MIT6_079F09_hw06

MIT6_079F09_hw06 - 6.079/6.975, Fall 2009-10 S. Boyd &...

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Unformatted text preview: 6.079/6.975, Fall 2009-10 S. Boyd & P. Parrilo Homework 6 additional problems 1. Maximizing house profit in a gamble and imputed probabilities. A set of n participants bet on which one of m outcomes, labeled 1 ,...,m , will occur. Participant i offers to purchase up to q i > gambling contracts, at price p i > 0, that the true outcome will be in the set S i ⊂ { 1 ,...,m } . The house then sells her x i contracts, with ≤ x i ≤ q i . If the true outcome j is in S i , then participant i receives $1 per contract, i.e. , x i . Otherwise, she loses, and receives nothing. The house collects a total of x 1 p 1 + + x n p n , ··· and pays out an amount that depends on the outcome j , x i . j ∈ S i The difference is the house profit. (a) Optimal house strategy. How should the house decide on x so that its worst-case profit (over the possible outcomes) is maximized? (The house determines x after examining all the participant offers.) (b) Imputed probabilities. Suppose x maximizes the worst-case house profit. Show that there exists a probability distribution π on the possible outcomes ( i.e. , π ∈ R m + , 1 T π...
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

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MIT6_079F09_hw06 - 6.079/6.975, Fall 2009-10 S. Boyd &...

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