First-Price Auction¶

Definition¶

In a first-price sealed-bid auction, each of $n$ bidders submits a bid $b_i\ge0$ simultaneously and without seeing the others' bids. The highest bid wins (ties are broken at random) and the winner pays her own bid; losers pay nothing. In a private-value auction with valuation $\theta_i$, bidder $i$'s payoff is $\theta_i-b_i$ if she wins and $0$ otherwise. A strategy is a bidding function $b_i(\theta_i)$.

Symmetric equilibrium (uniform case). If $\theta_1,\dots,\theta_n$ are i.i.d. with the uniform distribution on $[0,1]$, the symmetric Bayesian Nash equilibrium with an increasing bidding function is

Interpretation¶

Raising the bid has two opposing effects: it raises the probability of winning and lowers the surplus $\theta_i-b_i$ obtained when winning. Bidding one's value guarantees a payoff of $0$, so in equilibrium every bidder shades her bid below her value. The amount of shading, $\theta/n$ in the uniform case, falls as competition increases.

The figure uses the $n=3$ numbers of the Examples: a bidder with $\theta=0.9$ whose two rivals bid $\tfrac{2}{3}\theta_j$ earns the expected payoff $\mathbb{E}[u]=(0.9-b)(1.5b)^2$ for $b\le\tfrac{2}{3}$$\frac{2}{3}$, and $0.9-b$ for higher bids, which win for sure. Low bids rarely win and high bids leave little surplus, so the curve peaks at the equilibrium bid $b(0.9)=0.6$ with payoff $0.243$. The deviations to $0.5$ and $0.65$ (orange) give $0.225$ and $0.238$, and bidding the value gives $0$.

Unlike the second-price auction, the first-price auction has no dominant strategy: the best bid depends on what the rivals bid, so a bidder needs beliefs about the rivals' values and strategies. This is why the equilibrium is computed as a BNE rather than by a dominance argument.

A useful reading of the equilibrium bid: a bidder bids the expected highest rival value, conditional on that value being below her own. In the uniform case the highest of $n-1$ rival values, given that all are below $\theta$, has expectation $\frac{n-1}{n}\theta$, which is exactly $b(\theta)$.

Key Results¶

• General formula for independent private values (IPV). For i.i.d. values with a continuous cumulative distribution function (CDF) $F$ on $[0,\overline\theta]$, the symmetric increasing BNE is $b(\theta)=\theta-\dfrac{\int_0^\theta F(x)^{n-1}\,\mathrm{d}x}{F(\theta)^{n-1}}$ for $F(\theta)>0$$0$; for $\operatorname{Unif}(0,1)$ values this gives $\frac{n-1}{n}\theta$.
• Win probability. Under a symmetric increasing strategy, a bidder of type $\theta$ wins if and only if she has the highest value, with probability $F(\theta)^{n-1}$ ($=$$=\theta^{n-1}$ for $\operatorname{Unif}(0,1)$).
• Expected revenue with $\operatorname{Unif}(0,1)$ values: the winner is the highest order statistic $\theta_{(n)}$ with $\mathbb{E}[\theta_{(n)}]=\frac{n}{n+1}$$\frac{n}{n+1}$, so revenue is $\frac{n-1}{n}\cdot\frac{n}{n+1}=\frac{n-1}{n+1}$$\frac{n-1}{n+1}$, the same as in the second-price auction.
• Interim expected surplus of type $\theta$ (uniform case): $\left(\theta-\tfrac{n-1}{n}\theta\right)\theta^{n-1}=\theta^n/n$${\theta }^n/n$.
• Any bid above one's value is weakly dominated by bidding the value, but bidding the value itself is not a best response to the equilibrium bids of the others when $\theta>0$$0$.
• The Dutch auction is strategically equivalent to the first-price auction.

Examples¶

Deriving $b(\theta)$ for $\operatorname{Unif}(0,1)$ values. Suppose the rivals use $b(\theta)=\beta\theta$ with $\beta>0$$0$. A bidder with value $\theta$ who bids $b\le\beta$ beats a rival $j$ when $\beta\theta_j<b$$b$, which has probability $b/\beta$, so her expected payoff $\mathbb{E}[u]$ is

The first-order condition $-b^{n-1}+(n-1)(\theta-b)b^{n-2}=0$ gives $b=\frac{n-1}{n}\theta$$\frac{n-1}{n}\theta$, independent of $\beta$. Consistency requires $\beta=\frac{n-1}{n}$$\frac{n-1}{n}$.

Numbers with $n=3$$3$. A bidder with $\theta=0.9$ bids $0.6$, wins with probability $0.9^2=0.81$ and expects $0.3\times0.81=0.243$$0.243$. Deviating to $0.5$ wins only if both rivals have $\theta_j<0.75$$0.75$: payoff $0.4\times0.5625=0.225$$0.225$. Deviating to $0.65$ wins if $\theta_j<0.975$$0.975$: payoff $0.25\times0.950625\approx0.238$$0.238$. Both deviations do worse. Expected revenue is $\frac{2}{4}=0.5$$0.5$.

Two bidders. With $n=2$ the bid is $\theta/2$ and expected revenue is $\frac{1}{2}\mathbb{E}[\max(\theta_1,\theta_2)]=\frac{1}{2}\cdot\frac{2}{3}=\frac{1}{3}$$\frac{1}{3}$.

Prerequisites¶

• Private-Value Auction
• Bayesian Nash Equilibrium
• Uniform Distribution
• Order Statistic

Related Concepts¶

• Second-Price Auction
• Dutch Auction
• Revenue Equivalence Theorem
• Density of an Order Statistic

Sources¶

• Game Theory · Week 6

Aliases: first-price sealed-bid auction, First-Price Auction Bidding, FPA, FPSB auction Keywords: bid shading, b(v)=(n-1)v/n, symmetric bidding function, expected revenue (n-1)/(n+1), pay-your-bid auction Symbols: beta, integral, theta

Last updated: 17 October 2026 21:53