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Which Auction Is Best? |
When economists began to turn the power of game theory on auctions, they started noticing that one economist, William Vickrey of Columbia University in New York, had already used game theory to analyze auctions several years before Harsanyi developed his theory. Vickrey’s brilliant study of auction strategies was ahead of its time: Written in 1961 when economists were only starting to get a sense of game theory’s importance, it was relegated to an obscure journal and overlooked for years. Today, however, it is seen as the pioneering paper in the field of auction theory.
 Vickrey, who earned the Nobel Memorial Prize in Economics in 1996 partly for his work on auction theory, studied what economists call “private value” auctions, in which each bidder’s value for the item for sale is independent of the values of the other bidders. For instance, if a Rembrandt painting is being auctioned and you want to buy it simply because you like it, then knowing how much your rivals value it won’t affect how much you value it yourself. Vickrey compared three of the most common auctions (English, Dutch, and sealed first-price auctions) and designed a fourth with some surprising properties.
An English auction is the familiar “going, going, gone” auction of such art houses as Sotheby’s and Christie’s, in which the price goes up until only one bidder remains. In a Dutch auction the price starts out high and drops until someone is willing to pay that price. In a first-price auction, participants submit sealed bids and the highest bidder wins, paying her bid. To these auctions Vickrey added what became known as the second-price auction, in which participants submit sealed bids and the highest bidder wins, but pays only as much as the second-highest bid.
Why would anyone use such an arbitrary-sounding rule? Although Vickrey’s auction seems the least natural of the four, it is the one with the simplest optimal bidding strategy: Just bid the amount at which you value the object.
Suppose, for instance, you’re willing to pay up to $100 for an antique doll. What will happen if you bid less than $100, say $90? If the highest rival bid is $80, you’ll win and pay $80; but the same thing would have happened if you had bid $100. If the highest rival bid is $120, you'll lose; and again the same thing would have happened if you had bid $100. But if the highest rival bid is $95, you’ll lose the auction, whereas if you had bid $100 you would have won the doll for $95. So bidding $90 never improves your situation, and sometimes makes you lose an auction you would have liked to win. In a similar way bidding more than $100 never improves your situation, and sometimes makes you win an auction you would have liked to lose. In a second-price auction, honesty is the best policy.
You might wonder, though, why a seller would ever use a second-price auction. Why should she let the winner pay the second-highest bid when she could make the winner pay the highest bid? Astonishingly, Vickrey proved that in a wide class of situations, the seller can expect the same amount of money regardless of which of the four auctions she uses. In 1981, game theorist Roger Myerson of the University of Chicago extended Vickrey’s result to show that all auctions bring in the same expected revenue, provided they award the item to the bidder who values it most, and provided the bidder who values it least doesn’t pay or receive any money, as would happen if there were a fee or reward simply for entering the auction.
It’s easy to see that an English auction produces the same revenue as a second-price auction: An English auction ends precisely when the second-highest bidder drops out (although in some English auctions bidders must raise the high bid by some definite increment, in which case the winner pays marginally more than the second-highest bid). The Dutch auction and the first-price auction are also equivalent to each other, since in a Dutch auction, the prize goes to the bidder willing to bid highest, and she pays what she bids.
But why doesn’t a first-price auction bring in more money than a second-price auction? The reason is that in a first-price auction, it doesn’t pay to bid honestly. If you bid $90 for the antique doll, and the second-highest bid is $80, then you’ll win the doll for $90. If you had bid $100, you would have won but paid more. So in a first-price auction the best strategy is to bid less than your value for the item—what auction theorists call “shading” your bid.
Vickrey figured out how much bidders should shade their bids by looking for the Nash equilibrium strategy. This best strategy varies depending on the circumstances of the auction—for instance, the more bidders in the auction, the less each bidder should shade his bid, since there is less room between the highest bidder’s value and the second-highest bidder’s value. But Vickrey found that no matter what the number of bidders, the shaded bids mean the seller takes home only as much money as in a second-price auction.
Vickrey and Myerson’s work would seem to be the end of the story. All auctions bring in the same revenue, and the second-price auction has the easiest strategy. So it seems that auctioneers should just always use second-price auctions.
But it’s not that simple. Vickrey’s work laid the foundations of auction theory, but it didn’t answer all the questions. His work didn’t cover auctions in which the bidder who values the item most doesn’t necessarily win it—for instance, auctions that give preference to disadvantaged bidders (such as small businesses bidding against huge corporations), or auctions in which the seller sets a reserve price below which no one will win the item at all. What’s more, Vickrey assumed bidders have private values—knowing how their rivals value the item wouldn’t change how they value it themselves. But in most auctions, bidders’ values influence each other in subtle ways. Even in an art auction, in which many collectors are motivated purely by how much they like the work, some bidders may be dealers. If they find out, for instance, that a savvy dealer values the item highly, they are more likely to value it highly themselves.
Understanding situations in which bidders care about the market value of an object, not just how much they like it, gave economists plenty to do in the next few decades after Vickrey’s work. The result would turn out to shed a crucial light on a wide range of auction environments, from government sales of oil drilling leases to airwave spectrum auctions. |
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