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A jar of coins is passed around a room and everyone bids for it. The average guess is close to the true value, but the winning bid is far above it. The winner is not the best guesser. The winner is the one who overestimated the most.

Common values and the winner’s curse

In a private-value auction, your value is your own. In a common-value auction everyone wants the same thing, an oil lease, a spectrum license, a company, but nobody knows exactly what it is worth. Each bidder sees a noisy signal, and the mere fact that you won is bad news: it means your signal was probably too high.

From earlier lessons

Two quick ones before we start.

In a second-price auction with private values, you should bid...

Under incomplete information, a rational bidder best-responds to...

Predict, then play

The item has one true value, unknown to everyone. You see a noisy signal of it, and so does each rival. Bid, then see what it was really worth. Try adding more rivals.

Why did that happen?

Take a guess, then reveal the reason

Your signal can be unbiased before you bid: on average it equals the true value. But condition on winning. You win when your signal is the highest of everyone’s, and the highest of several unbiased signals tends to sit above the truth. So among the times you actually win, your signal can systematically overstate the value, and bidding it can mean overpaying. The cure is to calculate value as if you already know you won, then bid from that conditional estimate.

The math

Definition, common-value auction and the winner's curse

All bidders share the same unknown value; each sees a private signal correlated with it. A naive bidder who treats an interim signal estimate as a final value can overpay, because winning is correlated with having the most optimistic signal. Correcting for this means conditioning the value estimate on winning.

Take a common value $V$ with $n$ bidders, each seeing a signal $X_i$ correlated with $V$ . Before the auction, a bidder forms an interim estimate $\mathbb{E}[V \mid X_i=x]$ . But a bid is paid only when it wins. In a symmetric first-price model with an increasing bidding rule, the marginal winning event is that the highest rival signal equals yours, $Y_i = \max_{j\ne i}X_j=x$ . The equilibrium bid is therefore

b(x) = \mathbb{E}[V \mid X_i=x,\;Y_i=x].

The second condition is the winner’s-curse correction. It is usually bad news that every rival’s signal was no higher than yours, so this conditional expectation is below the interim estimate. Its numerical size is not universal: it depends on the prior, signal distribution, number of bidders, and auction format.

A note from the real world

Reality check

The curse is not just a theorem; people fall for it. In classroom jar-of-coins auctions, winning bids can exceed the true value, and oil-lease auctions motivated the original analysis. Experienced bidders learn to account for the information in winning. The Behavioral unit returns to how bidders learn (or fail to learn) to escape the curse.

Work it out

Condition on winning

You see a signal $x$ . Naively bidding $x$ treats your interim estimate as if it were the value. But your bid is paid only if it wins. In a symmetric first-price auction, a marginal win means the best rival signal is about $x$ too. The relevant estimate is therefore $\mathbb{E}[V \mid X_i=x, Y_i=x]$ , not $\mathbb{E}[V \mid X_i=x]$ .

That second estimate needs the model. A prior over $V$ , a distribution of signal noise, the number of bidders, and the auction format all enter it. The common lesson is precise even when a number is not: calculate value conditional on winning, then bid from that estimate. Bidding a raw signal ignores information contained in the win itself.

Key idea

When bidders share a common value, winning is bad news: your signal may overstate the prize. Bid from your estimate conditional on winning, not from the raw signal. The size of that correction comes from the model.

Check yourself

Check yourselfapply

Your signal of a common value is unbiased. Why is bidding it a mistake?

Check yourselfanalyse

You move from 2 bidders to 8. Can the right numerical bid be inferred from that fact and your signal alone?

Check yourselfbrings back: auctions

How does optimal bidding here differ from the private-value second-price auction you saw before?

Sources

Wilson, R. (1969). Competitive bidding with disparate information. Management Science, 15(7), 446-448.
Milgrom, P. R., and Weber, R. J. (1982). A theory of auctions and competitive bidding. Econometrica, 50(5), 1089-1122.

Common values and the winner's curse