IBM Research | Ponder This | August 2006 challenges

# Ponder This

## August 2006

<<July August September>>

Ponder This Challenge:

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:

• Log refers to the logarithm function.
• Don't worry about whether the number of contracts is an integer.

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 ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

• Challenge: 08/02/2006 @ 03:30 PM EST
Solution: 09/01/2006 @ 10:00 AM EST
List Updated: 09/07/2006 @ 10:00 AM EST

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)
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)