Auctions: how ads and compute are priced

Posting a price needs an estimate of demand. Deep-dive sessions 1 to 3 built that estimate from months of repeated sales of the same product. An ad position on one search page at one second, or a block of spare compute for the next hour, has no such history: it is sold once, it is worth different amounts to different bidders, and those amounts are private. Post a price and one of two things happens. It is too low, and the buyer who valued it most pays less than she would have; or it is too high, and it goes unsold. Either way the seller has thrown away information that only the buyers had.

An auction asks the buyers instead. The bids are the information the seller lacked. But asking is not free of trouble, because a buyer knows the answer will be used against her. Under a first-price rule, where the winner pays her own bid, "what is this worth to you?" is a question every rational bidder answers low. Under a second-price rule, where the winner pays the highest losing bid, the same question can be answered honestly with no regret. The difference between the two rules is one line of code, and it changes what every bid on the page means.

The three bidders above are constructed. Ana values the item at 80, Bo at 65, Cy at 50, and none of them knows the others' numbers. The point of the drawing is the tag: Ana wins because her bid was highest, and she pays 65 because that was Bo's bid. Her own bid appears nowhere in the price.

First price: highest bid wins and pays its own bid. If Ana bids her true 80 she wins and pays 80, and her surplus is zero; she would have been just as happy losing. So she shades. Bid 70 and she still wins, paying 70, with 10 of surplus. Bid 66 and she wins with 14 of surplus, but she is guessing at Bo's bid, and if she guesses wrong she loses an item worth 80 to her. Every first-price bidder is playing a guessing game against the other bidders' unknown values, and the bids on the sheet are guesses, not values.

Second price: highest bid wins and pays the highest losing bid. Ana bids 80, wins, and pays 65. Now try to improve on that. Bid 90: she still wins and still pays 65, because Bo's bid did not move. Bid 70: still wins, still pays 65. Bid 60: she loses to Bo, and forgoes an item worth 80 that she could have had for 65. There is no bid that beats bidding her true value, and several that are worse. The bids on the sheet are values, which is what the seller wanted to know in the first place.

The same three people, the same three values, and under one rule the sheet is a guessing game while under the other it is a survey with no reason to lie. That is the sense in which the rules decide what the data means.

Vickrey 1961, "Counterspeculation, Auctions, and Competitive Sealed Tenders", Journal of Finance 16(1) 8-37, is the paper that made the second-price sealed-bid rule a named mechanism. Its result, in the form worth carrying: in a second-price sealed-bid auction for a single item, bidding your true value is a dominant strategy, meaning it is at least as good as any other bid no matter what every other bidder does. Not "in equilibrium", not "if the others are rational", not "on average". Whatever Bo and Cy bid, whatever they know, whatever they are trying to do, Ana's best move is 80.

The proof fits in four sentences. Let the highest rival bid be b. If Ana's value v is above b, any bid above b wins and pays b, so she gets v minus b, and any bid below b loses and gets zero; bidding v is in the first group. If v is below b, any bid above b wins and pays b, a loss of b minus v, and any bid below b loses and gets zero; bidding v is in the second group. In both cases bidding v does as well as the best available option, and it does so without Ana needing to know b, which is the whole point: she has to know only her own value, which is the one thing she does know.

The result is fragile in an instructive way. It needs one item, one winner, and a rule that charges the winner a price her own bid cannot influence. Change any of those, as the ad ladder in Part 2 does, and the dominance goes away.

The rule. Take the standing bids for a query, sort them, and hand out the positions in order. The advertiser in position k pays the bid of the advertiser in position k plus 1, per click. The top bidder pays the second bid, the second pays the third, and the last position holder pays the first losing bid. With one position this is exactly Vickrey. With several it looks like Vickrey extended, and that resemblance is why it was adopted and why it is misread.

Where the resemblance breaks. In Vickrey's single-item auction, a bidder who lowers her bid either keeps winning at the same price or loses everything; there is no third outcome. On a ladder there is: she can drop one rung. Dropping a rung means fewer clicks but a lower price per click, and whether that trade pays depends on the click counts and the bids beneath her. In the invented example, A values a click at 10. Truthful at the top she pays 7 per click on 100 clicks, for 300 of surplus. Bidding 6.9 puts her second, paying C's 4 on 80 clicks, for 480. She is better off lying, and she is better off lying by an amount that depends on numbers she has to guess. The sheet of bids is a set of strategic positions again, not a set of values.

The 2007 result. Edelman, Ostrovsky and Schwarz 2007, American Economic Review 97(1) 242-259, state it directly: truth-telling is not an equilibrium of GSP. The paper goes further and characterises what the equilibria look like, but the sentence to carry is the one above. GSP is not VCG, and a bid on a GSP ladder is not a willingness to pay.

The practical consequence is about interpretation, and it is the same lesson as deep-dive session 2 in a different costume. There, a slope fitted across products looked like an elasticity and was not one, because the products were not comparable. Here, a bid on a GSP ladder looks like a value and is not one, because the bidder had a reason to shade that depends on everyone else's bids. Anyone who takes a feed of GSP bids and reads it as "what advertisers think a click is worth" is reading a strategy as a measurement.

The honest reading of GSP bids is that they are bounded by values from above, a bidder will not bid more than a click is worth to her, and shaped by the ladder from below. What sits between the two is the bidder's guess about her rivals. That is enough to run the auction, which is all the platform needs, and not enough to recover the values, which is what an analyst wants. The values are recoverable only under a mechanism where honesty is dominant, and the ladder is not one.

The Vickrey-Clarke-Groves rule is Vickrey's idea carried to several items or positions. Instead of charging a winner the bid below her, it charges her the harm her presence does to everyone else: the total value the other bidders would have received had she not bid, minus the total value they actually receive with her in the auction. With one item that difference is exactly the second-highest bid, so VCG and Vickrey coincide. On the ladder it is different from GSP. In the invented example, A's presence pushes B from position 1 to 2, C from 2 to 3 and D out, and A is charged the sum of what those three lost, valued at their own bids. Because the charge depends only on the others' bids and the effect of A's position, A's own bid cannot lower her price without also changing her position, and the single-item argument goes through: truthful bidding is dominant again.

Why did the industry run GSP rather than VCG? The paper discusses it and the honest summary is that GSP arrived first, was simpler to explain, and was already in use when the difference was understood. The lesson for anyone designing a mechanism, an internal market for compute, a bidding round for a data budget, a ranking for who gets which project, is that "pay the next bid down" is intuitive and is not the truthful rule once there is more than one prize.

A reserve price is a bid the seller enters on her own behalf. In a second-price auction it works as a floor: if only one bidder is above it, that bidder wins and pays the reserve instead of the next bid, which would have been lower. Set well, a reserve raises the seller's revenue on average, because the second-highest bid is often well below the highest and the reserve fills the gap. Set too high, it leaves the item unsold in the draws where it would have sold at a lower price, and that is revenue lost.

The reserve does not disturb Vickrey's dominance for the bidders. Each bidder's price is still either the reserve or a rival's bid, and neither is something her own bid can move, so bidding her true value remains her best move. What the reserve changes is the seller's problem: she has to choose a number, and to choose it she needs an estimate of the distribution of values, which is exactly the thing she ran an auction to avoid estimating. In practice she learns it from past auctions, which is where auction data and the panel methods of deep-dive sessions 1 and 2 meet.

The mechanism, with no numbers. A provider has capacity that would otherwise sit idle. Rather than post a price for it, the provider lets buyers state a maximum per unit of time and serves them from the highest maximum down until the capacity is used up. The price everyone pays is set by the last bid that got in, the marginal one, which is a second-price rule spread across many units: your own maximum decides whether you are served, and someone else's decides what you pay. When new demand arrives with higher maxima, the marginal bid rises, and any running job whose maximum is now below it is stopped. That stopping is the price of the discount, and it is not a failure of the system; it is the system.

What it means for the bidder. The Vickrey logic applies per unit: the maximum you state should be what an hour of the job is actually worth to you, not what you hope the price will be, because your maximum does not set the price, it only sets whether you are in. Teams that state a low maximum to "save money" are not saving money; the price is the marginal bid whether their maximum is high or low. They are only raising the chance of being stopped. Teams that state a maximum above the on-demand price are paying nothing extra either, until the market clears above it, at which point they should have used on-demand anyway.

What the page does not say. No price, because it changes weekly; no vendor, because the mechanism is the same at all of them and the details differ in ways that do not touch the economics. The exercise is to read your own provider's documentation with the mechanism in hand and check which parts are Vickrey and which are not.

Everything in Parts 1 and 2 assumed private values: the item is worth 80 to Ana and 65 to Bo, and those are different facts about different people. Many auctioned things are not like that. A dataset to be resold, a block of compute for a job whose payoff is a market outcome, a licence to something whose value is set by the world and not the buyer: these have a common value, one number worth the same to whoever wins, and the bidders differ only in their estimates of it. The five dots in the figure are five honest, unbiased estimates of the same 100, invented for the drawing.

Now run the auction. The highest estimate wins. But the highest of five unbiased estimates is not unbiased; it is biased upward, by construction, because it was selected for being highest. The winner is the bidder whose error was largest in the optimistic direction. If she bid her estimate, she paid 121 for something worth 100. That is the winner's curse: the news that you won is bad news about your estimate. The cure is to bid as if you already know you have won, meaning to condition your estimate on being the highest of n, which shades it downward by more when n is larger and when the estimates are noisier. Bidders who do not do this lose money on average and do not notice for a long time, because each individual win looks fine.

The cause and the cure are standard economics with no single source, and the page claims no figure for how large the curse is in any real market; the size depends on the noise in the estimates, which is the one thing the bidders cannot see. The mechanism, though, is worth carrying into every "we won the bid" celebration on a common-value item: ask what the winning implies about the estimate before asking what to do with the prize.