Common-Value Auctions and the Winner’s Curse

Why Winning Changes What a Bidder Knows

In a private-value auction, a bidder’s value does not depend on information held by rivals. In a common-value auction, the object has the same value to every bidder, but that value is uncertain and bidders observe different signals about it. A bidder who wins has usually outbid rivals with lower signals. That event is informative: it changes what the winner should expect the object to be worth.

Ignoring this information can produce the winner’s curse. A bidder may estimate the value from their own signal alone, bid as if that estimate were the relevant expected value, and then discover that winning was evidence that the estimate was too high relative to the information held by others. Equilibrium bidding accounts for this selection event. The derivation below uses a deliberately simple model so that both the conditional expectation and the optimal bid can be calculated explicitly.

A Common-Value Model with Independent Signals

There are \(N\geq 2\) risk-neutral bidders in a first-price sealed-bid auction. The highest bidder wins and pays their own bid. Each bidder \(i\) observes a private signal \(X_i\), and the signals are independent and uniformly distributed on \([0,1]\). The object’s value is common to all bidders and is the average of their signals:

Thus, bidder \(i\) knows \(X_i=x\), but not the other signals that enter \(V\). Assume there is no reserve price, all \(N\) bidders participate, and ties have probability zero. We look for a symmetric, increasing bidding strategy \(b(x)\). Because it is increasing, a bidder who submits the bid \(b(z)\) wins exactly when all \(N-1\) rival signals are below \(z\).

The signals are independent before the auction outcome is observed. Conditional on \(X_i=x\), the other signals still have mean \(1/2\), so

But if bidder \(i\) wins by using the cutoff \(z\), every rival signal is below \(z\). A uniform random variable on \([0,1]\), conditional on being below \(z\), has mean \(z/2\). Therefore,

The probability that all rivals’ signals are below \(z\) is \(z^{N-1}\). Both the probability of winning and the value conditional on winning depend on the cutoff. That dependence is central to the bidding decision.

Deriving the Symmetric First-Price Bid

Suppose a bidder with signal \(x\) chooses the bid \(b(z)\), corresponding to cutoff \(z\). The bidder wins with probability \(z^{N-1}\). Conditional on winning, the expected value is \(x/N+(N-1)z/(2N)\), and the payment is \(b(z)\). The bidder’s expected profit is therefore

In a symmetric equilibrium, a bidder with signal \(x\) chooses \(z=x\). For an interior signal, differentiate expected profit with respect to \(z\) and impose the first-order condition at \(z=x\):

Dividing by \(x^{N-2}\) and collecting terms gives a differential equation for the symmetric bid function:

Multiply both sides by \(x^{N-2}\). The left-hand side becomes the derivative of \(x^{N-1}b(x)\). Integrating from \(0\) to \(x\), and requiring the bid to remain finite at zero, gives

The finite-at-zero condition rules out an additional term proportional to \(x^{-(N-1)}\), which would diverge as \(x\) approaches zero. The candidate equilibrium bid is increasing, as required for the cutoff interpretation.

Checking That the Bid Is a Global Best Response

A first-order condition alone does not establish that a strategy is an equilibrium. To check the candidate, substitute the proposed bid function into the bidder’s expected-profit function. Write \(k=(N-1)(N+2)/(2N^2)\), so \(b(z)=kz\). Then

For \(z>0\), differentiating yields

This derivative is positive for \(z<x\) and negative for \(z>x\). Thus expected profit increases up to \(z=x\) and decreases after it. The bidder’s best cutoff is \(z=x\), including the boundary choices \(z=0\) when \(x=0\) and \(z=1\) when \(x=1\). Since bids outside the range of the increasing strategy cannot induce a cutoff beyond the endpoints, this verifies the best response across the available choices. Every bidder making this choice confirms the symmetric equilibrium.

How the Winner’s Curse Shapes Bids

At the equilibrium cutoff \(z=x\), the expected common value conditional on winning is \((N+1)x/(2N)\). The expected value conditional on the bidder’s signal alone is \(x/N+(N-1)/(2N)\). Their difference is

For any \(x<1\), this quantity is positive: conditioning on winning lowers expected value. The effect is strongest for lower signals and vanishes at \(x=1\), where no rival signal can exceed the bidder’s signal. In this model the bid is also below the expected value conditional on winning. But that comparison alone is not a general bidding rule. The equilibrium bid balances the value conditional on winning against the probability of winning and the consequences of changing the cutoff.

The model makes the source of the adjustment especially clear. Raising a bid increases the chance of winning, but it also makes winning possible against more rival signals. Those newly included outcomes change the bidder’s conditional expected value. A sound bidding calculation therefore conditions on the event that the bid wins; it does not simply take an expected value based on the bidder’s own information and subtract an arbitrary amount.

Common Mistake

• Wrong reasoning: “My signal implies an expected value of \(0.6000\), so I should bid as if \(0.6000\) were the value whenever I win.” This ignores the information in winning.
• Correction: Calculate value conditional on both the bidder’s signal and the winning event. For the three-bidder example with \(x=0.8\), that expectation is \(0.5333\), not \(0.6000\). Then optimize expected profit, including the probability of winning and the payment.
• Another confusion: The winner’s curse and first-price bid shading are not the same thing. The winner’s curse is the selection effect in conditional value; bid shading is the strategic choice to bid below an appropriate value benchmark to trade off payment against winning probability.