Three Auctions, One Surprising Result

Auctions turn up in more of business than the word suggests. Art and antiques, obviously. But also government procurement, construction tenders, spectrum licences, freight capacity, and the advertising slot that filled while this page loaded.

Three formats cover most of it, and the differences between them are less important than almost everyone expects.

The three formats

English auction. The open one. Bidders call out rising offers, everyone sees everything, it ends when nobody will go higher. The winner pays their final bid, which in practice is just above the second-highest bidder's limit.

Vickrey auction. Sealed bids, opened together. Highest bid wins — and pays the second-highest bid. Named for William Vickrey, who showed in 1961 that this rule has a remarkable property: bidding your true value is the best you can do no matter what anyone else does. You can't gain by shading, because your bid determines whether you win but not what you pay.

Sealed first-price auction. Sealed bids, highest wins, winner pays their own bid. The standard format for government contracts and private tenders. Nobody bids their true value here — winning at your value earns you nothing, so everybody shades down, and how far you shade depends on how many rivals you think you're up against.

Run all three on the same room

Switch formats and the mechanics change completely. In the first two, people bid what the item is worth to them. In the third, every bar drops — that's the shading.

Now look at the bottom chart, averaged over 400 random rooms. The three bars land in almost the same place.

Revenue equivalence

That's not a coincidence in the simulation. It's the revenue equivalence theorem (Vickrey 1961, generalised by Myerson in 1981): under a specific set of conditions, every auction format that awards the item to the highest valuer and gives a bidder with the lowest possible value zero expected surplus produces the same expected revenue for the seller.

The intuition is that the formats trade off against each other. First-price auctions collect the winner's own bid — a bigger number — but bidders shade, so the bid is lower. Second-price auctions collect a smaller number, the runner-up's bid, but nobody shades. The two effects cancel.

This is genuinely one of the more surprising results in economics, and its practical value is mostly negative: it tells you to stop agonising over the format and go work on something that matters. Which raises the obvious question.

When it doesn't hold

Revenue equivalence rests on assumptions. Where those break, the format starts to matter a great deal — and those are exactly the situations worth recognising.

Common values instead of private values. If the item is worth roughly the same to everyone and nobody knows what that is — an oil lease, a construction job, a company — then bidders learn from each other's behaviour. An open English auction lets them see rivals dropping out, which is information. Sealed formats don't. This is the biggest exception and it's covered properly in the winner's curse.

Risk-averse bidders. In a first-price auction, shading less means a worse deal but a better chance of winning. A risk-averse bidder will pay for that certainty, so first-price auctions raise more than second-price ones when bidders dislike risk.

Asymmetric bidders. If one bidder is known to value the item much more than the rest, the equivalence breaks, and the seller's best format depends on the details.

Collusion. Open auctions make cartels easy to police — you can see whether your co-conspirators kept to the deal. Sealed bids make it much harder. If you're running a procurement with a small number of repeat suppliers, this is not a theoretical concern.

So what do you actually use?

Open English when the item is emotive and one-of-a-kind, when publicity is part of the point, and when you want bidders to feed off each other. Art, charity, houses.

Sealed first-price when speed and confidentiality matter, when you're running procurement, and when you want to make collusion awkward. It's also the most familiar format to corporate buyers, which is not nothing.

Second-price and its relatives when you want bidders to reveal what they think things are worth, especially in automated, high-frequency settings where nobody can sit and strategise. Online advertising ran on a generalised second-price auction for years for exactly this reason — although the major display exchanges, Google's among them, moved to first-price around 2019, mainly because layers of intermediaries had made the second-price promise hard to verify end to end.

That last detail is the real lesson. The theory says the format shouldn't matter much. It stopped being true in practice not because the maths was wrong, but because the assumptions quietly stopped describing the situation.