IBM Ponder This #100 · August 2006

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.

Solution

Best opened after a real attempt

To be added.