Private-Value Auction¶

Definition¶

In a private-value auction, one indivisible object is sold to one of $n$ bidders. Bidder $i$ has a valuation $\theta_i$ for the object that is known only to $i$ and that does not depend on the information of the other bidders. In a sealed-bid format the bidders simultaneously submit bids $b_i\ge0$ (open formats such as the English and Dutch auctions work analogously with planned drop-out or stopping prices); the auction rules map the bid profile $(b_1,\dots,b_n)$ to a winner and to payments $p_i$. Bidder $i$'s payoff is

(in first- and second-price auctions a loser pays $p_i=0$$0$). In the independent private-values (IPV) model the valuations $\theta_1,\dots,\theta_n$ are i.i.d. draws from a cumulative distribution function $F$ on an interval $[\underline\theta,\overline\theta]$, and $F$ is common knowledge.

Interpretation¶

An auction with private values is a Bayesian game: bidder $i$'s type is $\theta_i$, the distribution $F$ plays the role of the common prior, and a strategy is a bidding function $b_i(\theta_i)$ giving a bid for every possible valuation. The solution concept is Bayesian Nash equilibrium: every type's bid maximizes its expected payoff given the rivals' bidding functions.

The figure draws the two equilibrium bidding functions of the Examples ($n=$$n=2$$2$, values $\operatorname{Unif}(0,1)$). A strategy is the whole line, one bid for every type, not a single number: in the second-price auction each type bids its value, in the first-price auction each type shades its bid to half its value. The type $\theta_1=0.7$ therefore bids $0.7$ or $0.35$, yet both bids give the same interim expected payoff $0.245$.

"Private" refers to two things: each bidder knows its own value, and learning the rivals' information would not change that value. This is what distinguishes it from a common-value auction, where the rivals' signals are informative about one's own value. A frequent confusion is to think that private values must differ across bidders; they may coincide, the point is only that each value is a matter of personal taste rather than an estimate of a shared quantity.

The format matters for strategy but, under the conditions of the revenue equivalence theorem, not for expected revenue.

Key Results¶

• In a second-price auction, bidding $b_i(\theta_i)=\theta_i$ is weakly dominant for every type.
• In a first-price auction with IPV values from $\operatorname{Unif}(0,1)$, the symmetric BNE is $b(\theta)=\tfrac{n-1}{n}\theta$$\frac{n-1}{n}\theta$: bids are shaded below value.
• With IPV values from $\operatorname{Unif}(0,1)$, both formats give expected revenue $\tfrac{n-1}{n+1}$, the expected second-highest valuation.
• In the standard formats the object goes to the bidder with the highest valuation in the symmetric equilibrium, so the allocation is efficient.

Examples¶

Two bidders, uniform values. Let $n=2$ and $\theta_1,\theta_2\sim \operatorname{Unif}(0,1)$ independent. In a second-price auction, bidder 1 bids $\theta_1$. If $\theta_1=0.7$$0.7$, she wins with probability $0.7$ and pays $\theta_2$ when $\theta_2<0.7$$0.7$, so her expected payoff is $\int_0^{0.7}(0.7-y)\,\mathrm{d}y=0.7^2/2=0.245$$0.245$.

In a first-price auction the symmetric BNE is $b(\theta)=\theta/2$$\theta /2$. With $\theta_1=0.7$ she bids $0.35$, wins with probability $\mathbb{P}(\theta_2/2<0.35)=0.7$ and gets $(0.7-0.35)(0.7)=0.245$$0.245$, the same interim expected payoff.

Prerequisites¶

• Bayesian Game
• Type
• Common Prior
• Bayesian Nash Equilibrium

Related Concepts¶

• First-Price Auction
• Second-Price Auction
• Common-Value Auction
• Revenue Equivalence Theorem

Sources¶

• Game Theory · Week 6

Aliases: Private Value Auction, private-values auction Keywords: independent private values, IPV, valuation theta_i, bidding function b_i(theta_i), sealed-bid auction Symbols: integral, theta

Last updated: 14 October 2026 08:21