## Slot machines

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parallax
Posts: 157
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### Slot machines

A casino has two types of slot machines available: a nickel machine which takes 5 cent bets and pays out 40 cents every 1 in 10 games, and a quarter machine which takes 25 cent bets and pays out 225 cents every 1 in 10 games. As a casino owner, which machine should you install more of? As a player, which one should you play at?
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ConMan
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### Re: Slot machines

As the casino owner, you can assume that there are a limitless number of punters available, and so the only thing stopping you from taking all their money is the speed at which they can lose it. Hence, you're looking to minimise the payoff received per game played.
Spoiler:
Machine 1: E(payout / game) = 9/10*(-5c) + 1/10*(40c-5c) = -1 c
Machine 2: E(payout / game) = 9/10*(-25c) + 1/10*(225c-25c) = -2.5 c

So the quarter machine is better for you.

As a player, you can assume that you have a limited amount of cash, and possibly a limited amount of time. There are a few strategies you may wish to pursue:

Strategy 1: Spend only your starting money (i.e. keep starting money and wins separate), meaning you want to maximise the payoff for the amount of money you put in.
Spoiler:
Machine 1: E(payout / dollar) = E(payout / game) * (games / dollar) = -1 c * 20 = -20 c
Machine 2: E(payout / dollar) = E(payout / game) * (games / dollar) = -2.5 c * 4 = -10 c

In this case, the quarter machine is also better for you.

Strategy 2: Again spending only your starting money, you don't care about wins and you just want to play for as long as possible, so of course you play the nickel machine.

Strategy 3: Feeding your wins back into the machine, you just want to play for as long as possible until your money runs out.
Strategy 4: Feeding your wins back into the machine, you want to play until you either win a certain amount, or your money runs out.

The last two are bounded Wiener stochastic processes, so getting expectations is a little tricky and I need to double check whether there's even a way to do it.

Edit: Silly me, it's not a Wiener process, but it is stochastic.
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JBJ
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### Re: Slot machines

Need a bit of clarification on the payouts. Does the payout include the original bet? i.e. on the Nickel slots, I put in \$.05. Does the slot machine pay out \$.40 or \$.45 ?

Spoiler:
If the payout does not include the original bet, i.e. I put in \$.05, I get back \$.40, then as the casino owner, the quarter machine is best.
Over 100,000 plays the accumulated income for the casino is around \$250,000 +/- \$30,000 vs. \$100,000 for the nickel slot +/- \$6,000.

However, if the payout includes the original bet, i.e. I put in \$.05, I get back \$.45, then the quarter machine is right out...
If the payout percentage is even slightly about 10%, even by a few 1/100th's of a percent, the quarter machine loses money for the casino (and is obviously the best choice for the player) to the tune of around \$10,000 for every 5/100th of percent.

In this case though, the nickel slot still does fairly well, giving the house income of around \$50,000 per 100,000 plays.
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Goldstein
Posts: 985
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### Re: Slot machines

parallax wrote:As a player, which one should you play at?

If I'm not required to play any games, neither.
If I'm required to play one game, the nickel machine.
If I'm required to play until my original winnings run out, the quarter machine.
If I'm required to play until all my money runs out, either.

I am, of course, assuming the goal of maximising my expected money.

Regarding the casino's best setup, it seems at the moment to be a question of psychology and some rules regarding punters' behaviours could tighten this up and make possible a logical solution.
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fyjham
Posts: 240
Joined: Wed Nov 21, 2007 1:16 am UTC

### Re: Slot machines

I'll start with the assumption that it takes an average of "T" seconds to play a game on both machines - That is to say there is a fixed time it takes the person to put a coin in and pull the lever and it doesn't change for each machine.

Spoiler:
So to get us a rough baseline, the average return from 5c is 4c, average return from 25c is 22.5c.

For the owner I'm going to assume there's a constant stream of gamblers pouring in and a lineup to use his awesome machine (EG: If he robs you dry you stand up and give the next person the seat), and he cares only about his average return over D seconds per machine, which should be his overall goal. Assuming D is long enough to average out the returns (I hope he stays in business over a few days ):
Spoiler:
With the 5c machine this would be D/T * 1c, 25c machine it would be D/T*2.5c, so clearly the optimal solution for the owner is the 25c machine.

For you as an individual we'll take a different approach and look at X games played.
Spoiler:
Maximum return from X games:
5c machine: 40 * X - 5 * X = 35X, or 700% ROI
25c machine: 225 * X - 25 * X = 200X, or 800% ROI

Average return:
5c: 4X, or 80% ROI
25c: 22.5X, or 90% ROI

Maximum loss:
5c: 5X
25c: 25X

So the 25c game has higher maximum return and higher average return, but also higher loss chance. If you simply play 1/5th as many games though you get the same maximum loss, higher average and maximum results and also by lowering the number of games played your results will be further from the average (Which could turn out good or bad, but quite frankly if you play for the average you'll always lose so...)

In summary: Everyone loves a 25c machine assuming there's an infinite queue of plebs with money and no psychology is involved, meaning only if your pokies aren't in reality!

crzftx
Posts: 371
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Location: Rockford, IL

### Re: Slot machines

That makes a bit of sense, but not completely.
Whatever is best for the casino is worst for the player, right?
People should not look at this in terms of net gain for the casino and percent loss per round for the player, at least I don't think so.
I think it should be net gain for the casino and net loss for the player.
Spoiler:
The average player loses 1c every round on the 5c machine. The average player will also lose 2.5c every round on the 25c machine. The more the player loses, the more the casino gains. The 25c machine is obviously better for the casino. The 5c machine is better for the player I think. Every player comes with a set amount of cash, and playing on the 5c machine allows for much more play time, since you only lose 1c each time.
For example, say I come with 100 pennies. After 10 rounds, I either have 90c or 75c; the first choice seems clearly better. That is the choice on the nickel machine. Sure, I lost a larger percent of my bet each round, but I lost less overall.

fyjham
Posts: 240
Joined: Wed Nov 21, 2007 1:16 am UTC

### Re: Slot machines

Well crzftx, you're working around the assumption a player will play X rounds, what if they put in X cents? EG: You come in with your 100c and put it all in each machine. The 5c machine gives you back 80c and the other 90c. It all hinges on the playstyle of the player and what the player classes as "good" (EG: Maximum return on investment, maximum play time, maximum winning results...).

If the player doesn't care how quickly he plays and just wants maximum potential return and the casino doesn't mind higher player turnaround then they could easily both prefer the same one, but working off your assumptions of what each person wants (Which is perfectly fair given they weren't set explicitly) then I'd agree with you.

The big difference I think is that I assumed the casino's goal isn't net profit per game, but net profit over a day, whereas the individual punter probably cares more about how much he gets back for his dollar, but it is fair to say that some people would prefer to get a longer game out of the machine

quintopia
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### Re: Slot machines

Some people like to putter along and some like high stakes. They play for amusement, not for gain. The casino wants to maximize profit. Clearly, the casino's best choice is to rip out a bar stool to make room for both machines. Then they can have two players at once, and entice both sorts of psychologies.

qinwamascot
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### Re: Slot machines

There aren't enough specifications to solve this. For instance, if I'm a casino owner, I might install more 5 cent machines than 25 cent ones. The people on the 5 cent machines will be there longer assuming both games take the same amount of time. So I need more to accommodate everyone who wants to play 5 cent games. The 25 cent ones will end quickly, so fewer are necessary. It depends on quite a few factors, so it's not an easy question without all the information.
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crzftx
Posts: 371
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Location: Rockford, IL

### Re: Slot machines

fyjham wrote:Well crzftx, you're working around the assumption a player will play X rounds, what if they put in X cents? EG: You come in with your 100c and put it all in each machine. The 5c machine gives you back 80c and the other 90c. It all hinges on the playstyle of the player and what the player classes as "good" (EG: Maximum return on investment, maximum play time, maximum winning results...).

If the player doesn't care how quickly he plays and just wants maximum potential return and the casino doesn't mind higher player turnaround then they could easily both prefer the same one, but working off your assumptions of what each person wants (Which is perfectly fair given they weren't set explicitly) then I'd agree with you.

The big difference I think is that I assumed the casino's goal isn't net profit per game, but net profit over a day, whereas the individual punter probably cares more about how much he gets back for his dollar, but it is fair to say that some people would prefer to get a longer game out of the machine

Sure that's true. I'm not a slots guy, no strategy in that, so I don't know how people play them. I guess if I were to play a machine, I'd play till either I lost up what I'm willing to bet or gained from my original amount. The best machine, for me, would be the one that earns me money more often.

Spoiler:
After brute-forcing which machine would be best for me, it looks like the nickel machine if I'm betting less than \$6, else the quarter machine.
100000 trials
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