High School How Much Would You Pay for a Game with Negative Expected Value?

[batmantrippin]

Hi there, I was recently having a discussion with people and they were adamant they would always pay for a game with negative EV.

The example was "Would you pay 1p for £200 free play of a 97% RTP slot machine with 35x wagering?"
So you pay 1p and get given £200 risk free to play on a slot machine with a 97% return to player (for every £1 you stake you expect to get 97p back with an infinite amount of spins).

After you wager 35x the £200 = £7000 through the machine you get to keep anything that's left (if you haven't already bust out.)

The deal with this risk free £200 is that even if you blow the lot you're only down the 1p you paid to play the game.

Clearly the £200 has an EV of nothing by the time you've finished wagering, but this doesn't mean sometimes you won't get lucky and win something.

My question is should you ever pay for this game? And if so, how much should you pay? If you'll pay 1p should you pay 2? 3? How about £100?

How do you value something with negative EV if it is given to you for free?

Thanks!

 

[Hornbein]

I studied this but was never any good at it. You do something called "stochastic integration." The easy way to do it is by computer simulation.

I bet that the expected value is positive. I'd play if the expectation is positive (and I had nothing better to do.).

You seem to be confusing expected value with the value with maximum likelihood. They are not the same.

 

[batmantrippin]

I studied this but was never any good at it. You do something called "stochastic integration." The easy way to do it is by computer simulation.

I bet that the expected value is positive. I'd play if the expectation is positive (and I had nothing better to do.).

You seem to be confusing expected value with the value with maximum likelihood. They are not the same.

Thanks for the response! So you're saying if you used stochastic integration you could work out exactly the amount you would pay to play this game? Or would you need more information about the game other than its Return To Player?

It's obvious that if you played through the £7000 with your own money at 97% RTP you'd expect to be £210 down. But don't know how to figure out how much you'd pay for a free shot at it. It's possible that it's a whole can of worms despite it sounding quite simple at first. (5 years since finishing my maths degree is just too long ago for my head)

 

[fresh_42]

Economists study such questions as a part of decision theory. They measure the risk by the standard abbreviation (or variance) and try to figure out the personal preferences of a decider depending on expectation value and variance. This indicates that different persons will have different answers to this question, depending on which extend someone is risk aversive.

This model is also used in risk minimization processes like the construction of portfolios. Here you will have additionally to take the covariances into account.

 

[batmantrippin]

Economists study such questions as a part of decision theory. They measure the risk by the standard abbreviation (or variance) and try to figure out the personal preferences of a decider depending on expectation value and variance. This indicates that different persons will have different answers to this question, depending on which extend someone is risk aversive.

This model is also used in risk minimization processes like the construction of portfolios. Here you will have additionally to take the covariances into account.

So ignoring any human's risk aversion, we need more information about the slot machines payout structure (not just the return to player) to figure out how much you would pay to play in my example?

 

[fresh_42]

So ignoring any human's risk aversion, we need more information about the slot machines payout structure (not just the return to player) to figure out how much you would pay to play in my example?

In the model mentioned above risk neutrality would mean that only the expectation value counts. If it is higher than the payment, one would play, otherwise not. But in reality, people estimate the chances they could fail to realize the expectation value and this is the variance.
A good example is lotto. People play lotto although their chances to win are more or less zero. On the other hand they daily drive their cars although there is a good chance for them to meet with an accident or even been killed.

 

[mfb]

Quite sure this game has a positive expectation value, at least for realistic unspecified game details. You have a high chance to end up with zero, but even a 1 % chance to end up with more than £1 would make it positive. We would need the distribution of expected outcomes. Everything else is a matter of your preference - and the value you assign to the entertainment or work of gambling.

 

[batmantrippin]

Quite sure this game has a positive expectation value, at least for realistic unspecified game details. You have a high chance to end up with zero, but even a 1 % chance to end up with more than £1 would make it positive. We would need the distribution of expected outcomes. Everything else is a matter of your preference - and the value you assign to the entertainment or work of gambling.

Thank you! 1p was the extreme example I was giving to try and see if it was even worth it for that. In the actual situation it could end up costing more like £20 to play which would start to make it a lot less likely to be worthwhile.

Sadly I don't have any data on the variance of the slot machine.

 

[Hornbein]

Thank you! 1p was the extreme example I was giving to try and see if it was even worth it for that. In the actual situation it could end up costing more like £20 to play which would start to make it a lot less likely to be worthwhile.

Sadly I don't have any data on the variance of the slot machine.

The higher the variance, the higher the payout will be, in this case anyway.

 

[fresh_42]

Sadly I don't have any data on the variance of the slot machine.

I just googled "variance of slot machines" and found a colourful microcosmos. One site had even examples of high, medium and low variance slot machines. I didn't know they had names! Seemingly this isn't the first time people think about risk at slot machines. One site has been named slotjunkies - felicitous.

 

[mfb]

Let's make a specific but unrealistic example: the machine only accepts games of £200 each, and gives back £200 with 97% probability and £0 with 0% probability. No one would gamble on such a machine - unless we get the money from somewhere else, like here. The probability to keep the money over 35 rounds is 0.97³⁵=34.4%. Our expectation value of money after playing is £69. It is not -10 as we stop playing most of the time.

We can push the expectation value as close to 0.97*£200 = £194 as we want, by making the game lotto-like: make sure we nearly always stop after a single game, but win a huge amount if we win in the first game.