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Math Help - [SOLVED] Probability Question

  1. #1
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    [SOLVED] Probability Question

    I am having a difficulty on this problem (part a ) and would appreciate if someone can help.

    Suppose you’re designing
    strategies for selling items on a popular auction Web site. Unlike other
    auction sites, this one uses a one-pass auction, in which each bid must be
    immediately (and irrevocably) accepted or refused. Specifically,
    • First, a seller puts up an item for sale.
    • Then buyers appear in sequence.
    • When buyer i appears, he or she makes a bid bi > 0.
    • The seller must decide immediately whether to accept the bid or not.
    If the seller accepts the bid, the item is sold and all future buyers are
    turned away. If the seller rejects the bid, buyer i departs and the bid
    is withdrawn; and only then does the seller see any future buyers.
    Suppose an item is offered for sale, and there are n buyers, each with a
    distinct bid. Suppose further that the buyers appear in a random order, and
    that the seller knows the number n of buyers. We’d like to design a strategy
    whereby the seller has a reasonable chance of accepting the highest of the n
    bids. By a “strategy,” we mean a rule by which the seller decides whether to
    accept each presented bid, based only on the value of n and the sequence of
    bids seen so far.
    For example, the seller could always accept the first bid presented. This
    results in the seller accepting the highest of the n bids with probability only
    1/n, since it requires the highest bid to be the first one presented.


    (a) Give a strategy under which the seller accepts the highest of the n
    bids with probability at least 1/4, regardless of the value of n. (For
    simplicity, you are allowed to assume that n is an even number.) Prove
    that your strategy achieves this probabilistic guarantee.
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  2. #2
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    This looks like another variation on the secretary problem, in which the optimal strategy selects the best applicant with a probability (asymptotically, as n increases) of 1/e.
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  3. #3
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    Thanks!

    Thank you very much for the response!

    I think I got the idea.
    Last edited by tk88; November 17th 2007 at 12:44 AM.
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