St. P Paradox

[DoomSlice]

Let's say I offered you the following game: I flip a fair coin. If it lands on heads, I give you one dollar and flip again. If it lands tails the game ends. If on the second flip it lands heads, I give you another 2 dollars. If it lands tails the game ends. Next is 4, 8, etc... until I have flipped ten times.

Q1: What is the expected value of this game?
Q2: What amount would you PAY in order to have the opportunity to play this game?
Q3: Now assume that all payouts are increased by 100-fold, what amount would you PAY in order to have the opportunity to play this game?

[Sephus]

$5.
$3.50.
$175.

i feel like im cheating though because apparently we're in the same financial econ class.

[DoomSlice]

Hah... yeah I am at Michigan.

[jedi]

SHould this be in the Probability forum?

[DoomSlice]

The paradox is that most people would pay a number that is significantly lower than the expected value of the game. The psychology behind this is 1) that people are risk averse and therefor require some sort of premium in order to take the "bet" and 2) that wealth has a diminishing marginal utility, so that you value the 9th and 10th flip of the coin less than the 1st-3rd,etc... flips of the coin, and thus will be willing to pay less for the total value of it.

Both of these points are what we strive to avoid when playing poker. 1) we should be looking to maximize expected value regardless of risk (through pounding the crap out of marginal +EV situations and having a large enough bankroll to survive the variance) and 2) we should be looking to further maximize profit even if we deem certain pots to be "big enough".

[bobbyi]

[ QUOTE ]
The paradox is that most people would pay a number that is significantly lower than the expected value of the game.

[/ QUOTE ]
That's not a paradox. In fact, you go on to explain why it makes sense.

The question really isn't very interesting if you are only flipping the coin ten times. The way it is interesting is if you are going to flip in an unlimited number of times (until it comes up tails) and pay the person 2^n dollars (or 2^(n-1) as you are dong here) where n is the number of flips. When you construct the game that way, the expected value is infinite since
1/2($1) + 1/4($2) + 1/8($4) + 1/16($8)...
isn't convergent. So based on classical economic ideas of rationality, there should be no limit on how much you would pay me to play this game. But paying $1B to play a game where half the time you are leaving with $1 doesn't seem very rational at all... which is why it is called a "paradox".

[Sephus]

correct.

[DoomSlice]

I was applying it to the poker context and explicitly assuming that there was some finite value to the game. If that made you feel a little uneasy then I apologize.

[Terrabon98]

hey me too, what class is this for?

[Go Blue]

What a coincidence. I made the same post in the probability forum. I'm guessing we're obviously all in the same class.

[ThinkQuick]

[ QUOTE ]
What a coincidence. I made the same post in the probability forum. I'm guessing we're obviously all in the same class.

[/ QUOTE ]

and you all posted your homework on 2+2. and now you've been exposed and won't do it again?