What would your guess be about the average donation to win $10?

Question

Here is the situation and this is for my graduate econ class, so I know that many of us have great sense of humor, but please help me out with a serious answer....

Facts:
1. You have a chance to win $10.
2. You can enter for free.
3. You can donate money.
4. Free entrants and donators have the same chance to win.
5. The average donation wins, whether guessed by free entrant or donator.
5. Only the donators affect the average donation - donations greater than 0 (zero) will be averaged out, and the one closest to the average will win.

How much would you donate?

Answer 1

Let's make a couple assumptions first:
1) $10 will be awarded regardless of how much is donated
2) your goal is to maximize your gain (if it were to minimize your loss then you would donate zero).
3) the only thing to gain is the $10 net you lost donation (that is, the glory of winning doesn't matter - otherwise you could donate $10,000 and almost certainly win)
4) Everyone else trying for the prize is as smart as you and logical (toss human 'weirdness' out the window)

Now, let p= probability that your donation is closest to the average. Then your gain function is p(10)-(d) where d is the amount you donate.

Let's also say that there are n total people vying for the prize and under the same assumptions you are. Then none of the n people (yourself included) will likely have donated zero since that cannot maximize the gain function if eeven one other person donated > zero.

If you know nothing about the psychology of the others you can assume your probability of being closest is aproximately the probability your donation is the median value of the group.

That is, p = 1/n .

So, your gain function becomes G =(1/n)(10) - d. Thus, if you donate over 10/n dollars, you have a negative expectation. Knowing this and assuming that the others can reasonably deduce this as well, the uniform distribution from 0
The max point would be the midpoint, so d=(1/2)(10/n) = 5/n dollars. For example, if 4 people total try for the prize, then donate 5/4 = $1.25.

If you don't know n in advance then you would have to take a best guess at it.

Again, there were assumptions at the beginning, but I would donate $(5/n).

Answer 2

it may extremely count number first on what the money replaced into being donated to. If it have been a reason that many people have been interested in donating, i think of it would be bigger, a minimum of in accordance with what the donation replaced into for and the place it replaced into being held... nonetheless, i could wager that the donation widely used could be very low with loose entries allowed. in all risk below $a million widely used. truthfully, permit me pass extra. it may additionally count number on if people could could donate/sign in for the drawing with a actual individual or if it replaced into purely a table with slips and a place to donate. i think of it would be bigger if somebody replaced into sitting on the table manning it, yet nonetheless not over a dollar. i could nonetheless wager in all risk $0.20 widely used...