If it's heads, you win $1 and the game ends.
If it's tails, I flip it again.

If it's heads, you win $2 and the game ends.
If it's tails, I flip it again.

If it's heads, you win $4 and the game ends.
If it's tails, I flip it again.

Heads on 4th flip: $8
Heads on 5th flip: $16
Heads on 6th flip: $32

And so on.

Question: What is the maximum amount of money you would be willing to pay to play this game?

controlfreak

Dec 16th, 2006, 10:56 PM

50 cents?

controlfreak

Dec 16th, 2006, 10:58 PM

you've already made $200,000 off these people, how about you cut me in and I'll arrange some advertising?

GoDominique

Dec 17th, 2006, 12:00 AM

you've already made $200,000 off these people, how about you cut me in and I'll arrange some advertising?
I wish it were so. :(

TomTennis

Dec 17th, 2006, 12:09 AM

surely, no matter how much money you pay, you will ALWAYS win it back, the minimum amount of winnings you will get (and that is gaurenteed) is what you started with, so in theory, play with every thing you own.

Is that correct or am I missing something?

King of Prussia

Dec 17th, 2006, 12:10 AM

surely, no matter how much money you pay, you will ALWAYS win it back, the minimum amount of winnings you will get (and that is gaurenteed) is what you started with, so in theory, play with every thing you own.

Is that correct or am I missing something?

:tape:

~Cherry*Blossom~

Dec 17th, 2006, 12:11 AM

LMAO, we did this in uni. It was an example. He started to go off about Utility Theory and I think he somehow may have used a Monte Carlo simulation. Or maybe a mathematical equation dealing with Utility and stuff.

GoDominique

Dec 17th, 2006, 12:19 AM

I just realise that I omitted a crucial information: I have an infinite amount of money and can pay out every possible win. :angel:

~Cherry*Blossom~

Dec 17th, 2006, 12:25 AM

ok, I just looked at my notes and it's called St Petersburg Paradox. I'll look for the solution afterwards ;) (I think we did it)

~Cherry*Blossom~

Dec 17th, 2006, 12:28 AM

lol ok, my notes say:

The paradox states that Expected Utility is infinity, so we should bet everything we have

King of Prussia

Dec 17th, 2006, 12:29 AM

lol ok, my notes say:

So you would bet everything you have to have a 50% chance to win 1$. :confused:

King of Prussia

Dec 17th, 2006, 12:31 AM

I don't get what would be the point to bet more than 1$???

GoDominique

Dec 17th, 2006, 12:32 AM

So you would bet everything you have to have a 50% chance to win 1$. :confused:
But, there's also a 25% chance to win $2, and a 12.5% chance to win $4, and a 6.25% chance to win $8, and ... :angel:

King of Prussia

Dec 17th, 2006, 12:33 AM

But, there's also a 25% chance to win $2, and a 12.5% chance to win $4, and a 6.25% chance to win $8, and ... :angel:

But you get the same odds by betting 1$... Or am I missing something?

GoDominique

Dec 17th, 2006, 12:34 AM

I don't get what would be the point to bet more than 1$???
You need to bet more than $1 if the game organiser (= me) demands it.

King of Prussia

Dec 17th, 2006, 12:36 AM

You need to bet more than $1 if the game organiser (= me) demands it.

Right, I thought it was us who choose how much we bet. :o

GoDominique

Dec 17th, 2006, 12:36 AM

But you get the same odds by betting 1$... Or am I missing something?
Nope. But that's not the question.

If you bet $1, the game is profitable of course.

The question is: Is the game still profitable for you if you have to bet $2, $5 or whatever?

King of Prussia

Dec 17th, 2006, 12:39 AM

Anyway my first impression without any doing any math is that I wouldn't be willing to bet a lot to play that game.

But the first impression is never the good one in probabilities.

~Cherry*Blossom~

Dec 17th, 2006, 12:40 AM

I don't get what would be the point to bet more than 1$???

lol me neither. I didn't really understand what my notes said.

At first we worked out the expected value of the game which was infinity.

Then our utility function = log (x), from that we saw that our game had a utility function of log(2)

Then we did something that I didn't understand why we did it:

U(M) = E[U(x)]
log (M) = log (2)

M=2

We said that the utility of a certain amount, M, is equal to expected utility of the game.

We did a graph

Said that the utility of winning a small amount with a high probability is higher than winning a large amount with low probability.

Then it came to that final statement of betting everything we have.

Finally I have : THIS IS NOT EXAMINABLE, SO YOU DON'T NEED TO KNOW THIS (yes, I wrote it in red writing)

My only guess is that when you win, you get your money back, plus the amount you win :shrug:

GoDominique

Dec 17th, 2006, 12:44 AM

No, what you bet is gone and you won't get it back. ;)

~Cherry*Blossom~

Dec 17th, 2006, 12:46 AM

No, what you bet is gone and you won't get it back. ;)

LOL, well I have no idea why my notes say I'll bet everything.

I guess it depends on whether you're a risk taker or risk averse. I myself would only bet $1.

King of Prussia

Dec 17th, 2006, 12:46 AM

Anyway, regardless of the mathematical expectancy and all that crap, the odds of winning a decent amount is extremely low so I wouldn't bother to play that game. :) Or play with one dollar only.

King of Prussia

Dec 17th, 2006, 12:51 AM

I guess this is the same kind of dilemna as "Deal or no deal" or? In theory, you're supposed to go on all the time as the banker offer is always too low, but in practice virtually no one does that.

GoDominique

Dec 17th, 2006, 12:55 AM

LOL, well I have no idea why my notes say I'll bet everything.
Because in theory it appears to be correct.

The expected value for each game is 1/2*$1 + 1/4*$2 + 1/8*$4 + 1/16*$8 + ....
which is 1/2 + 1/2 + 1/2 + 1/2 + ...
which is infinitely big.

So over the (very) long run, you will win an infinite amount of money, no matter how much you have to invest to play for each game.

In reality though it wouldn't work because you would have to play the game extremely often to show a decent profit. Even after 10,000 games your average win will usually be around $7 only. If you play more, the average keeps rising but very very slowly.

GoDominique

Dec 17th, 2006, 12:56 AM

Anyway, regardless of the mathematical expectancy and all that crap, the odds of winning a decent amount is extremely low so I wouldn't bother to play that game. :) Or play with one dollar only.
For once and for all, I will NOT do it for one dollar. :p

King of Prussia

Dec 17th, 2006, 01:00 AM

Because in theory it appears to be correct.

The expected value for each game is 1/2*$1 + 1/4*$2 + 1/8*$4 + 1/16*$8 + ....
which is 1/2 + 1/2 + 1/2 + 1/2 + ...
which is infinitely big.

So over the (very) long run, you will win an infinite amount of money, no matter how much you have to invest to play for each game.

In reality though it wouldn't work because you would have to play the game extremely often to show a decent profit. Even after 10,000 games your average win will usually be around $7 only. If you play more, the average keeps rising but very very slowly.

Geez weez, the math was simple, but I was too lazy to think. :o

The parallel with DOND is good, because the best mathematical option in games isn't always our best option because we have limit in time, money or the number of times we can play the game.

GoDominique

Dec 17th, 2006, 01:07 AM

Geez weez, the math was simple, but I was too lazy to think. :o

The parallel with DOND is good, because the best mathematical option in games isn't always our best option because we have limit in time, money or the number of times we can play the game.
Right.

Also, if I wasn't infinitely wealthy but only as rich as Bill Gates, the expected value would drop from http://upload.wikimedia.org/math/d/2/4/d245777abca64ece2d5d7ca0d19fddb6.png to $18. :o

drake3781

Dec 17th, 2006, 01:58 AM

This is an interesting question. I worked it out in Excel.

At 100 games, the player wins $3.05.

So I answered $3.

===========

To answer this question you have to assume:
- this is the only way you can ever get any money, and
- if someone outbids you for the amount paid to play the game, then you cannot play.
That's the only way to get people to pay up to the minimum amount between cost to play and average winnings.

drake3781

Dec 17th, 2006, 02:02 AM

Right.

Also, if I wasn't infinitely wealthy but only as rich as Bill Gates, the expected value would drop from http://upload.wikimedia.org/math/d/2/4/d245777abca64ece2d5d7ca0d19fddb6.png to $18. :o

Hm, I'm getting returns flattening out right around $3.05.