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MIT6_079F09_hw06 - 6.079/6.975 Fall 2009-10 S Boyd P...

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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 > 0 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 0 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 = 1) for which x also maximizes the expected house profit. Explain how to find π .
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