Among fair-division procedures, auctions are one of the oldest, dating back more than 2500 years. They are used in applications ranging from the determination of who gets to own a promising racehorse to the selection of a contractor to build a new science center at a local college. And, of course, the online auction site eBay has become a story unto itself.
But even the two examples above—an auction for a racehorse versus an auction to select a contractor—illustrate some fundamental differences between the kinds of auctions in use today. For example, is the winner the high bidder or the low bidder? With the racehorse, it’s the former, whereas with the contractor, it’s the latter. Are the bids oral (so everyone knows the last bid) or are they submitted in a sealed envelope (with no one knowing what anyone else bid)? Again, with the racehorse, it’s the former, whereas with the contractor, it’s the latter.
The subject of auctions is both large and important. Indeed, we could well devote an entire chapter (or book) to that topic alone. But we limit ourselves here to one particular kind of auction—known as a Vickrey auction—that is reminiscent of what is used today on eBay (and we will describe the latter momentarily as well). To avoid confusion, we will assume the high bidder wins (as opposed to the low bidder winning) in the auctions we are considering, and we will assume that ties in the bidding simply do not occur.
A Vickrey Auction PROCEDURE
In a Vickrey auction, bidders independently submit sealed bids for the object being sold. The winner is the high bidder, but he or she pays only the amount of the second-highest bid.
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For example, if there are four bids of $40, $50, $60, and $80, then the fourth bidder wins the auction with his bid of $80 but pays only $60 for the object being auctioned off. Vickrey auctions were introduced in a famous 1961 paper by William Vickrey, a Canadian economist and Nobel laureate. Vickrey spent his career at Columbia University and died in 1996, just three days after the announcement of his Nobel Memorial Prize in Economics.
Here is how an eBay auction typically works: When a seller places an item up for auction on the eBay site, he or she indicates a minimum sale price and sets the value for the bid increments. Bidders submit their bids independent of one another. As each new bid is submitted, bids are submitted automatically on behalf of the highest bidder, one increment above the “going price.” This continues until time expires and the person with the highest bid wins. In his text Introduction to Economic Analysis, R. Preston MacAfee explains the connection between a Vickrey auction and the system used on eBay.
The Vickrey auction underlies the eBay outcome because when a bidder submits a bid in the eBay auction, the current “going” price is not the highest bid, but the second-highest bid, plus a bid increment. Thus, up to the granularity of the bid increment, the basic eBay auction is a Vickrey auction run over time.
Vickrey auctions are interesting because of the answer they provide to the following question: If the object being sold is worth, say, $100 to a bidder, how much less than $100 should he or she bid?
Intuition suggests that in any auction situation, you should bid less than what the object being sold is actually worth to you. In fact, there are mathematical results that suggest that in a standard sealed-bid auction in which the high bidder wins and pays what he or she bid, you should bid only half of what the object is worth to you if there are two bidders, two-thirds of what it is worth to you if there are three bidders, three-fourths if there are four bidders, and so forth.
Remarkably, nothing like this is true with a Vickrey auction. Indeed, with a Vickrey auction, there is a very real sense in which “honesty is the best policy.”
Strategy for Bidding in a Vickrey Auction THEOREM
In a Vickrey auction, a bidder can never do better than that achieved by a bid of exactly what the object is worth to that bidder.
EXAMPLE 6 The Vickrey Auction
To see why this theorem is true, let’s assume that a lamp is being auctioned off and that it is worth $100 to a bidder named Bob. This means that Bob would prefer winning the lamp and paying less than $100 to losing the auction, but that he would rather lose the auction than wind up paying more than $100 for the lamp.
Let denote the highest of the bids other than Bob’s bid of $100. Either , in which case Bob wins the auction and pays dollars, or , in which case Bob loses the auction. We’ll consider these two cases separately and show that no bid for Bob can ever do better than his (sincere!) bid of $100.
Case 1: Bob wins the auction (so ). Any bid by Bob greater than yields the same outcome for Bob as does his bid of $100. He wins the auction and gets the lamp for dollars. So these bids are no better for Bob than his bid of $100. On the other hand, any bid less than is strictly worse for Bob than that achieved with his bid of $100 because he would lose the auction instead of getting the lamp for less than he actually thought it was worth.
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This completes the proof of the theorem. There is something quite satisfying in having a rigorous mathematical proof that establishes—at least in this one context—the fact that honesty is indeed the best policy.