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, the site works as follows.
• 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 If the seller rejects the bid, buyer departs and the bid is withdrawn; and only then does the seller see any future buyers.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.
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 may assume that n is an even number.) Prove that your strategy achieves this probabilistic guarantee.
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