Four buyers want the same item, and each knows what it is worth to them. The seller can sell it at first price, at second price, by English auction or by Dutch auction. The buyers are smart and adapt their bids to the rules. Which rules give the seller the most on average? Lock in your answer, then check it with a game, a simulation and the explanations.
At second price buyers bid honestly, but the winner pays someone else's lower bid. At first price the winner pays their own bid, but bids less than the item is worth to them. On average these two discounts are exactly equal: all four formats give the seller 60 out of 100. Only the spread differs. Check it with a game, a simulation and the explanations.
This is the classic model of auction theory. Here is everything you need to know about the lot, the buyers and the rules.
One item is for sale, say a vintage guitar. Each of the four buyers knows what it is worth to them personally: that is their value, a number from 0 to 100, with every value equally likely. A buyer knows their own value, not the others'. The values are independent: one buyer wanting the guitar badly says nothing about the others.
The seller picks one of four formats:
Each buyer wants to gain as much as possible: the difference between their value and the price they pay. They know the rules and know the others are smart too, so they pick the bid that is best for them under these rules. Nobody colludes, and nobody bids more than the item is worth to them.
If lots like this are sold over and over, which format brings the seller the most on average?
"First price, because the winner pays their own bid, not someone else's"
But the buyers know this too and bid less. A smart buyer at first price bids exactly what they would pay on average at second price. So all four formats give the seller 60.
"The seller's revenue" is the price the winner pays.
The same four smart buyers, values from 0 to 100. Which format brings the seller the most on average?
Pick an answer. A simulation of 10,000 lots in each format will appear here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Think about how buyers bid under each set of rules. Would a smart buyer bid their full value if, on winning, they pay exactly their own bid?
At second price the best bid is exactly your value, but the winner pays someone else's lower bid. At first price the bid is below the value, but the winner pays the bid itself. Which of the two discounts is bigger?
Take two buyers. At second price the seller gets the lower of the two values. At first price each bids half their value, and the higher one wins. Compare the average of the lower of two values with half the average of the higher.
Now you are a buyer. Each lot is sold at either first or second price, and you see which. Your value is random from 0 to 100, and so are the bots'. The bots are experienced: each bids in its own way and a little differently from lot to lot. The slider sets your bid as a percentage of your value, separately for each format. A series is 10 lots. Your gain is compared with "the textbook": the same lots played by the rule from theory.
The simulation sells 10,000 lots in each format, each format with its own independent buyers. The buyers bid as theory says: their value at second price and in the English auction, (n − 1)/n of it at first price and in the Dutch auction (round 4 explains why). The lines show how the average revenue changes as lots are sold; the dashed line is the exact value.
The first layer has no formulas, the second has an example, a table and formulas, the third covers the theorem and the people who proved it.
“In plain words” is hidden on all puzzles.
At second price the winner bids honestly, but pays not their own bid but a lower one from someone else. At first price they pay their own bid, but shade it themselves. In both formats the winner pays less than the item is worth to them; in one case the rule gives the discount, in the other the buyer takes it.
This card explains why the gap between formats is smaller than it seems, but not yet why it is exactly zero. Why the discounts match is shown by the card "Bid = the expected second price" in the next layer.
At second price your bid decides only one thing: whether you win. Someone else's bid decides what you pay. Bid above your value and you sometimes win at a price above what the item is worth to you, and lose money. Bid below and you sometimes lose a lot that would have paid. An honest bid makes neither mistake, whatever the others do.
In an English auction a smart buyer stays in until the price reaches their value. Bidding stops as soon as the second-last buyer drops out, that is, at the second-highest value. It is the same result as at second price.
In a Dutch auction the price falls, and each buyer has to decide in advance at what price to shout. They learn nothing new along the way, since the first shout ends it. It is the same sealed bid as at first price.
In a single lot the formats give different amounts. At second price the seller depends on whether a second buyer who also wants the item badly turned up, so the revenue jumps around more. At first price the winner "smooths" the bid, and the revenue is steadier. Same average, different risk.
“Rigorous” is hidden on all puzzles.
Say there are two buyers. Two random points on a segment from 0 to 100 split it on average into three equal parts, so the lower of two values averages 33.3 and the higher 66.7.
Second price: the winner pays the lower value, 33.3 on average. First price: each bids half their value, the higher one wins, and the seller gets half of 66.7, the same 33.3. In a single lot the amounts differ: with values 70 and 40, second price gives 40 and first price 35.
Your value is 80 and you have three rivals. If you win, all three have values below 80, uniform from 0 to 80. The largest of three such values averages ¾ · 80 = 60. That is what you would pay on average when winning at second price.
And that is exactly what a smart buyer bids at first price: 60. So first price takes from the winner, upfront and without risk, what second price would take on average. Hence the same revenue.
| Buyers n | Highest value | First-price bid | Revenue, 1st price | Revenue, 2nd price | Spread 1st / 2nd |
|---|
The average highest of n values is 100 · n/(n + 1). First price takes the share (n − 1)/n of it. Second price takes the average second-highest value, 100 · (n − 1)/(n + 1). Both routes give the same number. The spread is always smaller at first price.
Values Vi are independent and uniform on [0, 1] (in hundreds). The equilibrium first-price bid equals the average largest of the n − 1 rival values, given that all are below yours.
First-price revenue: a share of the average highest value. Second price: the average second-highest value.
The general theorem: if in any format the highest value wins and a buyer with value zero pays nothing, then a buyer with value v pays the same on average. P(v) is the chance of winning.
Spread of revenue: second price has a variance 2n/(n − 1) times larger. For n = 4 the standard deviations are 12.2 and 20.
“Deeper” is hidden on all puzzles.
In 1961 the Columbia University economist William Vickrey described the second-price auction in "Counterspeculation, Auctions, and Competitive Sealed Tenders" (Journal of Finance). In the same paper he first compared the revenue of different formats and found that under conditions like those in our puzzle they are equal.
In 1996 Vickrey received the Nobel prize in economics together with James Mirrlees. He died three days after the announcement, and the prize was presented posthumously.
The general theorem was proved independently by Roger Myerson ("Optimal Auction Design", Mathematics of Operations Research) and by John Riley with William Samuelson ("Optimal Auctions", American Economic Review), both in 1981. The same revenue comes not just from these four but from any format in which the highest value wins, even one where every bidder pays their bid, not only the winner.
The same papers answered how a seller can earn more: changing the rules won't help, a reserve price will (see round 5). In 2007 Myerson received the Nobel prize for the foundations of mechanism design theory.
Paul Milgrom and Robert Wilson of Stanford received the 2020 Nobel prize for improvements to auction theory and inventions of new auction formats. Together with Preston McAfee they proposed the simultaneous multiple-round auction, where many related licences are bid on at once and bidding closes for all of them together.
The US Federal Communications Commission ran the first one in July 1994: ten nationwide narrowband PCS licences raised $617 million in five days.
In 1999 David Lucking-Reiley tested the theorem in the field: he sold Magic: The Gathering collectible cards online in pairs of identical lots under different formats. The English auction and second price gave about the same revenue, as theory promises. But the Dutch auction brought in about 30% more than first price.
The price in his Dutch auctions fell slowly, about 5% a day, so buyers were not simply "bidding in an envelope". Real people and real rules add things the model leaves out.
First the maths (a reserve price that raises revenue under any rules), then the logic (which conditions the theorem rests on and what happens when one breaks), then practice (ad auctions where an honest bid loses).
A reserve price is a minimum below which the seller won't sell. It acts like an invisible extra buyer who bids honestly. If the highest value is below it, the lot goes unsold. If only one value is above it, the winner pays the reserve rather than a lower rival bid.
For values from 0 to 100 the best reserve is 50, and it does not depend on the number of buyers (as Myerson, Riley and Samuelson showed). With two buyers it lifts revenue from 33.3 to 41.7, even though one lot in four goes unsold. With four, only from 60 to 61.25: competition already holds the price up. Since the reserve works the same under all the rules, the equivalence survives it.
The same holds from another angle: an auction where everyone pays their bid, losers included, also gives 60 with four buyers. The bids there are far lower, but everyone pays. In the dollar auction, though, where bids are open and rise in turns, this logic leads to escalation.
Equal revenue is guaranteed only when the highest value wins, values are independent and private, buyers are alike in how their values are spread, and they are indifferent to risk. Break one condition and the formats part ways.
Fear of risk. A buyer who hates losing the lot bids higher at first price than a risk-neutral one, to lose less often. At second price an honest bid stays best, so revenue there does not change. The slider shows how far first price pulls ahead. Unequal buyers. When one buyer usually values the item more than the others, the ranking of formats depends on the details: sometimes first price wins, sometimes second (Maskin and Riley, 2000). Common value. If the item is worth the same to everyone but nobody knows how much, the winner's curse appears, and the equality breaks. Milgrom and Weber proved in 1982 that when buyers' estimates are linked to each other, an English auction, where you can see when the others drop out, brings in on average at least as much as the second price, and the second price at least as much as the first. The more buyers learn from the bidding, the bolder they bid.
Two ad slots above the search results: the first gets 200 clicks an hour, the second 199. Three advertisers value a click at $10, $4 and $2. The top bid gets the first slot and pays the next bid per click; the second gets the second slot at the third bid. The other two bid honestly. What do you bid, as the $10 advertiser?
Search ads were long sold exactly like this: Google launched the format in 2002, and in 2004 over 98% of its $3.189 billion revenue came from such auctions. It is called the generalized second price because the winner pays the next bid. But as Benjamin Edelman, Michael Ostrovsky and Michael Schwarz showed (American Economic Review, 2007), here, unlike in a Vickrey auction, an honest bid is not the best.
Check their example with the slider. An honest $10 bid gets the first slot at $4 and (10 − 4) · 200 = 1,200 an hour. A $3 bid gets the second slot at $2 and (10 − 2) · 199 = 1,592. One click an hour costs $2 too much. The reason is that your bid here decides not only whether you win but also which slot the others get, and with it your price. Once bids stop being honest, the simple "rules don't matter" no longer applies: in 2019 Google moved display ad sales in Ad Manager to a first-price auction.
Johann Wolfgang von Goethe asked the publisher Vieweg to name a price for the manuscript of his poem "Hermann and Dorothea", while his own demand lay sealed in an envelope with a go-between. If the offer was not below the demand, the deal went through at Goethe's figure. In 1998 the economists Benny Moldovanu and Manfred Tietzel showed this is a second-price auction, invented 164 years before Vickrey.
Stamp dealers ran mail-bid auctions under a rule close to second price from at least 1893, before Vickrey was born. David Lucking-Reiley showed this in 2000, tracing the story from 19th-century philately to eBay, where automatic proxy bidding works the same way.
William Vickrey was named a winner of the 1996 Nobel prize in economics for contributions to the economic theory of incentives under asymmetric information, auctions included. Three days later he died on his way to a research conference. A Columbia colleague accepted the prize on his behalf.
In July 1994 the US Federal Communications Commission sold ten nationwide licences for $617 million in five days. It was the first simultaneous multiple-round auction on the design by Milgrom, Wilson and McAfee, for which Milgrom and Wilson later won the Nobel.
In Lucking-Reiley's online field experiment with Magic cards, Dutch auctions earned about 30% more than first-price auctions, though theory promises the same. Earlier laboratory studies had found the opposite.
In 2004 Google earned $3.189 billion, and over 98% of it came from generalized second-price auctions for search ads. That is the count by Edelman, Ostrovsky and Schwarz in the paper that showed honest bidding is not the best in these auctions.
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