Thought Toys · Games & choices · Exhibit 127

The prize you only win when you have overpaid.

A sealed jar of coins is going to the highest sealed bid. Nobody knows what is in it; everybody gets a look and writes down an honest guess. The guesses are unbiased — as likely to be under as over — and nobody in the room is greedy. Each bids the number they wrote down, and the highest bid takes the jar and pays it. Watch what the winner pays. Nobody in the room is biased. The winning guess is.

One auction, and the law behind it ● a bidder's guess   ● the winner   ▮ what the winner overpaid

your profit on a win · by the law— your profit on a win · so far— how often you win— per auction entered—

—

Who bids that way
↳ your turn — drag How much you shade your bid to the fair shade, and watch how seldom you win anything

What you're seeing

The dashed line is what the jar is really worth. Every dot above it is one bidder's guess. The guesses scatter on both sides of the truth, and they scatter fairly: a guess is exactly as likely to be too low as too high, and no bidder is any better or any greedier than any other. Yours is the ringed one. Nothing in this room is broken.

Now look at which dot wins. It is not a dot picked at random. It is the one furthest to the right — and being furthest to the right is what winning means. So the winner's guess is not a fair sample of the guesses in the room. It is the highest of them, and the highest of eight honest guesses sits, on average, about 19 above the truth. The red bar along the axis is the gap. That is the winner's curse, and notice what is not in the explanation: nobody was over-eager, nobody miscalculated, nobody wanted it too much.

Drag How many people are bidding. This is the part that stings. With one bidder there is no curse at all — winning tells you nothing, because you were always going to win. Add rivals and the overpayment grows: with eight bidders and guesses that stray up to 25, the winner pays about 19.4 over the odds. More competition is supposed to be a good thing. Here all it does is reach further into the room for a bigger mistake. The chart underneath draws that curve for every room size from one to twelve, and the hollow dot is what has actually happened in the auctions run so far, wandering onto the line as the count grows.

Drag How far off a guess can be down to zero and the curse vanishes entirely. That is the tell. The curse is not made of competition, and it is not made of greed. It is made of error — and competition is only the machine that selects the largest one.

The cure, and the catch

If winning is evidence that you guessed high, then bid as though you had already heard that news: knock off the amount that winning is about to teach you. Bidding below your own guess like this is called shading it, and it is the third slider, measured in multiples of the fair shade — the exact expected overshoot. And now the two buttons matter, because they are two different games, and it is easy to think they are one.

With Everyone shades, all eight bidders knock off the same amount. The same person still wins — subtracting the same number from every bid changes nobody's rank — but the price falls by that amount. At the fair shade the winner breaks even on average, and you still take home one jar in eight. The curse is gone.

With Only you shade, the discount buys something else entirely. Your rivals still bid their guesses, so shading no longer just lowers the price: it costs you the auction. At the fair shade your win rate falls from one in eight to about one in 410. And your profit on those rare wins is still slightly negative. The only auctions you win now are the ones where your guess ran so far above the truth that it beat seven undiscounted rivals even after your discount. Push the slider anywhere you like: the best you can do, per auction entered, is almost exactly nothing. At these settings, bidding cleverly on your own does not beat the curse. It lets you dodge it, by hardly ever buying anything.

It is worth noticing who pays for the cure. When the whole room shades, the winner's rescue comes out of the seller's price, not out of thin air. A room that agrees to bid low together is doing something an auctioneer would recognise, and would not enjoy.

That gap between the two buttons is the real lesson, and it is not a lesson about arithmetic. The cure is cheap when the whole room takes it and nearly worthless when you take it alone. Which is part of why knowing about the curse is not the same as being safe from it.

The rule, exactly. One object is worth V to everyone, and nobody knows V. Bidder i sees si = V + ei, with the errors ei drawn independently and uniformly from −w to +w, so every estimate is unbiased. The uniform spread is a modelling choice, made because it gives a formula you can read; the control below shows what survives without it. The highest bid wins. The winner's estimate is the largest of n such draws, and

E[ largest of n ] = w (n − 1) / (n + 1)

which is the expected overpayment, and the fair shade. It is 0 at n = 1 and at w = 0, and it climbs toward w as the room fills. At the page's defaults, n = 8 and w = 25, it is 25 · 7/9 = 19.44 exactly.

When only you shade, there is no such tidy formula for the outcome, because the shade changes who wins as well as what they pay. The page computes it exactly all the same, by integrating over your own error: you win when your guess, less the shade, beats the best of the n − 1 rivals, so with F the uniform distribution function,

P(win) = ∫ f(x) F(x − shade)n−1 dx

and the same integral weighted by (shade − x) gives the profit. At a shade of zero the two games coincide and both reduce to the closed form above — which is how the proof file checks the integral.

Verified in node before this page existed (improve/verify/127-winners-curse.js, 17 checks, six of them controls):

  • An individual guess is unbiased: 400,000 draws, mean error within four standard errors of zero.
  • The winner's is not: for every room size from 2 to 12, 200,000 auctions each, the mean overpayment matches w(n−1)/(n+1) within four standard errors.
  • That closed form equals the order-statistic integral to 8×10−14.
  • The integral behind the Only you shade numbers matches simulation on six settings, 400,000 auctions each — worst residual 0.96 standard errors.
  • Shading by the law breaks the winner even at every room size, when everyone shades.
  • Shading alone is monotone in both directions: every step of the slider lowers the win rate and raises the profit on a win. At the defaults, 12.50% of auctions won at no shading, 0.24% at the fair shade.
  • And the point of the exhibit, gated because the page asserts it in prose: the best return per auction available anywhere on that slider, bidding alone against naive rivals, is 0.0014 — against 2.43 per bidder per auction when everyone shades twice the fair amount together.
  • Control one. With one bidder there is no curse: mean profit within four standard errors of zero, and the closed form exactly zero.
  • Control two. With w = 0 there is no curse at any room size: the worst absolute profit over 20,000 auctions is exactly 0, not merely small.
  • Control three, the known-bad reading. "Winners are just over-eager" is refuted, not hedged: over 300,000 auctions the highest guess runs +14.291 and the lowest −14.292, summing to zero. The bias is symmetric selection, and the loser is wrong by the same law in the other direction.
  • Control four. Competition alone is not the cause. Run the same room with private values — each bidder simply wants the thing a different amount, with no shared truth to be wrong about — and have the winner pay their own value: the profit is identically zero at every room size, so winning carries no bias whatever. The curse needs a common value, not merely rivals.
  • Control five, on the shape of the errors. The uniform spread is a modelling choice, not a fact about auctions, so it is tested as one. Try normal, triangular and two-point errors instead, all symmetric and unbiased at the same standard deviation. The winner's expected overpayment stays positive, and still grows at every room size from 2 to 10. The tidy formula w(n−1)/(n+1) does not carry over. Symmetry alone fixes the sign; the shape of the spread sets the size.
  • Control six. At a shade of zero the integral must return 1/n and −w(n−1)/(n+1) exactly, and it does, to 8×10−11 — while a deliberately shade-blind version of the same integral, a one-character slip, reports a 12.5% win rate where the truth is 0.24%. The check can tell them apart.

The curse was named in 1971 by three researchers at Atlantic Richfield — Ed Capen, Bob Clapp and Bill Campbell, writing in the Journal of Petroleum Technology — who had noticed that firms winning offshore drilling leases in the Gulf of Mexico kept earning less than they had forecast, year after year, without any obvious incompetence to blame. Their explanation was this one. Robert Wilson had modelled common-value bidding a few years earlier, and Wilson and Paul Milgrom shared the 2020 Nobel Memorial Prize in Economic Sciences partly for the auction theory that grew out of it. The nearest neighbour in this cabinet is Berkson's paradox, which is the same trick played with a different selector: condition on an extreme, and you inherit its bias.

Also in Games & choices: Nash equilibria →

All 12 in Games & choices
  1. 10The evolution of trust
  2. 100The measure went up. The thing barely moved.
  3. 106They meet in the middle. The beach walks twice as far.
  4. 109Twice the soldiers is four times the army
  5. 116He told them nothing they did not already know
  6. 127The prize you only win when you have overpaid — you are here
  7. 27Nash equilibria
  8. 37Braess's paradox
  9. 47The secretary problem
  10. 67Better at both, and still better off trading
  11. 69Everyone was consistent. The vote wasn't.
  12. 97Same votes. Different winner.

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