is the EV for this game really infinity?

dchz · #1 10-23-2007, 06:24 PM

just did this in econ class but i didn't think it was true....
i thought i might have missed something so 2p2, analyze this game with me

basically we'll flip coins:

If you get head on the first flip, i will give you 2 dollars

if you get tails on the first flip and get head on the 2nd flip, i will give you 4 dollars

if you get tails on the first 2 flips and get head on the 3rd flip i will give you 8 dollars

if you get tails on the first N-1 flips and get head on the nth flip, i will give you 2^(n) dollars

what is the EV for this game?

the answer i got in class was infinity

because the EV is (1/2)x2 + 1/4 (4)+ 1/8 (8) +.... + 1/(2^n) x 2^n

hence the sum of all the EVs = infinty

there seems to be a flaw in the logic here and i can't point it out and it's bugging the crap out of me

solve this for me 2p2

dchz · #2 10-23-2007, 06:26 PM

ahhh i just got it after i posted it

the EV for the game is not the sum of the EVs, but it's just 1 of the EVs, hence the EV of the game is 1 since you only get 1 pay out

dchz · #3 10-23-2007, 06:26 PM

LDO?

econ prof here i come

pzhon · #4 10-23-2007, 10:24 PM

This is the well-known Petersburg Paradox.

eviljeff · #5 10-23-2007, 11:31 PM

I will pay you $10 million to play this game

br.bm · #6 10-25-2007, 09:36 AM

Do I ever loose money?

dchz · #7 10-25-2007, 02:21 PM

[ QUOTE ]
I will pay you $10 million to play this game

[/ QUOTE ]

you realize there is only 1 pay out right?

i will play this game with you

when will you pay me?

WhiteWolf · #8 10-25-2007, 03:00 PM

[ QUOTE ]
ahhh i just got it after i posted it

the EV for the game is not the sum of the EVs, but it's just 1 of the EVs, hence the EV of the game is 1 since you only get 1 pay out

[/ QUOTE ]
No, your original calculation was technically correct. Read the linked wikipedia article for proposed explanations on why most rational people would not pay a lot of money for this lottery. For me, the most compelling was even if it was backed by Bill Gates' entire wealth, it wouldn't be worth more than $18 dollars.

LarryLaughs · #9 10-25-2007, 03:04 PM

Interesting proposition. Lets say that you'd have to pitch in 3$ to play. Would this mean negative profit?

According to the EV=1 statement, it would.

However, you always win at least 2$.
So, you always win 2 or 4 or 8 or 16... and so on. So the EV has to be more than 2.

I'd say that the EV is a bit more than 2 with high positive variance. Just by thinking about cases how it could end up.

Think about it this way.
Would you play it if your bet would be 3$?

Also, lets assume you play it 100 times. Assuming perfect distribution you would end up with:
50 times you win 2$
25 times you win 4 and so on.
If we stop the flips at 5 tails and one head, which you'll get once you'd make ~600$. Divided by 100 games that makes 6$ per game.

Repeat that, but now 10 000 times. Assuming you never get to more than 9 tails and one head (around 1/10000) you'd make ~10 $ per game, thus you'd make about 100 grand.

I am not good at this kind of math (sums over ranges) but I think you could simulate the EV with monte carlo simulation for several amounts of games. Like once for 10, 100, 1000, 10000 and so on, and get some kind of result.

dchz · #10 10-26-2007, 03:33 AM

[ QUOTE ]
Interesting proposition. Lets say that you'd have to pitch in 3$ to play. Would this mean negative profit?

According to the EV=1 statement, it would.

However, you always win at least 2$.
So, you always win 2 or 4 or 8 or 16... and so on. So the EV has to be more than 2.

I'd say that the EV is a bit more than 2 with high positive variance. Just by thinking about cases how it could end up.

Think about it this way.
Would you play it if your bet would be 3$?

Also, lets assume you play it 100 times. Assuming perfect distribution you would end up with:
50 times you win 2$
25 times you win 4 and so on.
If we stop the flips at 5 tails and one head, which you'll get once you'd make ~600$. Divided by 100 games that makes 6$ per game.

Repeat that, but now 10 000 times. Assuming you never get to more than 9 tails and one head (around 1/10000) you'd make ~10 $ per game, thus you'd make about 100 grand.

I am not good at this kind of math (sums over ranges) but I think you could simulate the EV with monte carlo simulation for several amounts of games. Like once for 10, 100, 1000, 10000 and so on, and get some kind of result.

[/ QUOTE ]

didn't really read your post becuase your avatar gave me night mares

i think the EV is infinity but the utility of this is less and it's the paradox

i hope this enlightened someone cause it sure made my head spin