The Jar on the Table
Fill a glass jar with coins — quarters, dimes, nickels, pennies — until they total exactly $8.00. Put the jar on a table in front of fifty people. Ask each one to write down what they'd pay for it. What happens next is one of the most instructive disasters in all of economics.
Here's what you'd expect if you've read anything about the "wisdom of crowds": the average guess will be surprisingly close to $8.00. Some people guess too high, some guess too low, and the errors wash out. Francis Galton saw this at a county fair in 1907, when the median guess of an ox's weight was off by less than one percent.1 Crowds are wise. That's the heartwarming story.
But the jar isn't being estimated. It's being auctioned. And in an auction, you don't care about the average guess. You care about the highest one. The person who writes down $12.37, wildly confident that she spotted a few more quarters than she did — she's the winner. She gets the jar. She also gets a jar worth four dollars less than she paid for it.
Congratulations. You've just met the winner's curse.
The insight is so clean it's almost mean: in any auction where the thing being sold has a single true value that nobody knows exactly, the winner is the person whose estimate was most wrong. Not the smartest bidder. Not the most informed. The most overconfident. Selection does the rest — out of fifty guesses scattered around the truth, the maximum is almost always above the truth. Winning the auction is evidence that you overpaid.
This is not a story about stupid people. It's a story about a statistical trap that catches smart people. The three petroleum engineers who first published on it — Ed Capen, Robert Clapp, and William Campbell, writing in the Journal of Petroleum Technology in 1971 — were trying to explain why major oil companies kept losing money on lease auctions, even though their geologists were world-class.2 The geologists weren't wrong on average. They were wrong at the margin, which is the only place that matters in an auction.
Fifty estimates scatter around the true value. The crowd is wise, but the winner is the rightmost dot — almost certainly above the truth.
Winning Is Bad News
Here's one way to think about it. Suppose the jar really does contain $8.00 in coins, and everybody's estimate is unbiased — on average, they'd guess $8.00. But estimates have noise. Some people guess $6, others guess $10, and the spread around the truth is determined by how hard it is to see the coins clearly. Call that spread σ (sigma), the standard deviation of the estimates.
Now: what's the expected value of the highest estimate out of N people?
- V
- True value of the item
- σ
- Standard deviation of estimates (how noisy the guesses are)
- N
- Number of bidders
That √(2 ln N) factor is the kicker. With 10 bidders, it's about 2.15 — so the winner overpays by roughly 2σ. With 100 bidders, it's 3.03. With 1,000 bidders, it's 3.72. The more people who show up to compete, the more spectacularly the winner gets fleeced.3
This isn't an exotic result from advanced statistics. It's the theory of order statistics — the study of what the biggest, smallest, or middle values of a sample look like. And it has a beautifully cruel implication: the more competitive the auction, the worse it is to win.
Oil, Blood, and Money
Capen, Clapp, and Campbell didn't discover the winner's curse in a classroom. They discovered it in the muddy, high-stakes world of offshore oil drilling. In the 1960s, the U.S. government auctioned leases for drilling rights in the Gulf of Mexico using sealed-bid, first-price auctions: you write down what you'll pay, everyone reveals simultaneously, highest bid wins and pays what they bid.
The oil companies had armies of geologists. They ran seismic surveys. They had decades of experience. And they were still losing money. Not because the geologists were bad, but because even a small variance in estimates, fed through an auction with a dozen competing firms, guaranteed the winner was the one whose geology team was most optimistic about that particular tract.
Capen and colleagues looked at actual rates of return on oil lease auctions in the Gulf of Mexico during the 1960s. The industry average return was below the risk-free rate of return — companies would have done better buying Treasury bonds. The companies with the most wins had the worst returns. Winning was literally anti-correlated with profitability.
The story gets better. Once the paper circulated, savvy companies started "shading" their bids — deliberately bidding below their best estimate. ARCO (Atlantic Richfield Company) was one of the first to take the math seriously, and their return on lease acquisitions improved dramatically.4 They didn't get better geologists. They got better at interpreting what winning would mean.
Try It Yourself
Enough theory. Let's play an auction game. Below is a jar of coins with a hidden value somewhere between $4.00 and $14.00. You'll bid against nine AI bidders. Play several rounds and watch what happens to your wallet.
🏺 The Jar Auction
Bid on a jar of coins. Beat the AI bidders. Try not to go broke.
The Curse Goes Corporate
If the winner's curse only afflicted oil companies and coin-jar enthusiasts, it would be a curiosity. But it shows up everywhere someone pays money for something whose value they can't precisely know.
Corporate acquisitions. When Company A announces it's buying Company B, what does Company A's stock price do? It drops. Not always, but on average, acquiring firms lose value at the moment of announcement.5 Why? Because the market is smarter than the CEO. The market sees a company that "won" a bidding war and infers — correctly — that the winner probably overpaid. The board thought the target was worth $40 a share; the market thinks the board is the highest bidder for a reason.
Free agency in sports. When baseball teams compete for a star free agent, the team that signs him tends to overpay relative to future performance. A comprehensive study by economists found that free agents systematically underperform their contracts, especially when many teams were bidding.6 The team that "won" Bryce Harper isn't necessarily the team that valued him most accurately. It might be the team whose spreadsheet had the rosiest projection.
eBay auctions. Researchers have shown that items on eBay with more bidders tend to sell for higher prices relative to their true market value — exactly what the winner's curse predicts. More competition doesn't just raise prices through normal demand; it raises prices because it amplifies the overestimation of the most extreme bidder.
On average, the acquiring company's stock drops on announcement day. The market reads "we won the bidding war" as "we overpaid."
The Math of Regret
Let's make the curse quantitative. The calculator below lets you explore how the expected overpayment scales with the number of bidders and the noisiness of estimates. Play with the sliders and watch the curse intensify.
Notice something? With just two bidders, the curse is mild — maybe half a standard deviation of overpayment. But ramp up to 50 or 100 bidders and the winner is paying multiples of σ above the true value. This is why spectrum auctions (where dozens of telecom companies bid for radio frequencies) and hot real estate markets (where fifteen offers come in on the same house) are such fertile ground for the curse.
What Can You Do About It?
The first defense is simply knowing the curse exists. Capen and his colleagues argued that many oil companies were losing money not because their engineers were incompetent, but because their bidding strategy was naive. If your geologists estimate a tract is worth $10 million, you should not bid $10 million. You should shade your bid downward, by an amount that depends on how many competitors you face.
If you know there are N bidders and estimates have standard deviation σ, you should reduce your bid by approximately σ · √(2 ln N) below your estimate. This is called "bid shading." If everyone does it, the curse goes away — but if only you do it, you'll sometimes lose auctions you should have won, and that takes discipline.
The second defense is to change the auction format. In a second-price (Vickrey) auction, the highest bidder wins but pays the second-highest bid.7 William Vickrey proved in 1961 that this format makes it optimal to bid your true estimate — no shading needed. The curse is reduced because you don't pay your own inflated estimate; you pay the next person's. But it's not eliminated entirely, because even the second-highest bid tends to be above the true value when there are many bidders.
The third defense is due diligence — reduce σ itself. If you can learn more about the true value before bidding, your estimates become tighter, the bell curve in our picture gets narrower, and the winner's extreme overshoot shrinks. This is why sophisticated acquirers spend millions on due diligence before making an offer. They're not just estimating the target's value; they're shrinking the variance of their estimate, which directly reduces the curse.
The Deeper Lesson
The winner's curse is really a lesson about conditional reasoning — the kind of thinking that humans find most unnatural. When you're deciding what to bid, you should not ask "What do I think this is worth?" You should ask: "Given that my bid is the highest, what should I think this is worth?"
These are very different questions. The first ignores the information contained in the act of winning. The second incorporates it. And the information contained in winning is almost always bad news: it means your estimate was more extreme than everyone else's.
Richard Thaler, who would go on to win the Nobel Prize in economics, ran a beautiful experiment in the 1980s to demonstrate this.8 He auctioned jars of coins to MBA students — people trained in economics, people who knew about the winner's curse. And they still fell for it. In twelve out of twelve auctions, the winning bid exceeded the true value of the coins. Knowing about a cognitive trap and avoiding it are two very different things.
In all twelve of Thaler's auctions, the winning bid exceeded the true value. Even MBA students who knew about the curse fell for it.
This is a pattern you see throughout the book. Mathematical reasoning often asks you to condition on events that feel irrelevant. "What's my estimate?" feels like the right question. "What should I believe, given that I won?" feels like a weird afterthought. But the weird afterthought is where the money is.
It's related to Bayesian reasoning — updating beliefs in light of evidence. Winning is evidence. Specifically, winning is evidence that you're at the high end of the estimate distribution, which is evidence that you've overestimated. A rational bidder updates on this evidence before the auction, by shading their bid. A naive bidder treats their estimate as gospel and pays the price — literally.
So the next time you find yourself in a bidding war — for a house, a painting, a company, a free agent — and you feel that rush of competitive excitement as you write down a number higher than you planned, remember: the feeling of winning is one of the most expensive emotions in economics. The crowd is wise. But the winner, almost by definition, is a fool.
Unless, of course, they've read this chapter.