An EV paradox

[zac777]

Suppose you have the opportunity to participate in the following lottery based on coin flips:

One fair coin is flipped repeatedly until tails comes up for the first time. The payout is made according to how many times the coin was flipped. If tails came up on the Kth flip, than each participant wins 2^K dollars. That is, if the first flip is tails, you win $2. If tails doesn't come up until the 4th flip, you win 2^4 dollars = $16. If tails doesn't come up until the 10th flip, you win 2^10 dollars = $1024.

What's the most you would be willing to pay in order to participate in this lottery, and why?

[well]

I'd pay anything, as long as I can keep playing this "lottery" until I am satisfied.
You might have liked to hear an infinite amount of dollars, but hey - this is not possible!

It's not really that much of a paradox if this answer was allowed.

If, however, the question was:

"If x is the maximum amount of dollars you'd pay to participate in this lottery, what would x be?"

Then there just was no answer, as the question was posed poorly.
For me, this would be equivalent to asking:
"What's the largest whole number?"

Still no paradox.

Regards.

[cepstrum]

Hi zac -

It's only a paradox if everyone has an infinite bankroll. Otherwise, the completely non-paradoxical result is that everyone will be willing to pay a different amount to play this game, and that amount will depend solely on the size of the player's bankroll.

Just another reminder that both the first _and_ second moments count.

Good Luck

Cepstrum

[Fiery Jack]

It doesn't state that you have to invest any money at all, therefore my answer is 0.

[zac777]

You can only play once. If you had a $10,000 bankroll and someone walked up and offered you the chance to play this lottery, would you really put all of your money down?

[well]

Of course not.

Would you risk it all to triple up on a 50-50 shot?

[zac777]

Triple-up? Nope.

But what if I could X-up, where you can make X as big as you want? Is there a prize large enough to offset the risk of ruin?

I think this problem brings up some interesting questions, a good portion of which are addressed here. Check it out if you're interested.

If I actually had a chance to play the game, I'd say that any prize over $100,000,000 would be the same to me, so I'd probably be willing to pay around $27.

[well]

[Thythe]

Clearly you just took a beginning game theory course. The problem outlined is known as the St. Petersburg Paradox. The idea of the whole paradox is that if someone is risk neutral, then they should be prepared to risk all they have on this lottery.

Of course none of us would do this which introduces the concept of utility curves. Each person will answer differently according to their utility curves which are in turn determined where a person is on the risk averse->risk neutral->risk loving scheme of things.

[zac777]

Sorry about the link, it should work now. And no, I didn't take a game theory course. A friend of mine (a game theory student) posed the scenario to me. It reminded me of the various threads about EV/bankroll, so I figured I'd toss it up here to see what everyone thought.