Ask the Slot Expert: A slot machine design puzzle

I was never asked to solve a puzzle during a job interview, fortunately. According to William Poundstone in How Would You Move Mount Fuji?: Microsoft's Cult of the Puzzle -- How the World's Smartest Companies Select the Most Creative Thinkers (There should be a limit on the number of sub-titles in a book's name.), one of the questions you might be asked in a Microsoft interview was: How would you move Mount Fuji?

I'm not interested in how you would move Mount Fuji. I have a different puzzle-like question to ask during your slot machine design interview with me.

First, some answers to the Fuji question. (Full disclosure: I googled the answer.) One, do nothing. The Earth is constantly rotating and revolving. Everything on it is constantly moving. Two, change your position. You will have a different perspective on the mountain's location.

Microsoft and other tech companies that asked puzzle questions weren't really interested in your solution, but rather in your thought process.

My interview question is inspired by a Mahjong tile game, a skill-based slot, I played a few years ago. The game was not Mahjong proper, but Mahjong matching tiles. If you were able to clear the board, you went to the bonus round. You had the choice of taking a guaranteed amount or choosing one of five gongs to reveal your bonus. Your gong may reveal a bonus amount below the guarantee.

Another example to set up the puzzle question. You win free spins in the bonus round. You have your choice between 30 spins at 1x, 15 spins at 2x, 10 spins at 3x, and 5 spins at 6x.

The puzzle you have to solve: How can you calculate the long-term payback of a game when the bonus round gives the player options and you don't know how the probability that a player will choose a bonus round option?

Need a hint? This isn't really a math question. There's no formula in the answer.

We can calculate the long-term payback of the reel-spinning game because we know the reel layouts, so we know the probability that a given symbol will land on a given reel. But we don't know the probabilities with which players will choose the bonus round options.

The way slot designers make it possible to calculate the long-term payback of a game that has a bonus round that gives the players options is to make it so they don't care what option the player chooses. And they do that by making the expected value of each option the same.

This way it doesn't matter which option the player chooses. The players' choices don't affect the expected value of the bonus round.

I don't remember exactly how the Mahjong game's bonus round worked. If the player's choice mattered in the gong-picking option, then the average of the values of the gongs was equal to the guarantee.

If the player's choice didn't matter -- the machine used the RNG to determine how much the gong the player picked would be worth -- then the sum of each value multiplied by the probability that it would be selected is the same as the guarantee.

This situation is like being given the option of a guarantee or spinning a wheel to get your bonus amount. Like the wheel on the Wheel of Fortune slot machines, the wheel slices are not equally like to land under the pointer.

On the machine with the choose-the-number-of-spins bonus round, I think I made the expected value of each choice the same by increasing the multiplier and decreasing the number of spins such that the product of the two was always 30. If that doesn't work, the slot designers could use different reel layouts for the options to get the expected values the same.

Designers can do things in a bonus round that wouldn't be permitted in the base game because the player has no money at risk in the bonus.

To quote Homer Simpson from The Simpsons slot machine when a player won free games: "Woohoo! I can't lose!"

If you would like to see more non-smoking areas on slot floors in Las Vegas, please sign my petition on change.org.