Auction Theory for Ad Buyers: What the Death of Second-Price Means for Bidding

TL;DR: For four decades auction theory told bidders something reassuring: in a second-price auction, bidding your true value is a dominant strategy, so you never have to guess what rivals will do. Vickrey (1961) proved it, and for one stretch of the programmatic era it described the machinery well. Almost no advertising auction you buy in today works that way. Google's search auction was a generalized second-price auction in which truthful bidding was never optimal, a result Edelman, Ostrovsky and Schwarz (2007) made precise. Programmatic display ran second-price until 2019, when Google Ad Manager moved to a unified first-price auction over roughly six months, chasing the header-bidding shift that Despotakis, Ravi and Sayedi (2021) tie directly to the format change. In a first-price auction the dominant-strategy comfort is gone: you must shade your bid below your value, and the right amount to shade depends on how many rivals you face. The symmetric benchmark says bid the fraction (n−1)/n of your value against n−1 competitors, so with three bidders you shave a third and with ten you shave a tenth. Revenue equivalence (Riley and Samuelson, 1981) tells you when the format should not matter, and every real ad market breaks its assumptions in ways that decide the price. Reserve prices (Myerson, 1981; Ostrovsky and Schwarz, 2023) and automated bidding are consequences of the same theory. A buyer who does not model the mechanism is quietly handing margin to the platform.

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The Bid You Were Taught Is the Wrong Bid

A performance marketer setting a maximum cost-per-click types in a number that stands for what a click is worth to the business. A programmatic trader configuring a demand-side platform sets a bid equal to the expected value of the impression. Both follow the same rule, and both learned it in the same place, whether or not they could name the source: bid your value, and let the auction sort out the rest. The rule is correct, dominant, and provably safe in exactly one setting. Most buyers have never bought media in that setting.

The setting is the second-price sealed-bid auction, the one William Vickrey analyzed in 1961 and the one that made his name. The highest bidder wins but pays the second-highest bid, and under that rule bidding your true value is a dominant strategy: it is your best move no matter what anyone else does. For roughly a decade of the programmatic era that rule described the machinery well enough. Display impressions cleared in second-price auctions, and the advice to bid your value was not folklore. It was a theorem, and it held.

Then the machinery changed, and the theorem stopped applying. Google's search auction, the largest advertising market on earth, was never a second-price auction in Vickrey's sense at all. It ran a generalized second-price auction, a different object in which truthful bidding is not optimal and never was. Programmatic display, which genuinely had been second-price, converted to first-price auctions in 2019. In a first-price auction the winner pays exactly what they bid, and the dominant-strategy guarantee evaporates. Bidding your value becomes the one thing you must not do, because you would pay your entire margin to the seller on every win.

This essay is about the gap between the bidding rule most buyers still carry in their heads and the auctions they actually face. The claim is cheap to state and expensive to ignore: the auction mechanism, not the size of your budget or the cleverness of your creative, decides how much of the surplus from a winning impression you keep and how much the platform takes. Bid shading, reserve prices, and automated bidding are not features bolted onto ad platforms. Each is a direct consequence of auction theory, and each is a place where a buyer who understands the mechanism keeps money a buyer who does not hands over. In what follows we build the theory in the order it was discovered, because that order is also the order in which the assumptions break, and because the rule in our heads has drifted from the auction in front of us.

Vickrey's Miracle: Why Second Price Rewards Honesty

Vickrey's result is one of those theorems that feels like a magic trick until you see the mechanism, at which point it feels inevitable. Start with the intuition, because the intuition is the whole thing.

In a second-price auction your own bid does one job and only one job: it decides whether you win. It does not set the price you pay, because the price is the highest of the other bids. Once you separate those two roles, your bid controls winning, someone else's bid controls the price, the right strategy writes itself. You want to win whenever the price would fall below what the item is worth to you, and to lose whenever the price would sit above it. Bidding exactly your value does both at once, automatically, without your knowing the price in advance.

Write vᵢ for what the impression is worth to bidder i and bᵢ for the bid submitted. Let p be the highest competing bid, which bidder i does not control. The payoff is the value minus the price when i wins and zero otherwise:

$$u_i = \begin{cases} v_i - p & \text{if } b_i > p \\ 0 & \text{if } b_i < p \end{cases}$$

Now we compare truthful bidding, bᵢ = vᵢ, against any deviation, checking every case. Suppose i bids above value, bᵢ > vᵢ. The only outcomes that change are those where the extra bid turns a loss into a win, meaning vᵢ < p < bᵢ. In every such case i now wins but pays p > vᵢ, for a negative payoff; truthful bidding would have yielded zero. Overbidding can only add losing trades. Suppose instead i bids below value, bᵢ < vᵢ. The only outcomes that change are those where the lower bid turns a win into a loss, bᵢ < p < vᵢ. In each case i now earns zero instead of the positive vᵢ - p that honesty would have captured. Underbidding can only forgo profitable trades. Whatever the competing bids turn out to be, no deviation from bᵢ = vᵢ ever helps, and some deviations hurt.

The property has a name that matters for everything that follows.

Two more properties travel with the result and explain why economists prized it. The auction is efficient: because everyone bids their value and the highest bid wins, the impression goes to the bidder who values it most. And the open ascending auction, the English auction, where bidders raise their hands until only one remains, is strategically equivalent, clearing at roughly the second-highest value. Real-time bidding with price feedback inherits some of that ascending character, which will matter when we reach common values. For now we hold onto the clean version: in Vickrey's world, honesty is not a virtue you choose, it is the move the mechanism pays you to make.

Revenue Equivalence: When the Format Should Not Matter

If honesty is dominant in a second-price auction and self-defeating in a first-price auction, a natural question follows. Which auction makes more money for the seller? The startling answer, under the right conditions, is neither. They tie.

The theorem and its four assumptions

The revenue equivalence theorem, proved in its modern form by Riley and Samuelson (1981, American Economic Review 71(3), 381-392) and, independently and more generally, by Myerson (1981, Mathematics of Operations Research 6(1), 58-73), says that a broad class of auctions all yield the same expected revenue to the seller and the same expected payment from each bidder. First-price, second-price, English ascending, Dutch descending, even the all-pay auction where every entrant forfeits their bid whether they win or not: identical in expectation.

The result sounds like magic and is really an accounting identity in disguise. What a bidder pays, on average, is pinned down by two things alone: the rule for who wins, and the promise that a bidder with the lowest possible value walks away with nothing. Fix those, and the expected payment is determined; the surface details of the pricing rule wash out. First-price bidders shade and pay their shaded bid on the wins; second-price bidders bid full value and pay the runner-up's bid; the two adjustments cancel exactly in expectation. Contrary to the intuition that a first-price auction must cost buyers more, the theorem says the format is a wash, until an assumption fails.

Four assumptions carry the theorem, and naming them precisely is the most useful thing an ad buyer can take from this section, because the assumptions are exactly what online auctions violate.

To make the theorem concrete, we take the textbook benchmark: n bidders whose values are drawn independently and uniformly on the interval from zero to one. The expected price the seller collects and the expected value of the winning bidder are both simple functions of n:

$$\mathbb{E}[\text{price}] = \frac{n-1}{n+1}, \qquad \mathbb{E}[\text{winning value}] = \frac{n}{n+1}.$$

The first expression is the seller's expected revenue in a no-reserve auction of any of the equivalent formats. The gap between the two, equal to (1)/(n+1), is the winner's expected information rent, the surplus the bidder keeps for having the highest value. Both curves climb toward one as bidders are added, and the rent between them shrinks.

The chart carries a lesson before we have said a word about reserves: the seller's revenue rises with the number of bidders and the winner's rent falls, both toward the full value of the item. A thick auction extracts almost everything on its own. A thin one leaves the winner most of the surplus. Everything a platform does about reserves, conflation, and demand recruitment is a fight over that gap.

Reserve prices: Myerson's thumb on the scale

Revenue equivalence covers auctions that sell to the highest bidder no matter how low the bids run. A seller who is willing to walk away can do better. Myerson (1981) worked out the revenue-maximizing auction and found that its central instrument is a reserve price: a floor below which the seller refuses to sell, credibly and in advance.

The optimal reserve solves a short equation. Writing F for the value distribution, f for its density, and v₀ for the seller's own value of keeping the item, the reserve r* satisfies:

$$r^{*} - \frac{1 - F(r^{*})}{f(r^{*})} = v_0, \qquad r^{*} = \tfrac{1}{2} \ \text{ for } v \sim U[0,1],\ v_0 = 0 .$$

Two features of this deserve an operator's attention. First, the reserve trades efficiency for revenue on purpose: it throws away some sales that would have happened at low prices in order to raise the price on the sales that do happen, and to threaten every bidder credibly. Second, and less intuitively, the optimal reserve does not depend on the number of bidders. For uniform values the seller sets a floor at one half whether two bidders show up or twenty. The reserve is the seller's substitute for competition when competition is thin, and its irrelevance to n is the seed of the next result.

Reserves are not a blackboard curiosity in advertising. Ostrovsky and Schwarz (2023, Journal of Political Economy 131(12), 3352-3376), building on a working paper first circulated in 2011, ran a large field experiment in sponsored-search auctions, setting reserve prices guided by Myerson's theory adapted to the search setting. The experiment was conducted with Yahoo!, across a large sample of keyword auctions. Consistent with the theory, revenues rose substantially once the new reserves were introduced. The point for a buyer is not the exact figure but the direction and the source: the floor under your search bids is not an accident of supply, it is a designed instrument tuned to extract revenue, and it was validated in the field on one of the largest auction platforms that ever ran.

One more bidder beats a smarter auction

Now let us put the two ideas side by side, the reserve that raises revenue and the competition that raises it, and ask which matters more. Bulow and Klemperer (1996, American Economic Review 86(1), 180-194) answered with one of the most quoted results in the field. A simple ascending auction with no reserve at all and n+1 bidders yields at least as much expected revenue as the cleverest revenue-maximizing auction, reserve and all, with only n bidders. Recruiting one more serious bidder beats designing the perfect mechanism. The lesson generalizes past this one model. Klemperer (1999, Journal of Economic Surveys 13(3), 227-286), in the standard survey of the field, argued that in real markets attracting entry and deterring collusion matter far more than the fine details of the pricing rule, and that is exactly the order of priorities a buyer should read into a platform's conduct.

Why the First-Price Bid Is a Shaded Bid

In a first-price auction the winner pays their own bid, so every dollar you bid above the price you needed is a dollar of margin handed to the seller. That single sentence contains the whole problem of first-price bidding and the whole reason the comfortable rule fails. Bidding your value guarantees zero profit on every win. You must bid below your value, you must shade, and the only question is by how much.

The equilibrium logic is a conditioning argument, and the conditioning is the subtle part. Your bid matters only in the states of the world where you win. Winning means your value is the highest of everyone's. So the bid to submit is your best estimate of what it takes to win, namely the highest of the other bidders' values, computed under the assumption that your own value is the top one. Formally, the symmetric equilibrium bid is the expected highest rival value conditional on that value being below yours:

$$b(v) = \mathbb{E}\!\left[\max_{j \neq i} v_j \;\middle|\; \max_{j \neq i} v_j < v\right], \qquad b(v) = \frac{n-1}{n}\,v \ \text{ for } v \sim U[0,1].$$

The second expression is the payoff for a media buyer. With values uniform on the unit interval and n bidders in total, the equilibrium bid is the fraction (n-1)/(n) of your value. Here n is the number of bidders competing for the impression, and it is the single number that sets how hard you should shade. With two bidders you bid half your value. With five you bid four-fifths. With ten you bid nine-tenths. As the auction thickens, the shading vanishes and first-price bidding converges toward paying full value, which is the same fact Bulow and Klemperer stated from the seller's side, seen now from the buyer's.

The number we almost never know directly is n. The platform knows it, or a close proxy. Bid-shading algorithms inside demand-side platforms estimate it from the distribution of minimum-winning-bid feedback, effectively reconstructing how much room there is between the winning price and the runner-up, and shaving your bid into that room. When a shading vendor reports that it saved you a given percentage, it is reporting an estimate of the gap between what you bid and what you needed, which, in the clean benchmark, is exactly the (1)/(n) your bid should have been below your value.

The auction mechanism, not the size of your budget, decides how much of each win you keep and how much the seller takes.

Common Values and the Winner's Curse

Everything we have covered so far assumed private values: your impression is worth what it is worth to you, and knowing a rival's value would not change your own. Advertising breaks that assumption in a specific and important way.

The value of an impression depends heavily on a quantity that is the same for everyone and unknown to all: will this user convert? Every serious bidder estimates the conversion probability with a model, and the true value the impression will deliver is, in large part, common across bidders rather than private to one. When the underlying value is common but each bidder sees only a noisy estimate of it, the strategic problem inverts. Winning stops being good news.

The mechanism is the winner's curse. Suppose every bidder submits their honest, unbiased estimate of a shared value. The bidder who wins is, by definition, the one whose estimate was highest, which means the winner is disproportionately the bidder who most overestimated the impression. Winning is itself evidence that your estimate ran hot. A bidder who does not correct for this will systematically overpay on exactly the impressions they win, not because the model is biased in general but because winning selects the cases where the model was most optimistic relative to the field.

The pattern was first named far from advertising. Capen, Clapp and Campbell (1971, Journal of Petroleum Technology 23(6), 641-653), three petroleum engineers, noticed that oil companies bidding for offshore drilling rights kept winning tracts and losing money, and traced it to bidders submitting their best estimate of a common reserve value without discounting for the fact that winning meant they had been the most optimistic. An ad exchange and an oil lease are the same auction wearing different clothes.

Milgrom and Weber (1982, Econometrica 50(5), 1089-1122) built the general theory of auctions with values that are affiliated, meaning bidders' estimates are positively correlated: when one bidder's signal is high, others' tend to be high too, which is exactly the situation with conversion estimates that all draw on similar features. Their central practical result is a revenue ranking. With affiliated values, the open ascending auction yields more revenue than the sealed second-price auction, which in turn beats the sealed first-price auction, because public bidding reveals information as it proceeds and softens the winner's curse, the linkage principle. Open competition lets bidders learn from each other and bid more aggressively without fear, which helps the seller.

Search Was Never Truthful: The Generalized Second-Price Auction

Let us return to the largest ad market of all, search, and to the claim that it was never a Vickrey auction. Sponsored search sells not one item but a ranked list of slots, the top slot drawing more clicks than the second, the second more than the third. Google's auction, inherited and refined from Overture's design, ranked advertisers by bid weighted by a quality estimate and charged each winner roughly the minimum needed to hold their position, which works out to about the bid of the advertiser ranked just below. It looked like a second-price auction stretched over many slots, and it was marketed with the Vickrey intuition: you do not pay your bid, you pay just enough to stay ahead of the next advertiser.

The generalized second-price auction is not the multi-slot Vickrey mechanism, and it is not truthful. Edelman, Ostrovsky and Schwarz (2007, American Economic Review 97(1), 242-259) and Varian (2007, International Journal of Industrial Organization 25(6), 1163-1178) established this at the same time and independently. Under GSP, bidding your true value is generally not optimal: an advertiser can often do better by shading down to land in a cheaper slot that captures nearly as many clicks at a much lower price per click. GSP has many Nash equilibria rather than one dominant-strategy solution, and the two papers pick out a refinement, Edelman and coauthors call it locally envy-free, Varian calls it symmetric, in which no advertiser would want to swap outcomes with the advertiser directly above. In that equilibrium GSP revenue coincides with the truthful benchmark, but reaching it requires every bidder to reason about the others. The dominant-strategy gift is gone.

The truthful alternative exists and has a name: the Vickrey-Clarke-Groves auction, which charges each winner the externality it imposes on everyone else, the value the other bidders lose by having this bidder present. Under VCG, reporting your true value is dominant again, exactly as in the single-item Vickrey auction. The two largest ad platforms made opposite choices here. Google kept GSP, familiar and not truthful. Facebook built its ad auction on VCG. Varian and Harris (2014, American Economic Review 104(5), 442-445) describe the theory and the practical headaches of running VCG at scale, including how to estimate the click-through rates the mechanism needs and how to handle broad-match substitution; the paper is the closest thing there is to a field manual from inside the two designs.

There is one more design choice hidden in search auctions that buyers rarely see and should. Levin and Milgrom (2010, American Economic Review 100(2), 603-607) named it conflation: the platform deliberately lumps heterogeneous queries and users into coarse categories rather than auctioning each precisely targeted impression on its own. Fine targeting would let advertisers cherry-pick and thin out each auction; conflation keeps auctions thick by forcing bidders to compete over bundles. The platform trades away some match precision to keep competition high, which, by Bulow and Klemperer, is where its revenue lives. Conflation is not a limitation of the technology. It is a lever, and it is aimed at your surplus.

2019: The Year Display Gave Up Second Price

Programmatic display was the one large market that genuinely ran second-price auctions, and in 2019 it stopped. Understanding why requires one piece of plumbing.

Publishers historically sold impressions through a waterfall: the ad server was asked first, then demand sources were called one at a time in a fixed priority order until one filled the slot. Google's ad server sat at the top with a structural advantage often called last look, it could see the competing bids that had already come in and win by beating them by a cent.

Header bidding, which spread across publishers between roughly 2015 and 2018, broke the waterfall. A snippet in the page header solicits bids from many exchanges at once, before the ad server is called, so demand sources compete in parallel rather than in sequence and the last-look advantage disappears. By March 2019, about 79 percent of the top thousand programmatic-selling websites had adopted header bidding, according to the Adzerk header-bidding index for that period.

Despotakis, Ravi and Sayedi (2021, Journal of Marketing Research 58(5), 888-907) tied the format change to the plumbing change with a clean argument. Once publishers moved from the waterfall to header bidding, so that every exchange bids simultaneously, an exchange that runs a second-price auction leaves money on the table relative to one that runs first-price, because the parallel structure changes what maximizes an exchange's clearing price. The exchanges therefore moved from second-price to first-price to raise their clearing prices, and Google Ad Manager, facing header-bidding competition it could no longer answer with last look, followed. Google announced the move to a unified first-price auction in March 2019, tested it on a rising share of traffic through the summer, and completed the roll-out to all Ad Manager partners by around September 2019, removing last look and, with it, the last corner of display where we could still bid as if the auction were second-price.

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Automated Bidding Is Delegated Bidding

The dominant way to buy paid media now is not to submit bids at all. You submit an objective, a target cost per acquisition, a target return on ad spend, a budget, and the platform's algorithm computes and submits the individual bids on your behalf, auction by auction, using conversion models and competing-bid data you never see. Google's Smart Bidding and Meta's automated bidding are the large examples. Whatever else it is, this arrangement is delegation: you have handed your bidding to an agent.

The awkward fact is who the agent works for. The algorithm that bids for you is built and run by the auctioneer, the same party that owns the auction and profits when clearing prices rise. That is the classic principal-agent conflict in its purest form, and the standard tools of mechanism design are exactly the tools for reasoning about it. A target-CPA system is nominally aligned with you: the platform has to hit your target or you leave. But within the wide set of bids that would hit your target, the platform has latitude, and it holds more information than you do about how far it could have shaded on your behalf. Whether it shades down toward the minimum you needed or bids up toward the ceiling your target implies is not something you can read off the outcome.

The value you should be willing to bid, your expected profit per conversion, net of margin and discounted for the customer's lifetime, is a separate problem from the auction, and an important one; it is the subject of our companion essay on customer lifetime value as a bid control variable. Automated bidding takes that value as its input, encoded in your target, and does the auction reasoning for you. The trouble is that it does only the auction reasoning the platform chooses to do. Getting the value right and getting the auction right are distinct tasks, and the platform will perform the second only as far as it keeps you spending, which is not the same as maximizing your profit. On a thin auction with no real rival to shade against, there is nothing for the agent to save you and delegation costs nothing. On a thick, competitive auction, the shading is the whole game, and the agent has little reason to hand the savings to you rather than to the seller.

What to Log, and How to Estimate Your Own Shading

The theory converges on a short list of measurements, and in our advisory work they are the first things we install. None of them require a model; they require instrumentation and the discipline to compare like with like.

Log three numbers on every auction the platform will expose: your bid, whether you won, and the price paid. In a first-price auction the gap between your bid and the clearing price is your realized overpayment, and the distribution of that gap across auctions is your shading opportunity, measured directly rather than assumed. A team that cannot produce those three columns is bidding blind and cannot tell whether its costs come from competition, from reserves, or from its own failure to shade. Where the platform also returns a minimum-bid-to-win signal, log that too; it is the closest thing to a direct readout of the price you needed.

Estimate your shading factor empirically rather than trusting the benchmark. Plot win rate against a bid multiplier on inventory you value, and find the multiplier where the win rate on valuable impressions stops rising; bidding above that point buys only marginal inventory at full price. The (n-1)/(n) curve tells you the ballpark to expect, against roughly five competitors, about four-fifths of value; against two, about half, but the empirical plateau tells you the truth for your auctions, and the two together tell you whether the platform's estimate of your competition matches reality.

The last discipline is the one most often skipped, and the brief singled it out: stop comparing average CPMs. Average CPM misleads across formats and across the 2019 boundary for three separate reasons, and they compound. Shading changes the mix of impressions you win, you win the cheaper ones and skip the expensive marginal ones, so your average CPM can fall while your cost per outcome rises, or the reverse, with no change in efficiency. First-price and second-price report structurally different prices, so a CPM before the switch and a CPM after it are not the same measurement wearing the same name. And the winner's curse means the average value of the impressions you win differs from the average value of the impressions auctioned, so a CPM computed over wins is a biased view of the market. Compare cost per incremental outcome, computed within a single format and, wherever possible, against a holdout, and treat any cross-format CPM comparison as the beginning of a question rather than the answer to one.

None of this asks us to out-compute a platform that sees the whole auction; the demand is narrower and achievable: to know which auction you are in, to shade when the mechanism charges your bid, to correct your estimate for the news that you won, to read a reserve for what it is, and to rebuild the counterfactual when you have delegated the bid. The platform designed the mechanism to work in its favor, which is the platform's job. Leaving the mechanism unmodeled, and paying full value into an auction built to make you shade, is a choice, and it is the one this essay was written to end.

Key Takeaways

1. Second-price auctions make truthful bidding a dominant strategy (Vickrey, 1961), and that is the only setting where "bid your value" is safe. Most ad auctions are not that setting, so the rule most buyers carry is the wrong rule for the auction they are in.
2. In a first-price auction you must shade below value. The symmetric benchmark is (n-1)/(n) of value against n bidders, half with two bidders, nine-tenths with ten, and the shading shrinks as competition thickens, which is why the platform wants your auction crowded.
3. Revenue equivalence (Riley and Samuelson, 1981; Myerson, 1981) says the format is irrelevant to expected revenue under risk neutrality and symmetric independent private values. Every real ad market breaks those four assumptions, and the breaks, not the pricing label, decide the price.
4. Google search ran a generalized second-price auction (Edelman, Ostrovsky and Schwarz, 2007; Varian, 2007) that was never truthful, while Meta adopted VCG, which is. The mechanism, not the platform's name, dictates whether to bid your value or shade it.
5. Ad impressions carry a common-value component, the conversion probability, so winning is adverse news about your own estimate (the winner's curse). A more accurate model helps only if you also discount the bid for having won (Milgrom and Weber, 1982).
6. The 2019 display switch to first price (Despotakis, Ravi and Sayedi, 2021) was theoretically revenue-neutral; the buyers who paid more mostly failed to shade. Bid shading is the required response to the mechanism, not an optional vendor feature.
7. Automated bidding is delegated bidding into an auction run by your counterparty, and you cannot audit a bid you did not submit. Reserves (Ostrovsky and Schwarz, 2023) and one-more-bidder competition (Bulow and Klemperer, 1996) are the seller's levers; log your bid, your win, and your price, or build the holdout, because otherwise the mechanism is invisible and it is not working for you.

Further Reading

• Customer Lifetime Value as a Control Variable: Re-Engineering Bid Strategies for Profitable Growth, where the value you bid comes from. This essay is where that value goes into the auction; the two problems are separate and both have to be solved.
• Mechanism Design and Marketplace Reputation Systems, the same incentive-compatibility toolkit turned on reviews and ratings rather than prices, a companion piece on designing rules that survive strategic participants.
• Attention Economics Quantified: Measuring the True CPM of Cognitive Load in Digital Advertising, what a CPM actually buys once you look past the auction to the attention it clears.
• The Micro-Economics of API Pricing: Marginal Cost, Value Capture, and Developer Elasticity, value capture and pricing power seen from the seller's side of the same table.