Button A or button B

[Lord Antares]

Suppose you are faced with a mysterious machine that has 2 buttons on it.

 

Button A has a 100% chance of giving you $100

 

Button B has an X% chance of giving you $1.000.000

 

What is the minimum value of X that would make you choose button B?

 

 

I can hardly imagine anyone would go for the mathematical solution. For example, if you gave A = 100% for $100, and button B = 50% for $202 to a non-pragmatic math machine, it would choose B, while around 100% of humans would choose B.

 

P.S. Obviously, it is implied you can only press the button once, so no B spamming until it rains money. (Or A spamming, depending on the value of x )

[Sensei]

P.S. Obviously, it is implied you can only press the button once, so no B spamming until it rains money. (Or A spamming, depending on the value of x )

That was my the first thought

Hit it 10k times could take 3h 1s per press.

 

What is the minimum value of X that would make you choose button B?

It's probably question how much somebody is desperated.

[Lord Antares]

Not neccessarily. Simply put, everyone would choose B if X = 50%. Everyone would choose B if X = 20%. Just find the value of X below which you would press button A and above which you would press button B.

[Sensei]

I don't think so billionaire, multi-millionaire, or millionaire would pick up $100 even if X would be very small, as these $100 would be meaningless for them.

[Prometheus]

So we want the expectation of winning B to be greater than £100, which occurs when the probability of winning the million is one in ten thousand.

 

This assumes the value of money is linear (i.e. that $1,000,000 is really 10,000 times more valuable than $100) - i don't think it is, but that's the human factor.

 

We should also mention the variance of the game as button A has the very desirable property of having none. This is why even at 1 in 10000 chance of winning with button B (where the expectation is the same as with button A), you'd be mad to pick B.

 

The variance is the true measure of risk here and how much one is willing to accept will change from person to person.

[Lord Antares]

Well that's exactly what I said in the first post: the mathematical answer is X = 0.0001 and above, but I wouldn't choose button B at such odds. That's why I gave the example of button A being much more desirable than B in the event of A = 100% for $100, B = 50% for $202, even though the mathematical choice here would be B.

So I was asking at what odds would YOU opt for button B.

 

It's not a puzzle, it's just a personal question.

[Delta1212]

I was initially going to say that $100 isn't exactly life changing money and I'm not desperate for cash by any measure, so I'd probably go with B regardless of what the odds were unless you got down to some incredibly miniscule percentage that really no longer made any mathematical sense.

 

Then I reframed the question in my mind as to what the odds of winning the million would have to be for me to be willing to purchase a ticket for $100.

 

And after reflecting on that, I think I actually converge to a similar answer about what "feels right" being the approximate break-even value.

 

I'd trade $100 for a 1 in 10,000 shot or better at a million dollars. Those are long odds but not so long I wouldn't take them. Up it by a factor of 10 to 100,000 and it no longer feels worth the money.

 

So anything better than a 0.01% chance and I'd probably hit B.

[Lord Antares]

1 in 10 000 does not equal 0.01 but yeah, that's another interesting way to phrase the question.

[StringJunky]

1 in 10 000 does not equal 0.01...

1/10 000 x100 = 0.01%

[Lord Antares]

?

 

1/1 000 000 x 100 = 0.0001

 

You divided by hundred twice (i.e. divided by 10 000)

[Delta1212]

Why are you diving by a million?

[Strange]

Not neccessarily. Simply put, everyone would choose B if X = 50%. Everyone would choose B if X = 20%. Just find the value of X below which you would press button A and above which you would press button B.

 

 

I think there has been quite a lot of research on exactly this sort of question. I'm not sure the results are quite as straightforward as you suggest (I can't remember any details of brief articles I have read - just that the results can be surprising).

 

The results might also be changed by priming(*) - e.g. if you ask after handing someone their salary versus just after giving them their electricity bill.

 

(*) Some priming experiments have turned out not to be reproducible; but I don't think there is much doubt that that the phenomenon itself exists.