bv_cvxbook_extra_exercises

# This form for is referred to as a k factor risk model

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Unformatted text preview: use proﬁt. Show that there exists a probability distribution π on the possible outcomes (i.e., π ∈ Rm , 1T π = 1) for which + x⋆ also maximizes the expected house proﬁt. Explain how to ﬁnd π . Hint. Formulate the problem in part (a) as an LP; you can construct π from optimal dual variables for this LP. Remark. Given π , the ‘fair’ price for oﬀer i is pfair = j ∈Si πj . All oﬀers with pi &gt; pfair will i i be completely ﬁlled (i.e., xi = qi ); all oﬀers with pi &lt; pfair will be rejected (i.e., xi = 0). i Remark. This exercise shows how the probabilities of outcomes (e.g., elections) can be guessed from the oﬀers of a set of gamblers. (c) Numerical example. Carry out your method on the simple example below with n = 5 participants, m = 5 possible outcomes, and participant oﬀers Participant i 1 2 3 4 5 pi 0.50 0.60 0.60 0.60 0.20 qi 10 5 5 20 10 Si {1,2} {4} {1,4,5} {2,5} {3} Compare the optimal worst-case house proﬁt with the worst-case house proﬁt, if all oﬀers were accepted (i....
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