31,541,447   564,813   310,933   5,368   464,556   5,072   2,590   36
04/13   31,076,891   559,741   308,343   5,332   480,061   5,321   2,986   57
04/06   30,596,830   554,420   305,357   5,275   448,935   7,124   3,084   38
03/30   30,147,895   547,296   302,273   5,237   439,510   6,793   939   63
03/23   29,708,385   540,503   301,334   5,174   388,928   7,446   1,863   53
03/16   29,319,457   533,057   299,471   5,121   381,695   8,362   3,078   81
03/09   28,937,762   524,695   296,393   5,040   480,902   11,573   2,413   83
03/02   28,456,860   513,122   293,980   4,957   463,356   14,129   2,835   75
02/23   27,993,504   498,993   291,145   4,882   451,083   13,923   2,406   162
02/16   27,542,421   485,070   288,739   4,720   602,906   21,411   4,149   198
02/09   26,939,515   463,659   284,590   4,522   779,305   21,828   5,444   244
02/02   26,160,210   441,831   279,146   4,278   1,007,777   22,004   7,249   249
01/26   25,152,433   419,827   271,897   4,029   1,312,565   23,385   10,324   250
01/19   23,839,868   396,442   261,573   3,779   1,317,119   21,318   11,324   279
01/12   22,522,749   375,124   250,249   3,500   1,790,345   22,660   17,217   294
01/05    20,732,404   352,464   233,032   3,206   1,499,561   18,435   14,655   233
12/29   19,232,843   334,029   218,377   2,973   1,258,540   15,460   12,493   186
12/22   17,974,303   318,569   205,884   2,787   1,656,411   18,537   16,472   239
12/15   16,317,892   300,032   189,412   2,548   1,494,763   17,247   18,825   229
12/08   14,823,129   282,785   170,587   2,319   1,375,502   15,483   18,418   175
12/01   13,447,627   267,302   152,169   2,144   1,114,175   10,286   15,942   121
11/24   12,333,452   257,016   136,227   2,023   1,197,199   10,784   14,130   106
11/17   11,136,253   246,232   122,097   1,917   1,099,790   8,501   11,115   65
11/10   10,036,463   237,731   110,982   1,852   767,645   6,838   8,868   68
11/03   9,268,818   230,893   102,114   1,784   588,207   5,809   5,936   35
10/27   8,680,611   225,084   96,178   1,749   492,026   5,585   5,238   (10)
10/20   8,188,585   219,499   90,940   1,759   401,037   5,053   4,501   48
10/13   7,787,548   214,446   86,439   1,711   351,270   4,886   3,910   48
10/06   7,436,278   209,560   82,529   1,663   306,965   4,962   3,232   36
09/29   7,129,313   204,598   79,297   1,627   303,616   5,136   3,058   54
09/22   6,825,697   199,462   76,239   1,573   288,070   5,370   2,196   82
09/15   6,537,627   194,092   72,043   1,491   250,265   5,404   1,825   65
09/08   6,287,362   188,688   72,218   1,426   282,919   5,638   2,734   92
09/01   6,004,443   183,050   69,484   1,334   251,790   5,291   3,237   104
08/25   5,752,653   177,759   66,247   1,230   330,411   7,889   4,076   125
08/18   5,422,242   169,870   62,171   1,105   358,071   7,463   4,973   114
08/11   5,064,171   162,407   57,198   991   365,353   7,203   5,776   117
08/04   4,698,818   155,204   51,422   874   418,683   7,532   7,367   109
07/28   4,280,135   147,672   44,055   764   460,996   7,042   7,130   91
07/21   3,819,139  140,630  36,195  674  463,682  5,395  8,181  57
07/14   3,355,457   135,235   28,744   617   422,861   5,102   5,607   57
07/07   2,932,596   130,133   23,137   560   351,367   3,394   5,006   24
06/30   2,581,229   126,739   18,131   536   278,941   6,406   4,367   26
06/23   2,302,288   120,333   13,764   510
John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots