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puzzle: the winning game piece

benzbubblecat

Joined
3/5/11
Messages
34
Points
18
There is a contest involving collectible game pieces. The probability of finding game piece A is 1 in 90,000,000. The probability of finding game piece B is 1 in 25. The person who possesses both pieces can exchange them for a $1,000,000 cash prize.

You find game piece A and decide to sell it to your friend who has game piece B. What is a fair price for your game piece?
 
Peice A is 3,600,000 times more valuable than peice B. A+B = 1 mil; A is worth 999999.72222229938269461591816224, B is worth 0.27777770061730538408183775504507
 
Although at first I wanted to agree with Alexei's argument, I think the mechanism for obtaining game pieces is important.
Suppose you can (like the McDonalds Monopoly) buy as many mystery game pieces as you want for a certain cost (say 1$). Then your friend won't sell it for 27 cents because it cost /at least/ 1$ to buy it! Rather, the average number of mystery game pieces needed to be bought to uncover piece B is 25 (negative binomial distribution with r=1) and thus B should cost 25$ (a good buy for yourself since A) you are guaranteed a net profit and B) You wouldn't be willing to buy the mystery game pieces yourself since if you are a typical consumer you are risk averse!)

There are many nuances in this problem. It's fair to say we need more information.

EDIT: Sorry, I was reading that as your friend is selling his game piece. Nevertheless, the issues I have with that sort of logic extend to that scenario too, since by no arbitrage (as Alexei implies) it seems that A+B=$1,000,000 (although food for thought: what happens if A+B>$1,000,000?!?!?!?)
 
The value ("fair price") of something is NOT the actual cost of said thing.

The piece of value $999,999.72 has an expected cost of $90,000,000 if you assume that a single piece costs a dollar (but alas it does not, nor does it ever say that anywhere. If you recalibrate your prices you might find yourself pleasantly surprised... ;) ).
 
Alexei Smirnov said:
The value ("fair price") of something is NOT the actual cost of said thing.

The piece of value $999,999.72 has an expected cost of $90,000,000 if you assume that a single piece costs a dollar (but alas it does not, nor does it ever say that anywhere. If you recalibrate your prices you might find yourself pleasantly surprised... ;) ).
Click to expand...
He asks for "a fair price for your game piece". That is a going price that the other player will accept.
 
without further information, this really is impossible. as mentioned above, the way in which pieces are obtained is important. if A has no ability to obtain another piece, the rational decision would be to sell for anything greater than or equal to $0.01 (or buy a B piece himself/herself for anything less than $999,999.99). surely, selling for that would be silly, hence the need for more information.
 
There's the distinction between market price and fair price. If markets are perfectly efficient, then Alexei's solution would be the market price. The market price for your piece would obviously change if you had the option to buy more pieces yourself. But I asked to see what valuation people might come up with if they where forced to sell their piece. That subjects you to charging a fair price:

So As the term is generally used, Fair Value can be clearly distinguished from Market Value. It requires the assessment of the price that is fair between two specific parties taking into account the respective advantages or disadvantages that each will gain from the transaction.
Click to expand...
 
EDIT: my original solution was way off: I assumed only one of each piece, which makes no sense since they have differing probabilities.

I would think that the "fair price" is dependent on the level of risk aversion in my friend. If he was risk neutral, he would pay me 25 times the going rate for unknown pieces.
 
I agree with Alexei, but I would give my friend a 50k extra. C'mon, he's a friend and I have 1 mln! :D
 
As said, it all depends on what are the possibilities now.
Can they redraw? If not than the fair price is 50/50 since none can win without the other one piece.
If yes than what is the price of each draw? If no price than the price is >0 since the other player can draw over and over until it gets A too.

There are more issues with this...

Too little data...I think it's impossible to solve it in one manner without more data.
 
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