That Damned Winner's Curse

It would not be better if things happened to men just as they wish.

~ Heraclitus of Ephesos

When reading Binmore’s (2007) very brief introduction to game theory, I was reminded of the winner’s curse, a classic anomaly in economic behavior. Binmore introduces the concept with two examples that were already classic when Richard Thaler (1988) wrote his famous essay on the topic. The first example is the auctioning of licences to drill for oil. The bidding companies have access to different surveys, with some making more optimistic estimates about the quantity of oil than others. The differences among the surveys are an index of their unreliability. There is only one true amount. The companies presumably bid in accord with the survey results that they have, and the company with the most optimistic survey will probably win the licence to drill. Unless all surveys underestimate the amount of oil, the winner will overpay, and he will recognize his curse while drilling. The classroom analog of this scenario involves a jar filled with coins, which the instructor offers to the highest bidder. This is an easy way for the instructor to get lunch money.

The winner’s curse is described as the result of a cognitive fallacy, a failure to think rationally, like a trained economist would (BTW, trained economists are often involved in high-stakes auctions or bidding wars in mergers & acquisitions). To me, the winner’s curse seems like a case of the familiar regression effect (Fiedler & Krueger, 2012). If there is one true value, such as the amount of oil in a field or money in a jar, estimates can be modeled as the true value + systematic bias (general over- or underestimation) + random error. The highest estimate is, by definition, the estimate with the largest positive error. If bias is smaller than half the range of the estimates, the average estimate is more accurate than the highest estimate. Analogously, the average bid in an auction is closer to the item’s true value than the highest bid is. A bidder who knows this will either not bid, or bid to approximate the average bid, which guarantees that he will not get the item. So why bid at all?

Thaler (1988) obliquely hints at a rational approach to bidding, but he does not reveal what that might look like. If optimal bidding involves integral calculus or anything that complex, there is no point accusing individuals of behaving myopically and irrationally. Nor can the recommendation be to not bid at all, because if that recommendation were followed by all but one, the one rebel could buy an oil field (or a jar of money) for a nickel. So what we have here is a non-cooperative game, in which bidding is an act of selfish defection, which ultimately brings on the winner’s destruction. Up until that point, bidding appears to be rational, and so we are thrown back on the question of when the rational bidder will stop.

Suppose the auction is set up such that each bidder makes an offer, gets to see all offers, and is allowed to submit a revised bid. Not knowing if there is systematic bias in the distribution of bids, bidders may assume that the item’s true value is the mean estimate. To make a profit, each bidder is motivated to make a second offer that is just below that mean. But they all do. So the winner in stage two may not be cursed by a loss but mocked by a microscopic gain. [Note, however, that only crazy sellers would allow this sort of thing.] Alternatively, bidders may assume that there is a negative bias in the initial distribution due to overall risk aversion. Bidders can now make offers greater than the mean, but how much greater? They know that they may enter winner’s curse territory at any moment – but when?

Bidders may also consider the number of other bidders. This number is positively associated with the expected size of the winner’s curse. A rational buyer may therefore look for items on which few others are bidding. However, a small number might also imply that the item in question is not worth having.

Finally, a rational course of action may be to enter the bidding by making an offer based on one’s information (perception or survey result) while heavily discounting that information by its (also estimated) unreliability. Doing so, no one can get away with stealing gold for a nickel. Assuming that not everyone will follow this strategy, you will almost never purchase anything when competitive bidding is involved, but on the rare occasion that your bid is the highest one (or the only one) you will profit. Otherwise, focus your efforts on unequal-value auctions, where you can win uncursed because you derive greater satisfaction from the item than your competitors would.

After this brief introduction to bidding and regressing, oil fields and jars of money, take a moment and contemplate the implications for mate selection, and particularly the games of rivalry played by men. Are you cursed with an overvalued mate?

Binmore, K. (2007). Game theory: A very brief introduction. Oxford, UK: Oxford University Press.

Fiedler, K., & Krueger, J. I. (2012). More than an artifact: Regression as a theoretical construct. In J. I. Krueger (Ed.). Social judgment and decision-making (pp. 171-189). New York, NY: Psychology Press.

Thaler, R. H. (1988). Anomalies: The winner's curse. Journal of Economic Perspectives, 2, 191–202.

Objections

One commentator writes:

"Much of [your analysis] turns on the notion of "true" value. Since economic value is typically defined as "what the market will bear" one cannot argue that something is over-valued (because the definition of value is exactly what you just paid)."

I disagree. I you make the winning bid for a jar of money, finding that you paid $11 when the jar "truly" contains $10, you have overpaid by $1. Now that the contents of the jar are common knowledge, you are stuck with your loss. To still assert that the $10 jar must evidently be worth $11 to you because that is what you paid, is to push the theory of revealed preference to tautological absurdity. Presumably, you would not offer $11 if you knew much much money the jar contained.The oil field example is analogous. The winner's curse strikes if there is unreliable information about the item's resale value, that is, its value after information has become more reliable and valid.

The commentator continues:

"This gets more complicated when we recognize different individuals have different values (that's why Michael Jackson could outbid Paul McCartney for the Beatles songbook--he was more willing to exploit the tunes for cash and thus he won the bidding because they were worth more to him."

It not only gets more complicated, but you are playing a different game, one in which the winner's curse, by definition, does not apply. The winner's curse is limited to auctions in which the item is worth the same to everyone once its "true value" (yes!) is known.