Why Is a Second Price Auction Truthful? The Vickrey Auction Explained

In 1893, mail-order stamp dealers were already running an auction that economists would not analyse for another sixty-eight years. The format was simple: collectors posted sealed bids by letter, the highest bidder won the lot, and the price charged was the second-highest bid, usually plus a small increment. David Lucking-Reiley documented this in the Journal of Economic Perspectives in 2000, in a paper whose title says the thing plainly — Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce. The dealers had found the rule independently, decades before William Vickrey was born, and Vickrey appears not to have known they existed.

So: why is a second price auction truthful? Because in a sealed-bid second-price auction for a single item, your bid determines only whether you win, never how much you pay — the price is set by somebody else’s bid. That severs the one link that makes bidding strategic. Bidding your true value is a weakly dominant strategy: it does at least as well as any other bid against every possible combination of rival bids, and strictly better against some. You do not need to know how many rivals there are, how rich they are, or what they think of you.

That last clause is the real content. A first price auction has a solution too, but the solution is a function of your beliefs about everyone else, and if those beliefs are wrong you lose money. The Vickrey auction does something different: it does not solve the strategic problem, it removes it. Truth-telling is optimal pointwise, state by state, not merely on average. Everything below is an unpacking of that claim, its proof, its exact limits, and the awkward fact that the industry most associated with second-price auctions has been quietly walking away from them.

The rule, stated precisely

One indivisible item. Bidders $i = 1, \dots, n$ submit sealed bids $b_1, \dots, b_n$ simultaneously. Bidder $i$ privately knows a value $v_i \ge 0$ : the most the item is worth to them in money. The highest bidder receives the item and pays $\max_{j \neq i} b_j$ , the highest of the competing bids. Ties are broken by a fair coin. Everyone is risk-neutral, so utility is just surplus.

Fix a bidder $i$ and write $b = \max_{j \neq i} b_j$ for the highest competing bid, which bidder $i$ does not observe. Then bidder $i$ ‘s payoff from bidding $b_i$ is

u_i(b_i, b) = \begin{cases} v_i - b & \text{if } b_i > b, \\ 0 & \text{if } b_i < b. \end{cases}

Stare at that for a moment. The variable $b_i$ appears only in the condition, not in the payoff. This is the whole mechanism in one line, and the proof that follows is just a careful audit of what that structural fact implies.

Why a second price auction is truthful: the dominance proof

To show weak dominance we must compare $b_i = v_i$ against every alternative, for every value of $b$ . There are only two kinds of deviation — bid high, or bid low — and for each one the relevant question is where $b$ falls relative to the two bids being compared. Take all amounts in abstract value units.

Case A: overbidding, $b_i > v_i$ . Three subcases.

$b < v_i$ . Both bids win, and both pay $b$ . Surplus $v_i - b > 0$ either way. No difference.
$v_i < b < b_i$ . The truthful bid loses and earns nothing. The inflated bid wins and pays $b$ , which exceeds $v_i$ , so surplus is $v_i - b < 0$ . Overbidding is strictly worse.
$b > b_i$ . Both bids lose. Zero either way. No difference.

Case B: underbidding, $b_i < v_i$ . Again three subcases.

$b < b_i$ . Both win, both pay $b$ . No difference.
$b_i < b < v_i$ . The shaded bid loses and earns nothing. The truthful bid wins and earns $v_i - b > 0$ . Underbidding is strictly worse.
$b > v_i$ . Both lose. No difference.

In table form, with $b$ ranging across the columns:

Deviation	$b$ below both bids	$b$ between the two bids	$b$ above both bids
Overbid $b_i > v_i$	tie: both win, pay $b$	truth gets 0; overbid gets $v_i - b < 0$	tie: both lose
Underbid $b_i < v_i$	tie: both win, pay $b$	truth gets $v_i - b > 0$ ; underbid gets 0	tie: both lose

Every cell is either a tie or a strict win for the truthful bid. That is weak dominance, and it required no assumption whatsoever about the distribution of $b$ — no independence, no symmetry, no common prior. The argument is not probabilistic. It compares outcomes state by state, which is why it survives even if your beliefs about your rivals are completely wrong.

A concrete instance. Suppose the item is worth 100 units to you. Overbid 120 and let the best competing bid be 110: you win and pay 110 for something worth 100, a loss of 10. Underbid 80 and let the best competing bid be 90: you lose an item you would have bought for 90 and valued at 100, forgoing 10. The two mistakes are mirror images, and both are invisible until it is too late, because you never see $b$ before committing.

Weak, not strict: where the indifference lives

It matters that this is weak dominance. Look again at the table: most cells are ties. If the highest competing bid is far below your value, any bid above it wins at the same price, so all those bids are payoff-equivalent. Truth-telling is never beaten, but it is frequently matched.

The consequence is that truthful bidding is not the only Nash equilibrium. Consider two bidders with values 100 and 60 units. Have the first bid 100 and the second bid 0; neither can profit from a change — that is the truthful equilibrium. But now have the first bidder bid 55 and the second bid 101. The second bidder wins and pays 55, earning 5. The first bidder cannot profitably respond: beating 101 would mean paying 101 for an item worth 100. This absurd configuration is a Nash equilibrium of the second-price auction, and it is inefficient — the item goes to the bidder who values it less.

Such equilibria are artefacts, sustained by bids nobody would risk making, and they are ruled out by the dominance criterion itself: bidding 101 with a value of 60 is weakly dominated. But their existence is a reminder that “truthful” here is a statement about dominance, not about uniqueness of equilibrium. That distinction is precisely the one that gets lost when the result is quoted loosely, and it becomes load-bearing later, when we get to generalised second-price auctions. Readers who want the general machinery of equilibrium notions will find it in our discussions of game theory and strategic decisions and mixed-strategy Nash equilibrium.

Why first price auctions force you to guess

Now change one rule: the winner pays their own bid. The payoff becomes

u_i(b_i, b) = \begin{cases} v_i - b_i & \text{if } b_i > b, \\ 0 & \text{if } b_i < b, \end{cases}

and the structural fact that drove the whole argument above is gone. Your bid is now in the payoff, not just the condition.

Immediately, bidding your value is pointless: if $b_i = v_i$ you earn zero whether you win or lose. Every bidder must shade downward, and shading creates a trade-off. Shade a little and you probably win, but with a thin margin. Shade a lot and the margin is fat but rarely collected. The optimum balances the two, and it depends on the chance that a given bid beats everyone else — which depends on how many rivals there are and how their values are distributed.

That is the difference between first price and second price auctions, compressed: in one format the right bid is a fact about you, in the other it is a bet on other people. The first-price bidder is solving an inference problem under uncertainty, in the spirit of Bayesian inference; the second-price bidder is solving nothing at all.

Deriving the symmetric first-price equilibrium

Let values be independent and identically distributed on $[0,1]$ with cumulative distribution $F$ , continuous with density $f > 0$ . Look for a symmetric equilibrium in which every bidder uses the same strictly increasing, differentiable bid function $\beta$ .

Suppose bidder $i$ has value $v$ but bids as though their value were $z$ , submitting $\beta(z)$ . Since every rival uses $\beta$ and $\beta$ is increasing, bidder $i$ wins exactly when all $n-1$ rival values fall below $z$ , an event of probability $F(z)^{n-1}$ . Write $G(z) = F(z)^{n-1}$ , the distribution function of the maximum of the $n-1$ rival values — a straightforward order statistic, of the kind that also drives extreme value theory. Expected surplus is

U(z, v) = \bigl(v - \beta(z)\bigr) G(z).

In equilibrium the best misreport is no misreport, so $\partial U / \partial z$ must vanish at $z = v$ :

-\beta'(v) G(v) + \bigl(v - \beta(v)\bigr) G'(v) = 0 .

Rearranged, this is a linear first-order differential equation,

\beta'(v) + \frac{G'(v)}{G(v)} \beta(v) = v \, \frac{G'(v)}{G(v)} .

Its integrating factor is $\exp\!\int \frac{G'}{G} = G(v)$ , which collapses the left side into an exact derivative:

\frac{\mathrm{d}}{\mathrm{d}v} \bigl[\beta(v) G(v)\bigr] = v \, G'(v).

Integrating from $0$ to $v$ , and using the boundary condition $\beta(0)G(0) = 0$ — a bidder with the lowest possible value cannot win with positive probability and earns nothing — gives the general solution

\beta(v) = \frac{1}{F(v)^{n-1}} \int_0^v x \, \mathrm{d}\!\left(F(x)^{n-1}\right).

The formula has a clean reading. The quantity $\mathrm{d}G(x)/G(v)$ is the conditional density of the highest rival value given that it lies below $v$ , so the integral is a conditional expectation:

\beta(v) = E\bigl[\max_{j \neq i} v_j \;\big\vert\; \max_{j \neq i} v_j < v\bigr] = E\bigl[\text{second-highest value} \mid v \text{ is the highest}\bigr].

In a first price equilibrium you bid what you would have paid in a second price auction, conditional on winning. The two formats aim at the same number; one computes it for you, the other makes you estimate it.

The uniform case

Take $F(x) = x$ on $[0,1]$ , so $G(x) = x^{n-1}$ and $G'(x) = (n-1)x^{n-2}$ . Then

\int_0^v x (n-1) x^{n-2} \, \mathrm{d}x = (n-1)\int_0^v x^{n-1}\,\mathrm{d}x = \frac{n-1}{n} v^{n},

and dividing by $v^{n-1}$ :

\beta(v) = \frac{n-1}{n}\, v, \qquad \text{shading} = v - \beta(v) = \frac{v}{n}.

The shade is exactly $v/n$ : one bidder’s worth of restraint, split off the top. It vanishes as the field grows, which is the formal version of the intuition that competition squeezes out margin. At a value of 80 units:

$n$	$\beta(v) = \frac{n-1}{n}v$	bid at $v = 80$	shading $v/n$
2	$0.500\,v$	40.0	40.0
3	$0.667\,v$	53.3	26.7
5	$0.800\,v$	64.0	16.0
10	$0.900\,v$	72.0	8.0

Note what the table demands of a bidder: to use the right row you must know $n$ . Get it wrong — believe you face two rivals when you face nine — and you leave a third of your value on the table. No row of any table is needed in a Vickrey auction.

Revenue equivalence, computed both ways

If the two formats produce different bids, do they produce different revenue? The revenue equivalence theorem says no, under hypotheses worth stating carefully, because each one does work:

Independent private values. Each $v_i$ is drawn independently and known only to bidder $i$ .
Risk neutrality. Bidders maximise expected surplus, not expected utility of surplus.
Symmetry. All values come from the same distribution $F$ .
Same allocation rule in equilibrium. Both mechanisms, in the equilibria compared, award the item to the bidder with the highest value.
Same payoff for the lowest type. A bidder with the minimum possible value expects zero.

Grant these and expected revenue is identical. Let us check it for the uniform case rather than take it on trust.

Second price. Revenue is the second-highest of $n$ draws. For $n$ independent uniforms on $[0,1]$ , the $k$ -th smallest order statistic has mean $k/(n+1)$ . The second-highest is the $(n-1)$ -th smallest, so

E[\text{revenue}_{\text{SP}}] = \frac{n-1}{n+1}.

First price. Revenue is the winning bid, $\beta(\max_i v_i) = \frac{n-1}{n}\max_i v_i$ . The maximum of $n$ uniforms has mean $n/(n+1)$ , hence

E[\text{revenue}_{\text{FP}}] = \frac{n-1}{n} \cdot \frac{n}{n+1} = \frac{n-1}{n+1}.

The $n$ cancels and the two expressions agree exactly. A Monte Carlo run with 400,000 trials per field size reproduces both to three decimals: for $n = 5$ , both formats return 0.6667 against the predicted $2/3$ ; for $n = 10$ , both return 0.8181 against $9/11 = 0.8182$ .

$n$	$E[\max]$	first-price revenue	second-price revenue	$\frac{n-1}{n+1}$
2	$2/3$	$0.3333$	$0.3333$	$1/3$
3	$3/4$	$0.5000$	$0.5000$	$1/2$
5	$5/6$	$0.6667$	$0.6667$	$2/3$
10	$10/11$	$0.8182$	$0.8182$	$9/11$

The mechanics differ sharply — a first-price seller collects a shaded top bid, a second-price seller collects an unshaded runner-up bid — and the two effects cancel to the last decimal. Shading is not revenue leaking away; it is bidders anticipating, and pricing in, exactly the discount the second-price rule would have granted them anyway. For the seller, then, the choice between formats is not about revenue under these hypotheses. It is about everything the hypotheses exclude: risk attitudes, asymmetry, collusion, and trust.

Vickrey, Goethe, and the stamp dealers

The theory is due to William Vickrey, in Counterspeculation, Auctions, and Competitive Sealed Tenders, published in the Journal of Finance in 1961 (volume 16, issue 1, pages 8–37). The paper’s starting point was not auction design but a proposal of Abba Lerner’s about how a public agency might restore efficient allocation in imperfect markets. Section III is where Vickrey suggests that a sealed tender be awarded at the second-highest bid, and observes that each bidder then has a dominant strategy of bidding true value.

Vickrey shared the 1996 Sveriges Riksbank Prize in Economic Sciences with James Mirrlees, for work on incentives under asymmetric information. The announcement came on 8 October 1996. Vickrey died three days later, and did not live to collect the award; sources differ slightly on the date, the Nobel Foundation recording 10 October and Britannica 11 October. He had also, decades earlier, proposed the congestion-pricing schemes that eventually became electronic road tolling.

The historical irony is the one we opened with. Lucking-Reiley’s 2000 paper argues that second-price sealed bidding was the predominant format for mail-order stamp auctions and that dealers were using it as early as 1893 — before Vickrey was born, and without any of the theory. There is an earlier isolated instance too: Moldovanu and Tietzel documented that in 1797 Goethe sold a manuscript by a sealed-bid second-price procedure. The theory did not create the practice. It explained a practice already in use, in a corner of the economy no economist was reading, which is a recurring shape in applied mathematics and one also visible in the history of voting mathematics.

The ad-tech story: GSP is not Vickrey

Most present-day interest in second-price auctions comes from online advertising, and most of what circulates about it is wrong in an interesting way.

Search engines sell several ad slots at once, ranked by desirability. The mechanism that became standard is the generalised second-price auction (GSP): rank the bidders, give slot $k$ to the $k$ -th highest bidder, and charge each winner the bid of the bidder immediately below them. For a single slot this is exactly the Vickrey rule. For two or more slots it is not — and the difference is not cosmetic.

The Vickrey principle generalises to the Vickrey–Clarke–Groves (VCG) mechanism, in which each winner pays the externality they impose: the loss in value suffered by everyone else because this bidder is present. GSP keeps VCG’s allocation rule but replaces its payment rule with the simpler next-bid-down charge. Edelman, Ostrovsky and Schwarz showed in 2007 that GSP is not truthful: it generally has no dominant-strategy equilibrium, and bidding true value is not an equilibrium at all. The reason is positional. A bidder in the top slot may prefer to lower their bid, drop to the second slot, and pay substantially less for only slightly less exposure. In the Vickrey auction this manoeuvre does not exist, because there is nothing to drop to.

What Edelman, Ostrovsky and Schwarz did establish — and Varian independently, the same year — is weaker and subtler: GSP has equilibria whose payoffs match the VCG outcome, and a corresponding ascending auction has a unique equilibrium with those payoffs. So the claim “ad auctions are Vickrey auctions, therefore bid your value” is a double error: the mechanism is not Vickrey, and its good behaviour is an equilibrium property requiring strategic bidders, not a dominance property requiring nothing.

The industry has since moved further away. On 6 March 2019 Google announced on its Ad Manager blog that it would shift publisher display and video inventory to a unified first price auction, in which the winning buyer pays what it bid, with the transition expected to complete by the end of 2019. Rival exchanges had largely moved to first pricing from 2017 onward; the stated rationale was simplicity and compatibility with header bidding. A documented reversal, then: the format whose shading problem the Vickrey auction was designed to eliminate became the exchange standard.

Search advertising is murkier and must be reported carefully. In testimony in the United States antitrust trial in September 2023, Google’s advertising vice-president Jerry Dischler said the company frequently makes adjustments — internally called tunings or knobs — to its search ad auctions, including raising reserve prices, and that advertisers are typically not told. He confirmed earlier sworn statements that prices had been raised by as much as 5 percent for some advertisers, with around 10 percent possible on some queries. Testimony and exhibits also described a mechanism labelled RGSP, randomised generalised second price, under which the runner-up could be placed in the top slot and the nominal winner in the second. Google’s position, as reported, is that such mechanisms prevent runaway auctions and weight long-term quality; the Department of Justice characterised them as price levers. Much of the testimony was sealed, so the mechanics are known publicly only through redacted exhibits and press accounts. What can be stated as established is the fact of the 2019 first-price migration for Ad Manager and the published 2007 result that GSP is not truthful. The search-auction details belong in the category of reported-under-oath rather than documented-in-specification. The broader machinery by which search results themselves get ranked is a separate story, told in our article on PageRank and Google.

Where the dominance argument stops working

The proof in the third section is airtight, which makes its hypotheses worth naming, because each is a place where practice departs from it.

One indivisible item. Multi-unit and combinatorial settings inherit the idea through VCG, and inherit trouble with it: computing VCG prices over bundles is generally intractable, and VCG revenue can be perverse — adding bidders can lower it, and the mechanism is vulnerable to bidders splitting themselves across identities.

Private values. If bidders are partly guessing at a common underlying value, as in a mineral-rights sale, winning itself is bad news: it means you were the most optimistic. That winner’s-curse correction has no place in the dominance argument, which assumes $v_i$ is known.

No collusion. A ring of bidders can exploit the second-price rule neatly: one member bids high, the rest bid near zero, and the item is bought at the suppressed runner-up price. Second pricing makes the payoff from a cartel unusually large, a coordination logic adjacent to the one in the prisoner’s dilemma.

A seller who does not cheat. This is the sharpest objection, and it is well documented. The winner pays a number they cannot verify. A seller who opens the bids can insert a shill just below the top bid and pocket the difference, undetectably, so long as the winner never learns the true second-highest bid. Rothkopf and Harstad modelled exactly this in 1995, concluding that an honest bid-taker would not choose a Vickrey auction in a one-shot setting, and that a seller who cheats when it pays eventually destroys the reputation the format depends on. Rothkopf, Teisberg and Kahn had earlier asked directly why Vickrey auctions are rare, and put fear of cheating near the top of the list. Note the asymmetry: in a first price auction the same manipulation gains the seller nothing, since the winner’s payment is their own bid.

Bidders who behave as modelled. They do not. Laboratory second-price auctions show persistent overbidding above value, and subjects do not learn their way out of it with experience, as Kagel and Levin reported in 1993. The same experiments found truthful behaviour emerging quickly in ascending English auctions, which are strategically equivalent in theory — so the equivalence fails empirically, with higher revenue in the sealed-bid format. Later reanalysis by Garratt and Wooders found efficiency improving over time while the frequency of overbidding stayed flat, the gain coming from less variable overbidding. A study of highly experienced eBay traders found no net bias in either direction, but still found bidders not bidding their values. Weak dominance, it turns out, is a thin incentive: when most states of the world leave you indifferent, there is little feedback to teach you the optimum.

None of this touches the mathematics. It relocates the interest. The Vickrey auction is a precise demonstration that a strategic problem can sometimes be dissolved by changing the rules rather than solved by out-thinking rivals — the central promise of mechanism design, and a close relative of the constraint-shaping arguments in linear programming and the unintended-consequence arguments in Braess’s paradox. That the dissolution is only as robust as the auctioneer’s honesty, the bidders’ arithmetic, and the single-item assumption is not a flaw in the theorem. It is a reasonably complete list of what to check before trusting it.

Frequently asked

What is a second price auction?

A sealed-bid auction for one item in which the highest bidder wins but pays the second-highest bid submitted. It is also called a Vickrey auction, after William Vickrey's 1961 analysis of it.

Why should you bid your true value in a second price auction?

Because your bid only decides whether you win, never how much you pay. Bidding above your value can win you the item at a price above what it is worth; bidding below can lose you a profitable win. Bidding your value avoids both and is weakly dominant.

What is the difference between a first price and second price auction?

In a first price auction the winner pays their own bid, so bidding your value guarantees zero surplus and every bidder shades downward by an amount that depends on beliefs about rivals. In a second price auction the winner pays the runner-up bid, so truthful bidding is optimal without any guessing.

What is bid shading?

Bidding deliberately below your true value to leave yourself a margin if you win. It is necessary in a first price auction, and with n bidders and uniform values the equilibrium shade is exactly v divided by n.

Do real ad auctions use second price?

Not in the textbook form. Search ads historically used a generalised second-price auction, which is not truthful, and Google moved its Ad Manager display exchange to a first price auction starting in 2019. Testimony in the 2023 antitrust trial described further departures from a plain second-price rule.