Understanding the winner’s curse

Sometimes winning comes at too high a cost. In such cases, it is the winners that are the real losers. The phenomenon is known as the “winner’s curse” and it affects a wide variety of situations, from baseball free agency signings to stock market IPOs.

Whether you are an executive in a big auction or a participant on eBay, it’s important to understand the winner’s curse. Here are a few thoughts on how and why the winner’s curse happens.

What is the definition?

The winner’s curse describes a pattern in auctions. In certain auctions, the winner will tend to overpay.

A more precise definition comes from leading auction scholar Paul Milgrom of Stanford University:

The winner’s curse in auction theory traditionally refers to the selection bias that arises because a bidder tends to win more often when his value estimate is too high than when it is too low.

Source: Putting Auction Theory to Work, page 256

Intuitively, why does the winner’s curse happen?

The winner’s curse happens because the auction winner tends to be the bidder that overestimates value.

A simple game will illustrate. Imagine an economics professor auctions off a jar of coins to a classroom. The rules are that students must guess the value of the coins, and the professor will sell the jar to the highest bidder. What can we expect out of this auction?

The student guesses will be distributed in some fashion. If students are guessing honestly, typically the average of the guesses will be very close to the true value of the coins. Averaging the guesses will average out errors and exhibit the phenomenon dubbed “the wisdom of crowds.”

This turns out to be bad news for the auction winner. If the average guess is close to the true value of the coins, then the highest guess must be larger. The winner of the auction will be the student that overestimated the jar of coins and he will lose money on average.

When does the winner’s curse happen in general?

The key characteristic is the jar of coins was worth the same thing to every bidder. This feature is called a common value. In general, auctions where the winning prize has a common value are susceptible to the winner’s curse.

Common value auctions happen on big stages too. Think about a sports league auctioning off the broadcast rights for a sporting event. The broadcast rights have a common value which is equal to the amount of advertising revenue the event can bring in. Different stations will estimate the revenues differently, just like students bidding on a jar of coins. We might expect this auction to exhibit the winner’s curse.

A real-life example? In 2005, the NFL auctioned off the rights to broadcast Monday Night Football. The winner was ESPN, which paid 1.1 billion dollars for the broadcasting rights over eight years. This bid was roughly double what ABC was previously paying. Either ESPN saw revenue that other stations didn’t or it ended up overpaying.

There are other auctions that are susceptible to the winner’s curse. Here is a good explanation relating baseball free agency to oil field economics through the winner’s curse.

Does the winner’s curse affect every auction?

No. Many auctions are immune from the winner’s curse. One notable example is auctions where bidders have an independent private value for the winning prize. That is, the prize has an intrinsic value to each individual person rather than a common value to everyone.

A good example would be an eBay auction for a Batman collector’s item. The item may be worth $10 to one person and $100 to another. The values are completely individual and dependent on personal preferences.

In such an auction, it is easy to avoid overpaying. All one has to do is set a maximum bid less than the individual estimation of worth. The winner can always get the item at a price below his individual value and the winner’s curse should never happen.

Can you illustrate the winner’s curse mathematically in a game?

The following game mathematically illustrates the winner’s curse.

Consider two companies who are bidding for the rights to drill on an oil field reserve of unknown value. Experience indicates the rights could either be worth 10, 20, or 30 million dollars with probabilities 25 percent, 50 percent, and 25 percent, respectively.

Each company buys equipment to evaluate the field. The equipment, however, is crude and can only give readings of “low” or “high” value. The equipment manufacturer promises the equipment has the following accuracy for two separate readings:

• if field is worth 10, then two separate readings will be “low, low”
• if field is worth 30, then two separate readings will be “high, high”
• if field is worth 20, then two separate readings will either be “low, high” or “high, low” with equal probability

Unfortunately, each reading is very expensive so each company can only afford to do one. While they jointly have enough information to determine the oil field value, neither is willing to share that knowledge for competitive concerns.

This means each company has to bid based on a single reading. By Bayes Rule, we can deduce that a single reading of “low” means the field is either worth 10 or 20, with equal probability. Similarly a single reading of “high” means the field is either worth 20 or 30, with equal probability.

Each company has to bid on its expected value of the field, which will be 15 million for a “low” reading and 25 million for a “high” reading.

The problem is that such a strategy will tend to overvalue the field! Here is why.

Suppose you get a “low” reading. What is your expected outcome for bidding 15? It depends on your opponent who will either bid 15 or 25. If your opponent bid 25, then you lose the auction and you get nothing. But if your opponent bid 15, then you are in a tie to win. However, the fact your opponent bid 15 means the other reading was also a “low” one. This means both of you had “low” readings, and hence the field was really worth 10 million. Both of you are overbidding by 5 million, so it would be better not to win. If we assume ties are assigned randomly, then a bidder would lose 1.25 million in this case.

A similar argument can show why a bid of 25 with a “high” reading will lose on average, so I’ll omit the details.

In summary, the winner in this auction tends to lose money.

Can the winner’s curse be avoided?

The winner’s curse happens because players overbid based on their value. One way they could avoid this is by bid shaving, the practice of bidding a lower amount to account for risk of overpaying. In the oil field example, we could imagine the companies shaving their bids by some percentage, like 15 percent or 25 percent.

Unfortunately experience suggests that people don’t learn to bid appropriately:

After the winner’s curse phenomenon became better understood, oil companies were more cautious and started bidding fractions of what they thought the drilling rights were worth in order to avoid the curse. If all of the oil companies were bidding .75 what they thought the true value of the drilling rights were worth, they were much less likely to be bitten. A higher-risk operator might be willing to bid .8 the estimated value, whereas a more conservative player might only bid .65. The problem that quickly surfaced is that newcomers to the industry often did not factor in the fractional weighting when making their bids. Well, these newcomers were often bitten by the winner’s curse, and while the newcomers might have annoyed the older companies, in the long run it was mainly the newcomers that were hurt. (source)

Another workaround is to design better auctions to ease buyer concerns while still raising good revenue.

Key points

• The winner’s curse affects auctions of common value
• Examples include baseball free agency and oil fields valuation
• The key problem is the winner will have the highest valuation and overbid. The fact others don’t want to pay is a bad sign, a form of adverse selection
• Auction design and bid shaving are two ways to deal with the winner’s curse