# 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 such an auction, the goods being sold have a similar value for all bidders, but players are uncertain of this value when they bid. Each player independently estimates the value of the good before bidding.

The winner of an auction is, of course, the bidder who submits the highest bid. When each bidder is estimating the good's value and bidding accordingly, that will probably be the bidder whose estimate was largest. If we assume that on average the bidders are estimating accurately, then the person whose bid is highest has almost certainly overestimated the good's value. Thus, a bidder who wins after bidding what they thought the good was worth has almost certainly overpaid.

More formally, this result is obtained using conditional probability. We are interested in a bidder's expected value from the auction (the expected value of the good, less the expected price) conditioned on the assumption that the bidder won the auction. It turns out that for a bidder bidding their true estimate, this 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 what they believe the good is worth. This may make it less likely that the bidder will win the auction, but it also protects them from overpaying in the cases where they do win. A savvy bidder knows that they don't want to win if it means they will pay more than a good is worth.

The winner's curse gets stronger as the number of bidders increases. This is because the more bidders there are, the more likely it is that some of them have greatly overestimated the good'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.

## 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 '50s, when the term 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 M, oil companies might guess its value to be anywhere from \$5 M to \$20 M. The company who wrongly estimated at \$ 20 M 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.

Topics in game theory
Evolutionarily stable strategy - Mechanism design - No-win - Winner's curse - Zero-sum
Games: Prisoner's dilemma - Chicken - Stag hunt - Ultimatum game - Matching pennies ...
Related topics: Mathematics - Economics - Behavioral economics - Evolutionary biology - Evolutionary game theory - Population genetics - Behavioral ecology
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