Winner's Curse¶

Definition¶

The winner's curse is the systematic overpayment that occurs in a common-value auction when bidders bid on the basis of their naive estimate $\mathbb{E}[v\mid\theta_i]$ of the common value $v$ , which ignores the information contained in the event of winning. Because a bidder tends to win precisely when her signal $\theta_i$ is among the most optimistic, the value conditional on winning is lower:

\mathbb{E}[v\mid\theta_i,\ i\text{ wins}]\ <\ \mathbb{E}[v\mid\theta_i]

in the usual models, so a naive winner expects to pay more than the object is worth on average.

Interpretation¶

Each bidder's estimate may be unbiased, yet the winning estimate is biased upwards: the winner is selected as the bidder with the largest estimate, and the largest of several noisy estimates overshoots the truth on average. Winning is bad news about what the other bidders observed.

Densities of one bidder's estimate, uniform on 80 to 120, and of the largest of three such estimates, with the true value 100 and the mean 110 of the largest estimate marked.

In the figure (the jar-of-coins example below), each estimate $\theta_i=100+\varepsilon_i$ is uniform on $[80,120]$ and centred on the true value $v=100$ $100$ . The largest of three estimates has density $3(x-80)^2/40^3$ for $80\le x\le120$ $120$ , which piles up near $120$ , and its mean is $\mathbb{E}[\max_i\theta_i]=110$ $110$ . If every bidder bids her estimate, the bidder with the largest estimate wins, so on average the winner pays $10$ more than the object is worth.

Add bidders: the largest estimate moves further above the true value, so a naive winner's expected loss $w(n-1)/(n+1)$ grows. With $n=1$ there is no curse.

The curse is a mistake of naive bidders, not a feature of equilibrium. A rational bidder computes the conditional expectation of $v$ given her own signal and the event that she wins, and shades her bid accordingly. In a Bayesian Nash equilibrium of a common-value auction, winners do not lose money on average. The downward adjustment for this effect is larger with more bidders, because winning against many rivals is worse news; stronger competition pushes bids in the other direction, so the net effect of $n$ on bids depends on the model.

The curse does not arise with private values: a rival's signal then says nothing about one's own value. The same logic of "the event that the trade happens is informative" appears in adverse selection, for example in the market for lemons.

Key Results¶

If $n$ estimates are $\theta_i=v+\varepsilon_i$ with i.i.d. errors of mean $0$ , then $\mathbb{E}[\max_i\theta_i\mid v]=v+\mathbb{E}[\max_i\varepsilon_i]>v$ for $n\ge2$ (non-degenerate errors), and the bias grows with $n$ .
Rational bidding conditions on winning: bid less than $\mathbb{E}[v\mid\theta_i]$ in a first-price auction; in a second-price auction, bid $\mathbb{E}[v\mid\theta_i,\ \text{highest rival signal}=\theta_i]$ in the symmetric equilibrium.
In equilibrium the winner's expected profit is nonnegative; only bidders who ignore the information in winning suffer the curse.

Examples¶

Jar of coins. The true value is $v=100$ $100$ . Each of $n=3$ bidders observes $\theta_i=100+\varepsilon_i$ with $\varepsilon_i$ i.i.d. $\operatorname{Unif}(-20,20)$ , so each estimate is unbiased. The largest error is the maximum order statistic of three uniforms: $\mathbb{E}[\max\varepsilon_i]=-20+40\cdot\frac{3}{4}=10$ $10$ . If everybody bids her estimate in a first-price auction, the winner pays $110$ on average for an object worth $100$ : an expected loss of $10$ .

Two-bidder common value. Let $v=\theta_1+\theta_2$ with $\theta_1,\theta_2$ i.i.d. $\operatorname{Unif}(0,1)$ , and bidder 1 observes $\theta_1=0.6$ $0.6$ . Her unconditional estimate is $\mathbb{E}[v\mid\theta_1=0.6]=0.6+0.5=1.1$ $1.1$ . Suppose both bidders naively bid their unconditional estimate $\theta_i+0.5$ in a first-price auction. Bidder 1 then wins exactly when $\theta_2<0.6$ $0.6$ , and $\mathbb{E}[v\mid\theta_1=0.6,\ \theta_2<0.6]=0.6+0.3=0.9$ $0.9$ . She pays $1.1$ , so she loses $0.2$ on average whenever she wins. The equilibrium bid in this model is $b(\theta)=\theta$ $θ$ , here $0.6$ , which leaves an expected margin of $0.3$ when winning.

Prerequisites¶

Sources¶

Game Theory · Week 6

Aliases: Winner’s Curse, winners' curse Keywords: overbidding in common-value auctions, conditioning on winning, E[v | signal, win], jar of coins, oil-field auction Symbols: epsilon, theta

Last updated: 27 October 2026 11:52