Definitions

# Winner's curse

The winner's curse is a phenomenon akin to a Pyrrhic victory that occurs in common value auctions with incomplete information. In short, the winner's curse says that in such an auction, the winner will tend to overpay. However, an actual overpayment will generally occur only if the winner fails to account for the winner's curse when bidding. So despite its dire-sounding name, the winner's curse does not necessarily have ill effects.

## Explanation

In a common value auction, the auctioned item is of roughly equal value to all bidders, but the bidders don't know the item's market value when they bid. Each player independently estimates the value of the item before bidding.

The winner of an auction is, of course, the bidder who submits the highest bid. Since the auctioned item is worth roughly the same to all bidders, they are distinguished only by their respective estimates. The winner, then, is the bidder making the highest estimate. If we assume that the average bid is accurate, then the highest bidder overestimates the item's value. Thus, the auction's winner is likely to overpay.

More formally, this result is obtained using conditional expectation (which employs conditional densities). We are interested in a bidder's expected value from the auction (the expected value of the item, less than the expected price) conditioned on the assumption that the bidder wins the auction. It turns out that for a bidder's true estimate the expected value is negative, meaning that on average the winning bidder is overpaying.

Savvy bidders will avoid the winner's curse by bid shading, or placing a bid that is below their ex ante estimation of the value of the item for sale — but equal to their ex post belief about the value of the item, given that they win the auction. The key point is that winning the auction is bad news about the value of the item for the winner. It means that he/she was the most optimistic and if bidders are correct in their estimations on average, that too much was paid. Therefore savvy bidders revise their ex ante estimations downwards to take account of this effect.

The severity of the winner's curse increases with the number of bidders. This is because the more bidders, the more likely it is that some of them have overestimated the auctioned item's value. In technical terms, the winner's expected estimate is the value of the first order statistic, which increases as the number of bidders increases.

There is often confusion that winner's curse applies to the winners of all auctions. However, it is worth repeating here that for auctions with private value (i.e. when the item is desired independent of its value in the market), winner's curse does not arise. Similarly there may be occasions when the average bid is too low relative to exterior market conditions e.g. a dealer recognising an antique or other collectable as highly saleable elsewhere when other bidders do not have the necessary expertise.

## Examples

Since most auctions involve at least some amount of common value, and some degree of uncertainty about that common value, the winner's curse is an important phenomenon.

In the 1950s, when the term winner's curse was first coined, there was no accurate method to estimate the potential value of an offshore oil field. So if, for example, an oil field had an actual intrinsic value of \$10 million, oil companies might guess its value to be anywhere from \$5 million to \$20 million. The company who wrongly estimated at \$20 million and placed a bid at that level would win the auction, and later find that it was not worth as much.

Other auctions where the winner's curse is significant:

• Spectrum auctions in which companies bid on licenses to use portions of the electromagnetic spectrum. Here, the uncertainty would come from, for example, estimating the value of the cell phone market in New York City.
• IPOs, in which bidders need to estimate what the market value of a company's stock will be.
• Pay per click advertising online, in which advertisers gain higher ranking if they bid higher amounts per click from a search engine user.