The Verification Venue · an institution you can operate
The Price You Didn't Bid
Change one rule, that the winner pays the runner-up's bid rather than their own, and lying stops paying. Not "usually". Not "on average". There is no value your rivals could hold, no bid they could submit, under which any bid other than your honest one does better for you. You can check every case at once, below.
William Vickrey wrote this down in 1961 and it earned a share of the 1996 Nobel Memorial Prize in Economics. The mechanism is trivial to state: everyone submits a sealed bid, the highest bid wins, and the winner pays the second-highest bid. The reason honesty is safe is one sentence long, and once you have seen it you cannot unsee it: your bid never sets your price. It only decides whether you win. So raising your bid can only buy you auctions you did not want, and lowering it can only lose you auctions you would have profited from.
Layer one · the whole strategy space, at once
Every lie you could tell, priced
Set your private value for the item. The map below shows your payoff for every bid you could make (left to right) against every highest rival bid that could turn up (bottom to top). Green is profit, red is a loss you paid for, grey is going home empty-handed with nothing lost.
The most you would pay and still be glad. Prices here run 0 to 1; call it a fraction of your budget.
Drag it away from your value in either direction and watch the worst-case column below go negative. It is exactly zero at one point only.
Worst case, your bid
0.00
over all rival bids
Worst case, honest bid
0.00
always exactly zero
Rival bids where your bid beats honesty
0
out of 101 checked
The picture is the proof. The green triangle is where you win and profit; the red triangle above it is where you win and it costs you. Bidding above your value does nothing except widen the red. Bidding below it does nothing except shave green off the bottom of your own column. The honest bid is the unique column that has all of the green and none of the red, and that is what "weakly dominant" means: never worse, sometimes strictly better, against every possible opponent.
Seven candidate strategies, scored against all 101 rival bids on the grid.
| strategy | bid | worst | best | mean | beats honest |
|---|
Two failure modes, and only two. Overbid and you buy auctions you should have lost: value 0.40, bid 0.70, a rival at 0.55, and you have just paid 0.55 for something worth 0.40 to you. Underbid and you lose auctions you should have won: value 0.80, bid 0.50, a rival at 0.65, and you walked away from 0.15 of free surplus. Honesty avoids both without needing to know anything at all about your rivals. That last clause is the whole point: a dominant strategy needs no beliefs.
Layer two · the answer to "so what, everyone knows this"
Four auctions, one number
The usual dismissal is that this is just eBay, that everyone already bids their value, and that the choice of format is decoration. The second half of Vickrey's 1961 paper is the reply, and it is much stranger than the first half. Under independent private values, risk-neutral bidders, and a common value distribution, the four classic formats raise the same expected revenue. Not similar. The same.
The first-price bidder shades: with 5 bidders each bids 0.800 of their true value. That shading is not a guess. It is the unique symmetric equilibrium, and the arena below finds it by brute force as well as quoting it: it sweeps 2001 candidate shading factors for your own bid, holding the other bidders at (n−1)/n, scores each one by exact expected payoff, refines the winner by golden section, and reports the argmax next to the quoted formula so you can watch the two agree to within 10−7. Shade less and you win too often at prices you dislike; shade more and you lose auctions you wanted. The two effects cancel exactly at (n−1)/n, which is what makes it an equilibrium: the best answer to everyone else playing it is to play it yourself.
Values are drawn independently, uniform on [0,1], fresh each auction.
The clocks tick in real steps of 1/4096, so the two clock formats carry a real, named granularity error.
The generator is mulberry32, the same one the offline verifier runs, so the numbers here and there match digit for digit.
Every auction is simulated as a mechanism, not as a formula. The English and Dutch prices come from a clock that actually ticks.
Closed form (n−1)/(n+1)
0.6667
the number all four should hit
Widest gap, simulated vs exact
·
across the four formats
Best shading found by search
·
searched, not quoted
Two of those four equalities are not statistical at all. A Dutch clock and a sealed first-price auction give a bidder exactly the same information at the moment of decision (none), so the strategies coincide and the price is identical on every single draw, not merely on average. The same holds for the English clock and the sealed second-price rule under private values. The arena confirms this: the per-draw gap never exceeds one clock tick, which is a discretisation artefact and nothing else. The genuinely surprising equality is the one between the pairs, first-price against second-price, and that one holds only in expectation.
Layer two, continued · the theorem that ignores the room
The reserve price that refuses to move
Now change hats. You are the seller. You may set a public reserve: bids below it are refused and the item goes unsold. What reserve maximises your revenue, and how should it depend on how many bidders show up?
The intuition almost everyone has is that a crowded room needs no reserve (competition will do the work) and a thin room needs a high one. The intuition is wrong, and Roger Myerson's 1981 paper says why. Write the virtual value ψ(v) = v − (1−F(v))/f(v). The optimal mechanism sells to the bidder with the highest virtual value, provided it is positive, so the reserve is wherever ψ crosses zero. For values uniform on [0,1] that is ψ(v) = 2v − 1, zero at v = 1/2. The number of bidders does not appear in that equation. It cannot, because the equation is about a single bidder's own distribution.
You do not have to take that on faith. Below is the seller's expected revenue as a function of the reserve, assembled from scratch (no bidder clears the reserve: revenue 0; exactly one does: revenue equals the reserve; two or more do: revenue is the second-highest value), then grid-searched over 1001 candidate reserves. Move the room size and watch the peak stay put.
Two bidders or forty, the grid search returns the same answer.
Drag it and watch what the seller gains, and what the room loses.
Grid-search optimum r*
0.500
over 1001 reserves
Seller's gain over no reserve
·
Sales destroyed on purpose
·
item goes unsold
That third readout is the part worth sitting with. The optimal reserve is not a safety margin or a bargaining posture. It is a threat that must sometimes be carried out, and with two bidders it is carried out a quarter of the time. A quarter of all auctions end with a buyer who wanted the item, a seller who wanted to sell it, and no sale, and that is the arrangement that provably maximises the seller's revenue. Efficiency and revenue are different objectives, and here they visibly pull apart: the reserve destroys 0.083 of expected surplus at n=2 to gain the seller 0.083.
And then the second thing. Watch the gain column as the room fills.
Optimal reserve, computed independently for each room size.
| n | r* | no reserve | at r* | gain | unsold |
|---|
The location of the optimum is stubbornly independent of the crowd. The value of the optimum is not: it collapses roughly like the probability that the reserve binds at all, which is rn. By ten bidders the seller is fighting for the fifth decimal place. This is the honest reading of Myerson for anyone running a real auction: the theorem is exactly right and increasingly beside the point, and the reliable way to raise revenue is to attract one more bidder rather than to tune the reserve. Bulow and Klemperer (1996) made that precise: a plain English auction with n+1 bidders beats the optimal mechanism with n bidders.
Layer two, closing · where the folklore is simply false
The auction that isn't Vickrey
The most repeated claim about second-price auctions is that Google's ad auction is one. It is not, and the difference has been worth a great deal of money. Google Search ads run a generalised second-price auction (GSP), in which the advertiser in each slot pays the bid of the advertiser in the slot below. With one slot that is exactly Vickrey. With two or more it is a different mechanism, and truthful bidding is no longer dominant, which Edelman, Ostrovsky and Schwarz proved in 2007 with an instance you can operate here.
Two ad slots with click-through rates you can set, and three advertisers valuing a click at 10, 4 and 2. Bid honestly and the top advertiser wins slot one. Shade the top bid down and watch what happens.
Slide it down to 3 and the advertiser gives up slot one on purpose.
Slot 3 is the fall-off-the-page slot: zero clicks. Ranking is by bid alone, which is the textbook GSP; Google's live auction also multiplies in a quality score, and a randomised variant (RGSP) disclosed in the 2023 antitrust trial. Neither change makes it truthful.
GSP, bidding honestly (10)
1200
slot 1 at price 4
GSP, bidding 10.0
1200
GSP, best bid found
1592
searched 0 to 20
VCG, same instance, honest
1598
pays 402 for slot 1
VCG, same shading deviation
1592
At 200 and 199 clicks the shading pays: 1592 against 1200, a 33% improvement on telling the truth. Pull slot 2's clicks down and the incentive reverses and honesty wins again. That reversal is the theorem. A dominant strategy is one that wins under every configuration; a strategy that wins under some click-through rates and loses under others is not dominant, and an advertiser in GSP therefore has to reason about the other advertisers. Swap in VCG on the identical instance, charging each advertiser the harm they impose on the others, and honesty is restored: 1598 beats 1592.
Google has known this since at least 2007, and Hal Varian, its chief economist, wrote his own analysis of the position auction the same year. The company nonetheless kept GSP for search, through 2026, while moving other inventory to first-price outright: Google Ad Manager's display exchange in 2019, AdSense announced in October 2021 and completed by the end of that year. (Meta is widely described as running a VCG-style auction instead, but we found no primary source from the company stating its pricing rule, so we do not assert it here.) There is no consensus that any one of these is best; there is only a well-understood tradeoff between truthfulness, revenue, and being explicable to an advertiser.
The check: every number on this page, recomputed here
Nothing above is quoted from a table. Each figure is computed in your browser at read time by the same routines the offline verifier runs. What follows is the live state of that computation, plus every assumption and free choice that could move a number.
1 · the dominance grid
2 · the four formats
3 · the reserve
4 · the position auction
free choices, named
- Independent private values. Each bidder knows their own value and learns nothing about it from anyone else's bid. Everything on this page is conditional on that. Under common values (an oil tract, a company) the analysis changes completely: winning is itself bad news about your estimate, the winner's curse appears, and second-price truthfulness fails. This page deliberately contains no common-value auction.
- Risk neutrality. Bidders maximise expected surplus. Revenue equivalence fails under risk aversion: risk-averse bidders shade less in a first-price auction to secure the win, so first-price and Dutch then raise more than second-price and English. This is a known, standard exception, not a fringe objection.
- Symmetric uniform[0,1] values. The specific numbers (n−1)/(n+1), (n−1)/n and r* = 1/2 are all consequences of that distribution. Revenue equivalence itself is far more general (any common, atomless, regular distribution), but the values shown here are uniform-specific. With asymmetric bidders, revenue equivalence fails outright.
- The tie rule. When your bid exactly equals the highest rival bid we break the tie with a fair coin, so the expected payoff is (v−r)/2. Any other convention (always lose, always win, lowest index) leaves the dominance result intact; we checked this convention and state it because a tie on a continuous grid is a measure-zero event that the discretisation makes visible.
- Grid resolution. The dominance check uses 101 bids by 101 rival bids by 101 values, which is 1,030,301 cases. That is exhaustive on the grid, not on the continuum. The continuum statement is the theorem; the grid is the demonstration.
- Clock granularity. The English and Dutch clocks tick in steps of 1/4096. That is why their simulated prices differ from the sealed formats by up to one tick per draw, and the arena reports that worst-case gap rather than hiding it.
- The best-response search is a best response, not a fixed-point proof. The 2001-point sweep finds the shading factor that maximises your expected payoff while the other bidders are held at (n−1)/n. That it returns (n−1)/n shows the profile is an equilibrium; it does not by itself show the equilibrium is unique, and it searches only over strategies of the linear form b(v) = s·v. Uniqueness among all increasing strategies is the textbook result, and we cite it rather than compute it. The reported agreement is to within 10−7: golden section on a curve this flat near its maximum stalls at about the square root of double precision, which is why we state a tolerance instead of claiming an exact match.
- Monte Carlo error. The simulated revenues are sample means over a finite number of auctions. They will not equal the closed form exactly, and should not. The arena shows the gap; with 20,000 auctions it typically sits in the third decimal.
- Floating point at large n. By n ≈ 50 the seller's gain from the optimal reserve is around 10−16, smaller than the rounding error of the revenue formula itself. The table reports it as below double precision rather than as a number, because the arithmetic genuinely cannot resolve it. The analytic derivative, dR/dr = n rn−1(1−2r), still pins the optimum at exactly 1/2 for every n.
- The seller's own value is zero. If the seller values keeping the item at v₀ > 0, the optimal reserve rises to the solution of ψ(r) = v₀, that is (1+v₀)/2 for the uniform case. It is still independent of n.
what is contested, and what is not
Not contested. Weak dominance of truthful bidding in the sealed second-price auction; the revenue equivalence theorem under its stated hypotheses; the independence of Myerson's optimal reserve from the bidder count; the failure of truthfulness in GSP with two or more slots. These are theorems with published proofs.
Contested or unsettled. Whether real bidders actually bid their values in second-price auctions is an empirical question with a substantial literature finding systematic overbidding in laboratory second-price auctions, with competing explanations (a "joy of winning", confusion about the pricing rule, spite) and no settled resolution. Whether first-price or second-price raises more revenue in practice is likewise an empirical matter that turns on which assumption fails first. And whether the display advertising industry's move to first-price improved outcomes for publishers is actively disputed. This page shows the mathematics; it does not claim the mathematics describes any particular market.
what this page does not claim
eBay is not a second-price auction, though it is close and is often described as one. Its proxy-bidding system does implement an ascending auction where the winner pays a small increment above the runner-up's proxy maximum, which is the second-price idea. But eBay uses a hard close at a fixed time, which makes last-second sniping a real strategy and breaks the strategic equivalence with the open English auction. Amazon's former soft-close (extending the deadline after a late bid) produced far less sniping, which is the cleanest available evidence that the closing rule, not the pricing rule, is doing the work. Ockenfels and Roth documented the difference.
We also make no novelty claim of any kind here. Everything on this page is textbook mechanism design from 1961, 1981 and 2007. What is ours is the instrument.
Run the whole thing offline: node research/the-price-you-didnt-bid/verify-the-price-you-didnt-bid.mjs (154 checks, exit 0).
Where the clean theorems break, and what replaces them
Collusion. The second-price auction is dominant-strategy truthful for an individual, and catastrophically vulnerable to a ring. A cartel of bidders in a second-price auction can have one member bid high and everyone else bid zero, buying the item at the reserve, and no member has an incentive to defect from that arrangement: a defector who outbids the designated winner then pays the designated winner's inflated bid, which is above their own value. The very rule that makes the auction safe for an honest individual is what makes the cartel self-enforcing. Robinson set this out in 1985. The first-price auction is less robust for the individual and more robust against the ring, which is one honest reason procurement agencies use it.
The seller's own incentive. A seller who observes the bids in a sealed second-price auction has an obvious temptation: invent a losing bid just below the winner's. This is the standard argument, made at length by Rothkopf, Teisberg and Kahn in 1990, for why second-price sealed-bid auctions are rare in practice and ascending open auctions are common, since an open auction's price is witnessed by everyone in the room. The theorem assumes a seller who commits.
Budgets. Truthfulness assumes you can pay what you bid. A bidder with a hard budget below their value has no honest bid to make, and dominance fails at the budget line. Ad auctions are budget-constrained by construction, which is a second reason the folklore transfer from Vickrey to advertising does not hold.
Common values. Deliberately absent above. When bidders are estimating the same unknown quantity, the highest estimate is systematically too high, so the winner tends to overpay: the winner's curse. Rational bidders shade to correct for it, and second-price truthfulness is not a dominant strategy in that setting at all. Everything on this page lives strictly in the private-values world, where your value is yours and no one else's bid tells you anything about it.
Multiple identical items. The Vickrey generalisation (pay the externality you impose) remains truthful but has known pathologies: non-monotone revenue (adding a bidder can reduce revenue to zero), vulnerability to shill bidding, and low revenue relative to alternatives. Ausubel and Milgrom's "The Lovely but Lonely Vickrey Auction" is the standard catalogue of why the beautiful mechanism is so rarely deployed. It is the best short answer to why the world does not simply run VCG everywhere.