Puzzle for August 2006.
Prediction markets trade in contracts that pay off according to whether some future event occurs. If the event occurs the contract pays off 1 unit otherwise the contract becomes worthless. The price of a contract can be interpreted as the market's opinion on the probability that the future event will occur.
Suppose individuals A and B think the probability that the future event will occur is p or q respectively. Suppose A and B have risk capital X and Y respectively. Suppose A and B invest so as to maximize the expected log of their capital. What price r for the contract will clear the market (ie assure that the sum of the demand for the contract from A and B is 0). We are assuming contracts can be sold short which corresponds to risking (1-r) units to gain r units if the event does not occur. We are also ignoring all frictional costs like commissions or interest.
I came up with this problem myself but I doubt it is original. Please don't submit solutions you haven't derived yourself.
NOTE - Some clarifications:
We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!
We invite visitors to our website to submit an elegant solution. Send your submission to the webmaster.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: firstname.lastname@example.org
08/02/2006 @ 03:30 PM EST
Solution: 09/01/2006 @ 10:00 AM EST
List Updated: 09/07/2006 @ 10:00 AM EST
People who answered correctly:
Ross Millikan (08.02.2006 @04:18:54 PM EDT)
Joe Fendel (08.02.2006 @06:10:04 PM EDT)
V. Balakrishnan (08.02.2006 @10:44:56 PM EDT)
Vincent Vermaut (08.03.2006 @02:44:04 AM EDT)
Alan Murray (08.03.2006 @10:21:02 AM EDT)
Dion K. Harmon (08.03.2006 @06:00:45 PM EDT)
Joseph DiVincentis (08.03.2006 @07:12:35 PM EDT)
Mark Perkins (08.03.2006 @07:15:48 PM EDT)
Hagen von Eitzen (08.04.2006 @12:50:20 AM EDT)
James Dow Allen (08.04.2006 @09:28:38 AM EDT)
Jan Rubak (08.04.2006 @01:52:06 PM EDT)
Rob Berger (08.04.2006 @04:51:38 PM EDT)
Ashish Mishra (08.04.2006 @09:32:10 PM EDT)
Ed Sheppard (08.05.2006 @06:24:52 PM EDT)
Jay Sounderpandian (08.05.2006 @06:54:37 PM EDT)
Se Kwon Kim (08.06.2006 @12:07:46 PM EDT)
Firat Solgun (08.06.2006 @03:01:41 PM EDT)
John Fletcher (08.06.2006 @07:23:21 PM EDT)
Siddharth Joshi (08.07.2006 @04:09:08 AM EDT)
Frank Mullin (08.07.2006 @08:35:57 PM EDT)
Paulo Natenzon (08.08.2006 @02:53:39 AM EDT)
Phil Muhm (08.08.2006 @09:18:10 AM EDT)
Dion K. Harmon (08.08.2006 @11:14:44 PM EDT)
Mark Pilloff (08.09.2006 @02:48:27 AM EDT)
Arthur Breitman (08.09.2006 @03:22:30 PM EDT)
Grey Zone (08.10.2006 @04:17:02 PM EDT)
Didier Bizzarri (08.11.2006 @05:05:49 AM EDT)
Yoav Raz (08.13.2006 @02:37:53 AM EDT)
Aleksandar Tamburov (08.13.2006 @04:14:57 PM EDT)
Dan Dima (08.14.2006 @05:20:07 PM EDT)
Wolf Mosle (08.15.2006 @12:35:26 PM EDT)
Dan Stronger (08.16.2006 @03:26:25 AM EDT)
Dick King (08.16.2006 @02:14:16 PM EDT)
Du Yang (08.17.2006 @12:53:32 PM EDT)
Pawan Agarwal (08.18.2006 @09:10:46 AM EDT)
Oleg Trott (08.18.2006 @06:42:50 PM EDT)
David McQuillan (08.19.2006 @04:41:00 PM EDT)
Arjun Acharya (08.28.2006 @12:36:03 PM EDT)
Ben North (08.29.2006 @05:09:01 AM EDT)
Michael Quist (08.29.2006 @01:28:35 PM EDT)
Mahmoud Elhaddad (08.30.2006 @12:34:24 AM EDT)
Fatih Mehmet Dogu (08.30.2006 @05:00:54 AM EDT)
Attention: If your name is posted here and you wish it removed please send email to the webmaster.