Chapter 15 discusses the relationship between the VCG principle and second-price auctions. In particular, we saw that the VCG principle is a generalization of the idea behind second-price auctions to a setting in which there is more than one object being sold. In this problem we explore this relationship in an example. Suppose that a seller has one item, which we call item x. There are three buyers, whom we call a, b, and c. The values that these buyers (a, b, and c) have for the item are 6, 3, and 1, respectively.
(a) Suppose that the seller runs a second-price auction for the item. Which buyer will win the auction and how much will this buyer pay?
(b) Now let’s suppose that the seller uses the VCG mechanism to allocate the item. Remember that the first step in the running the VCG mechanism when there are more buyers than items is to create fictional items, which each buyer values at 0, so that the number of items to be allocated is the same as the number of bidders. Let’s call these additional (fictional) items y and z. Find the allocation that results from running the VCG mechanism. What are the prices charged to each buyer for the item that they receive? Explain why the price that buyer a pays is the harm that he causes to the remaining bidders by taking the item he is assigned.
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