Second-Price Auction¶

Definition¶

In a second-price sealed-bid auction, each of $n$ bidders submits a bid $b_i\ge0$ simultaneously and privately. The highest bidder wins (ties broken at random) and pays the second-highest bid, that is, the highest bid among the other bidders; losers pay nothing. In a private-value auction with valuation $\theta_i$ , bidder $i$ 's payoff is $\theta_i-\max_{j\ne i}b_j$ if she wins and $0$ otherwise.

Interpretation¶

The winner's own bid decides only whether she wins, not what she pays. Let $c=\max_{j\ne i}b_j$ be the highest rival bid. Winning is worth $\theta_i-c$ $c$ , which is positive exactly when $c<\theta_i$ $θ_{i}$ . Bidding $b_i=\theta_i$ wins in precisely those cases, so no other bid can do better.

Bidding $b_i>\theta_i$ gives a lower expected payoff than truth-telling when $\theta_i<c\le b_i$ $b_{i}$ : she wins for sure for $c<b_i$ $b_{i}$ , or with the tie-winning probability at $c=b_i$ $b_{i}$ , and pays $c>\theta_i$ $θ_{i}$ .
Bidding $b_i<\theta_i$ gives a lower expected payoff when $b_i\le c<\theta_i$ $θ_{i}$ : she loses for sure for $c>b_i$ $b_{i}$ , or loses with positive probability at $c=b_i$ $b_{i}$ , instead of winning a positive surplus. At $c=\theta_i$ the surplus is zero, so ties do not change the payoff.

Hence truthful bidding is a weakly dominant strategy; it is not strictly dominant, because against many rival bids (for instance $c$ far above or far below $\theta_i$ ) other bids give the same payoff. The argument uses no beliefs about the rivals and no distributional assumptions, but it does rely on private values: with common values, a rival's bid carries information about one's own value.

Key Results¶

With private values, $b_i(\theta_i)=\theta_i$ is weakly dominant for every type, so the truthful profile is a Bayesian Nash equilibrium for any distribution of values.
In the truthful equilibrium the highest-value bidder wins and pays the second-highest valuation.
With $\theta_i$ i.i.d. $\operatorname{Unif}(0,1)$ , expected revenue is the expectation of the second-highest order statistic: $\mathbb{E}[\theta_{(n-1)}]=\frac{n-1}{n+1}$ $\frac{n - 1}{n + 1}$ , the same as in the first-price auction.
The truthful equilibrium is not the only BNE: for $n=2$ with values on $[0,1]$ , "bidder 1 bids $1$ , bidder 2 bids $0$ " is also an equilibrium, but it uses weakly dominated strategies and can give the object to the lower-value bidder.
The English auction with private values leads to the same outcome as the truthful second-price equilibrium.

Examples¶

Case check. A bidder values the object at $10$ . Compare bids $8$ , $10$ and $12$ against highest rival bid $c$ :

$c$	bid $8$	bid $10$	bid $12$
$7$	$10-7=3$	$3$	$3$
$9$	$0$ (loses)	$10-9=1$	$1$
$11$	$0$	$0$	$10-11=-1$
$13$	$0$	$0$	$0$

Bidding $10$ is never worse than $8$ or $12$ and is strictly better against $c=9$ (versus $8$ ) and $c=11$ (versus $12$ ).

Payoff of a bidder with value 10 against the highest rival bid c: bidding 10 gives max{10 − c, 0}, bidding 8 gives up the shaded gain for 8 < c < 10, bidding 12 loses money for 10 < c < 12.

The figure draws the payoff of the bidder in the case check, value $10$ , as a function of the highest rival bid $c$ . Truthful bidding earns $\max\{10-c,0\}$ $c, 0}$ . The shaded interiors show the lost gain for $8<c<10$ when bidding $8$ (e.g. $1$ at $c=9$ $9$ ), and the loss for $10<c<12$ when bidding $12$ (e.g. $-1$ at $c=11$ $11$ ). Random ties also matter at the alternative bids: with one rival, $c=8$ gives bid $8$ expected payoff $1$ versus $2$ from truth-telling, and $c=12$ gives bid $12$ expected payoff $-1$ versus $0$ from truth-telling. Thus the payoff differences occur on the intervals $[8,10)$ and $(10,12]$ , including the tie endpoints; elsewhere all three bids give the same payoff.

Revenue with three bidders. With $n=3$ and $\operatorname{Unif}(0,1)$ values, expected revenue is $\frac{2}{4}=\frac{1}{2}$ $\frac{1}{2}$ . For a bidder of type $\theta$ the expected payment is $\mathbb{E}\big[Y\,\indicator_{\{Y<\theta\}}\big]$ (she pays $Y$ when she wins, that is when $Y<\theta$ $θ$ , and nothing otherwise) with $Y$ the highest of the two rival values, which has density $2y$ on $[0,1]$ : $\int_0^\theta y\cdot2y\,\mathrm{d}y=\frac{2}{3}\theta^3$ $\frac{2}{3} θ^{3}$ . Averaging over $\theta$ and summing over the three bidders gives $3\cdot\frac{2}{3}\cdot\mathbb{E}[\theta^3]=3\cdot\frac{2}{3}\cdot\frac{1}{4}=\frac{1}{2}$ $\frac{1}{2}$ , as stated.

Prerequisites¶

Sources¶

Game Theory · Week 6

Aliases: second-price sealed-bid auction, Vickrey auction, SPA, SPSB auction Keywords: truthful bidding, bid your value, highest losing bid, second-highest bid Symbols: integral, theta

Last updated: 27 October 2026 11:52