Common-Value Auction¶

Definition¶

In a common-value auction the object has the same value $v$ for every bidder, but no bidder knows $v$ . Each bidder $i$ privately observes a signal $\theta_i$ that is informative about $v$ , and her payoff from winning at price $p$ is $v-p$ $p$ . Bidder $i$ 's expected value therefore depends on the other bidders' signals as well as on her own.

A two-bidder example with this structure has $v=\theta_1+\theta_2$ $θ_{2}$ , where bidder $i$ observes only her own component $\theta_i$ , and $\theta_1,\theta_2$ are independent.

Interpretation¶

The contrast with a private-value auction is about information, not about who values the object more. With private values, learning a rival's signal would not change one's own valuation; with a common value it would. Classic examples are drilling rights to an oil field of unknown size, or a jar of coins whose total is estimated by eye.

Because the value is shared, winning is itself informative: a bidder wins in a symmetric equilibrium when her signal is the highest, which is bad news about the others' signals. A rational bidder therefore bids on the basis of the conditional expectation of $v$ given her signal and given that she wins (or, in a second-price auction, given that she is just tied). Ignoring this produces the winner's curse.

Truthful bidding results for private values do not carry over directly. In a second-price auction the case argument for bidding "one's value" fails, because one's value is not known; the equilibrium bid must instead be computed as a Bayesian Nash equilibrium.

Key Results¶

In a symmetric equilibrium with increasing bids, $\mathbb{E}[v\mid\theta_i]$ overstates what a bidder can expect when she wins; the relevant quantity is $\mathbb{E}[v\mid\theta_i,\ i\text{ wins}]$ $θ_{i}, i wins]$ .
For $v=\theta_1+\theta_2$ with $\theta_i$ i.i.d. $\operatorname{Unif}(0,1)$ : the symmetric BNE of the second-price auction is $b(\theta)=2\theta=\mathbb{E}[v\mid\theta_1=\theta_2=\theta]$ $θ]$ , and that of the first-price auction is $b(\theta)=\theta$ $θ$ .
In that example both formats give expected revenue $\frac{2}{3}$ .
The Dutch–first-price equivalence holds with common values; the English–second-price equivalence can fail with three or more bidders, because drop-out prices reveal signals.

Examples¶

Take $v=\theta_1+\theta_2$ with $\theta_1,\theta_2$ independent and uniform on $[0,1]$ .

Second-price. Let bidder 2 bid $2\theta_2$ . Bidder 1 with signal $x$ who bids $b\le2$ wins when $\theta_2<b/2$ and pays $2\theta_2$ , so her expected payoff $\mathbb{E}[u]$ (written $u$ because $v$ is the common value) is

\mathbb{E}[u]=\int_0^{b/2}\big(x+y-2y\big)\,\mathrm{d}y =\int_0^{b/2}(x-y)\,\mathrm{d}y.

The integrand is positive for $y<x$ and negative for $y>x$ $x$ , so the payoff is maximized at $b/2=x$ $x$ , i.e. $b=2x$ $2 x$ . Revenue is $2\,\mathbb{E}[\min(\theta_1,\theta_2)]=2\cdot\frac{1}{3}=\frac{2}{3}$ $\frac{2}{3}$ .

First-price. Let bidder 2 bid $\theta_2$ . Bidder 1 with signal $x$ bidding $b\le1$ wins when $\theta_2<b$ $b$ , with expected payoff

\mathbb{E}[u]=\int_0^{b}(x+y-b)\,\mathrm{d}y=xb-\tfrac{1}{2}b^2,

maximized at $b=x$ $x$ . Revenue is $\mathbb{E}[\max(\theta_1,\theta_2)]=\frac{2}{3}$ $\frac{2}{3}$ .

Numbers. With $x=0.6$ $0.6$ , $\mathbb{E}[v\mid\theta_1=0.6]=1.1$ $1.1$ , but $\mathbb{E}[v\mid\theta_1=0.6,\ \theta_2<0.6]=0.6+0.3=0.9$ $0.9$ . The first-price bid $0.6$ leaves an expected margin of $0.3$ when winning.

Value 0.6 plus theta_2 plotted against bidder 2's signal for a bidder with signal 0.6; the winning region theta_2 below 0.6 is shaded and the levels 1.1, 0.9 and the bid 0.6 are marked.

The figure fixes bidder 1's signal at $x=0.6$ $0.6$ , so the value is $0.6+\theta_2$ $θ_{2}$ . Averaged over all $\theta_2$ it is $1.1$ , but in the first-price equilibrium bidder 1 wins only in the shaded region $\theta_2<0.6$ $0.6$ , where the average value is $0.9$ . A bid computed from $1.1$ would overpay on average, which is the winner's curse; the equilibrium bid $0.6$ sits $0.3$ below the conditional value.

Move the bid $b$ : a winning bid means $\theta_2<b$ $b$ , so it buys only the green average value. The naive bid $1.1$ overpays, while $b=x=0.6$ maximizes the expected payoff.

Prerequisites¶

Sources¶

Game Theory · Week 6

Aliases: Common Value Auction, pure common-value auction, common-values auction Keywords: private signal, interdependent values, wallet game, v = theta_1 + theta_2, conditioning on winning Symbols: integral, theta

Last updated: 22 October 2026 19:04