Ponder This Challenge - January 2007 - Future contracts - how many to buy?

Ponder This Challenge:

Puzzle for January 2007.

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 estimate of the probability that the future event will occur.

Suppose some future event (such as a horse race) has n possible outcomes each with associated contract a(i) (which can be bought and sold) which will pay off 1 unit if the ith possibility occurs. The other n-1 contracts will become worthless. Suppose contract a(i) has market price p(i). Then p(i) can be interpreted as the market's estimate that the ith event will occur. In an efficient market (which we assume) the sum of the p(i) will be 1.

Suppose you believe the true probability of the ith possibility occurring is q(i) (for i=1,...,n). Clearly the q(i) should also sum to 1 and we assume this. Suppose you have X units to invest and you wish to do this in such a way that the expected value (ie the weighted average over the possible outcomes) of the logarithm of your payoff after the event is maximized. How many of each of the n contracts should you buy? Assume X is large enough that you don't have to worry about buying an integral number of each contract.

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Solution

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  Answer:

  We may assume all of X is invested since if we buy an equal number of each of the n contracts we will break even (since by the assumption of an efficient market the p(i) sum to 1). Let a(i) be the amount invested in the ith contract. Since the price of the ith contract is p(i) we are buying a(i)/p(i) contracts. If event i occurs these contracts will each be worth 1 unit otherwise they will be worthless. Hence the payout if the ith event occurs is a(i)/p(i). We believe the ith event will occur with probability q(i) so the expected log value is the sum over i of q(i)*log(a(i)/p(i)). We wish to maximize this over the constraint that the sum of the a(i) is X. So using the method of Lagrange multipliers we look at the derivatives of (Sum q(i)*log(a(i)/p(i))) - y * (Sum a(i)) with respect to a(i). The ith derivative is q(i)/a(i) - y. So setting the derivatives to 0 we have a(i)=q(i)/y. So Sum a(i) = Sum q(i)/y = 1/y. So to satisfy the constraint we should set y=1/x. So a(i)=q(i)*X. So in other words we should spread our capital X over the n contracts in proportion to our estimate of the probability that each of the n associated mutually exclusive events will occur.

  The problem as stated asks for the number of each contract we should buy. Let the number for the ith contract be n(i). Then n(i)=a(i)/p(i)=X*q(i)/p(i).

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Solvers

• V. Balakrishnan (01.03.2007 @04:29:59 PM EST)
• Se Kwon Kim (01.03.2007 @11:51:47 PM EST)
• Michael Quist (01.04.2006 @12:01:35 AM EST)
• Ashish Srivastava (01.04.2007 @03:42:58 AM EST)
• Christian Blatter (01.04.2007 @06:17:31 AM EST)
• Aditya Kamat (01.04.2007 @07:29:28 AM EST)
• Joseph DeVincentis (01.04.2007 @11:14:43 AM EST)
• Phil Muhm (01.04.2007 @11:22:24 AM EST)
• Arthur Breitman (01.04.2007 @12:21:56 PM EST)
• Kerry M. Soileau (01.04.2007 @05:14:26 PM EST)
• dineshk (01.04.2007 @08:03:51 PM EST)
• Lukasz Bolikowski (01.04.2007 @09:51:35 PM EST)
• Ariel Flat (01.04.2007 @10:19:58 PM EST)
• Du Yang (01.04.2007 @10:57:57 PM EST)
• Chris Messer (01.05.2007 @04:36:21 AM EST)
• Jayavel Sounderpandian (01.05.2007 @09:14:41 AM EST)
• Brian Larva (01.05.2007 @01:51:37 PM EST)
• John Robinson (01.05.2007 @06:55:25 PM EST)
• Matthias Nagel (01.06.2007 @11:31:50 AM EST)
• James Dow Allen (01.06.2007 @12:23:55 PM EST)
• Dan Stronger (01.06.2007 @04:06:12 PM EST)
• Yoav Raz (01.07.2007 @08:25:20 AM EST)
• John Tromp (01.07.2007 @10:47:49 PM EST)
• Dion Harmon (01.08.2007 @12:21:20 AM EST)
• Eric Cope (01.08.2007 @07:31:56 AM EST)
• Mithil Ramteke (01.08.2007 @07:59:52 AM EST)
• Ashish Mishra (01.08.2007 @06:41:08 PM EST)
• Tom Gutman (01.08.2007 @06:55:44 PM EST)
• srangana (01.09.2007 @12:46:05 PM EST)
• Eugene Vasilchenko (01.09.2007 @03:15:26 PM EST)
• Madhukar Korupolu (01.09.2007 @05:10:32 PM EST)
• Dharmadeep Muppalla (01.10.2007 @12:13:20 PM EST)
• John Fletcher (01.10.2007 @01:22:10 PM EST)
• Yurun Liu (01.10.2007 @04:20:11 PM EST)
• Rajeev Rao (01.10.2007 @05:17:43 PM EST)
• Amit Prakask (01.11.2007 @12:30:20 AM EST)
• David McQuillan (01.12.2007 @06:28:42 PM EST)
• teodora.baeva (01.14.2007 @10:41:24 AM EST)
• Zhou Guang (01.14.2007 @09:32:25 PM EST)
• Dan Dima (01.17.2007 @08:05:31 AM EST)
• Ankit Aggarwal (01.18.2007 @03:23:01 AM EST)
• Martin Thiim (01.18.2007 @08:59:44 AM EST)
• Matthew Samuel (01.18.2007 @10:38:20 PM EST)
• Lawrence Hon (01.18.2007 @11:53:48 PM EST)
• David Liu (01.19.2007 @03:47:53 PM EST)
• Jakub Łopuszański (01.21.2007 @06:35:44 PM EST)
• Beata Stehlikova (01.22.2007 @03:30:44 PM EST)
• Maciej Wisniewolski (01.23.2007 @02:41:28 AM EST)
• Kenneth McKay (01.24.2007 @08:15:40 AM EST)
• Alexander Bleck (01.24.2007 @08:35:56 AM EST)
• M. Haase (01.26.2007 @04:19:51 PM EST)
• Ed Sheppard (01.27.2007 @01:27:24 PM EST)
• Reiner Martin (01.29.2007 @06:37:17 PM EST)
• Amine Belemlih (01.30.2007 @10:29:53 AM EST)
• Adriano A. Batista (01.31.2007 @10:32:24 PM EST)