Auction prices

How can the rules of an auction change the price a thing finally sells for?

We speak of an auction as if it discovers a price, the way a thermometer discovers a temperature. But hand the same painting to the same room of bidders four times, changing nothing but the rulebook, and the hammer can fall at four different numbers. If the object and the people are fixed, what is left for the rules to act on?

Only one thing: what each bidder chooses to say. What the painting is worth to her is settled before she walks in. She decides what number to announce, and the gap between those two is where the whole subject lives.

Take the familiar auction, the one with the ascending chant. Below your value, dropping out costs you a bargain; above it, winning costs you money. So you stay in until the price passes your value, and so does everyone else. Who is left, and at what price? The bidder with the highest value, at roughly the moment the second-highest bidder quits. The winner never pays her own number — the price is set by the person who lost.

Now seal the bids in envelopes and let the highest bid win at the price written on it. If you write your true value and win, what have you gained? Nothing at all — you paid exactly what it was worth to you. So you write something lower. How much lower? That depends on how many rivals you imagine and what you think they are worth, which is guesswork about people you cannot see. The rule has turned an honest report into a strategic one.

Vickrey's move in 1961 was to keep the envelope but change the payment: highest bid wins, and pays the second-highest bid. Your own number now decides only whether you win, never what you pay. Can shading it ever help? It can only cost you a win you would have been glad to have. Bidding your true value becomes the best you can do whatever anyone else does — a dominant strategy, and a rare thing in economics. The Dutch auction runs a price downward until someone shouts stop, and choosing when to shout is exactly choosing a number in advance — so it behaves like the sealed first-price rule wearing a clock.

Here is the surprise. Under demanding assumptions — each bidder's value private and drawn independently, bidders alike and indifferent to risk — all four give the seller the same expected revenue, the shading in the sealed auction cancelling the discount of the open one. That is the revenue equivalence result, and it matters chiefly for how its assumptions fail. Where the value is largely common — an oil tract holding an unknown quantity of oil — the winner is whoever overestimated it most. Bidders who anticipate that shade further, and open bidding, where you watch rivals drop out, supplies information that dampens the fear; Milgrom and Weber showed that under such conditions the ascending auction tends to raise more.

Two corrections. The rules never change the object's value to anyone; they change the relationship between that value and the announced bid, and therefore the price. And "the auction found the market price" hides which price — the ascending auction reveals roughly the second-highest value in the room, not the highest. That is why sellers care about a reserve price, which is not a trick but the seller entering the room as a bidder.

If the second-price rule is so elegant, why is it rare? Partly because it asks bidders to write down their true valuation and trust the seller with it, and partly because the seller can profit by inventing a losing bid just under the winner's.

• Why selling many identical items at once breaks these formats, and what spectrum auctions do instead.
• How collusion between bidders survives an ascending auction more easily than a sealed one.

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