Two auctions that earn the same

The reading that is almost right gave each chooser a private reading of a shared quantity, slightly off, and found that private information can turn a band of equilibria into a single one. An auction is the cleanest game built on private information. Each bidder knows what the object is worth to itself and nothing about what it is worth to the others, except the odds; each submits a sealed bid; the rules decide who wins and what is paid. The seller chooses the rules, and the question every seller asks is which rules collect the most.

Two rules are the classical candidates. In a first-price auction the highest bid wins and pays its bid. In a second-price auction the highest bid wins and pays the second-highest bid. The second sounds like a gift to the winner, and the first like the seller’s natural choice. William Vickrey showed in 1961 that, under the conditions this essay assumes, they earn the seller exactly the same on average. This essay measures that equality, the ways bidders behave to produce it, and the two ways it fails.

Bidding one’s value, and bidding less #

The model is the simplest one with real private information. There are $n$ bidders. Each bidder’s value for the object is drawn independently and uniformly between 0 and 1; each knows its own value and knows that the others’ are drawn the same way. A bidder that wins gains its value minus what it pays; a bidder that loses gains nothing.

In the second-price auction the best bid needs no calculation: bid exactly one’s value. If the bid wins, the price is set by somebody else’s bid, so raising one’s own bid changes nothing about the price, only about whether one wins — and winning is worth it exactly when the second-highest bid is below one’s value, which is exactly when bidding one’s value wins. Bidding more risks winning at a price above one’s value; bidding less risks losing an object one would gladly have paid for. Truthful bidding is best whatever the others do, the property a lie that pays found that most voting rules lack. The seller’s revenue is then simply the second-highest value.

In the first-price auction, bidding one’s value is pointless: winning at one’s value gains nothing. So bidders shade, and how much depends on what the others do — this is a game, not a calculation.

Suppose every rival bids a fixed fraction $k$ of its value. A bid $b$ then beats a given rival with probability $b/k$, and all $n - 1$ of them with probability $(b/k)^{n-1}$, so the expected profit is $(v - b)(b/k)^{n-1}$. Bidding high wins often and gains little; bidding low gains a lot when it wins and rarely wins. The peak is at $b = \tfrac{n-1}{n} v$, whatever $k$ is. With $k = \tfrac{n-1}{n}$ itself, everyone’s best reply is to use the same fraction: with three bidders, everyone bids two thirds of their value, and nobody gains by changing. That is the symmetric equilibrium, the solution concept the value from both sides introduced for two players with known payoffs, here in a game where each player’s payoff is private.

The same average, by two different routes #

The seller’s revenue in the first-price auction is $\tfrac{n-1}{n}$ times the highest value; in the second-price auction it is the second-highest value. Those are different random quantities with different distributions, as the hero figure shows: the second-price revenue ranges over almost everything from nought to one, and the first-price revenue bunches below two thirds. Their averages are 0.4988 and 0.4995 in the simulation — both $\tfrac{n-1}{n+1} = \tfrac12$ for three bidders.

The exact calculation is a fact about the largest values in a sample. For $n$ values drawn uniformly, the largest has average $\tfrac{n}{n+1}$ and the second-largest $\tfrac{n-1}{n+1}$ — the order statistics split the interval into $n + 1$ pieces of equal average length, the fact behind the values drawn from the same hat. The first-price revenue averages $\tfrac{n-1}{n} \cdot \tfrac{n}{n+1} = \tfrac{n-1}{n+1}$, and the second-price revenue averages $\tfrac{n-1}{n+1}$ directly. The shading in the first auction is precisely enough to make up for paying the highest bid instead of the second.

The third rule in the figure makes the equality look like more than a coincidence. In an all-pay auction every bidder pays its bid, and only the highest wins — the shape of a lobbying contest or a race to a patent. Bidders shade far more, since losing bids are paid too: the equilibrium bid is $\tfrac{n-1}{n} v^n$, nearly nothing for low values. And the seller’s average revenue is, again, $\tfrac{n-1}{n+1}$. Three rules with completely different bids, payments and losers, and one average.

A bid is a forecast of the second-highest value #

There is a cleaner way to see the shading, and it explains the equality before any theorem. In the second-price auction, a bidder with value $v$ that wins pays the second-highest value, whatever it is. Given that $v$ is the highest of $n$ uniform values, the others are uniform below $v$, and the largest of those $n - 1$ has average $\tfrac{n-1}{n} v$. So the price a winner expects to pay in the second-price auction is exactly $\tfrac{n-1}{n} v$ — the bid it makes in the first-price auction.

A first-price bidder, in equilibrium, bids its forecast of what it would have paid under the other rule. That is why the averages agree: whenever the bidder with value $v$ wins, the first-price auction charges the forecast and the second-price auction charges the outcome, and a forecast and its outcome have the same average. The first-price rule charges it with certainty; the second-price rule charges it with noise around it. The spread figure is that noise, and the risk-aversion figure is what happens when bidders will pay to avoid it.

The same reading covers the all-pay auction. There, a bidder pays its bid whether or not it wins, so its bid must equal its forecast price times its chance of winning: $\tfrac{n-1}{n} v$ times $v^{n-1}$, which is the equilibrium bid $\tfrac{n-1}{n} v^n$. Every rule charges the same expected amount; the rules differ only in whom they charge it to and when. And the reasoning fails exactly where the theorem’s hypotheses fail — when the winner is not the bidder with the highest value, or when a bidder cares about the noise, or when its forecast of the others depends on its own information.

The truthful bidding of the second-price auction has a further property that the first-price equilibrium lacks: it does not depend on knowing the distribution of the others’ values. A first-price bidder who misjudges how many rivals there are, or how their values are spread, shades by the wrong amount; a second-price bidder never has to judge either. That robustness is what made Vickrey’s rule attractive to designers of automated markets, and it is the same property that no stable matching rule can have for everyone: a rule under which telling the truth is best whatever the others do is rare, and auctions for a single object are one of the few places it is achieved without giving anything up.

Why the average cannot differ #

The revenue equivalence theorem, proved in general by Roger Myerson and independently by John Riley and William Samuelson in 1981, explains it. Suppose bidders are indifferent to risk, values are independent and drawn from the same distribution, the object always goes to the bidder with the highest value, and a bidder with the lowest possible value expects to pay nothing. Then every auction rule with an equilibrium gives the seller the same expected revenue.

The proof is a short argument about a single bidder. In any such equilibrium, a bidder with value $v$ wins with probability $v^{n-1}$ — the chance its value is the highest — whatever the rules. Its expected profit, as a function of its value, must then grow at exactly the rate of that winning probability: a bidder with value $v + \varepsilon$ could always pretend to be $v$ and win as often, at the same expected price, and gain $\varepsilon$ more each time it wins, and in equilibrium it cannot do better than that by more than a vanishing amount. So the expected profit is pinned down by the winning probabilities, and since the expected payment is value times winning probability minus expected profit, the payment is pinned down too. The rules can redistribute who pays what and when; they cannot change how much is paid on average.

That argument is the same shape as the curve of an average and the average of a curve: the profit as a function of value is convex, its slope is the winning probability, and fixing the slope everywhere fixes the function. It is also why the ascending auction of the auction house, where the price rises until one bidder remains, earns the same: with private values it is strategically the same as the second-price auction — the last rival drops out at its value — and the descending Dutch auction, where the price falls until somebody calls out, is strategically the same as the first-price auction.

Same mean, different risk #

Equal averages do not make the auctions equal for a seller who cares about anything besides the average.

The second-price revenue is the more variable, by a factor of 2.00 with two bidders and 1.45 with twenty. With two bidders it is the lower of the two values, which is anything from nought to one; the first-price revenue is half the higher value, which can never exceed one half. A seller who would rather have a sure 0.33 than a gamble averaging 0.33 prefers the first-price rule. As the number of bidders grows both spreads shrink, because the top two values of a large sample crowd towards the top of the range, and the difference between the rules shrinks with them.

When bidders dislike risk #

The theorem assumes bidders care only about expected profit. Real bidders often prefer a sure modest gain to a gamble with the same average, and that changes the first-price auction but not the second.

In the second-price auction risk does not matter: bidding one’s value is best whatever one’s attitude, because the bid never affects the price. In the first-price auction a cautious bidder faces a real trade-off between profit and the chance of getting any, and caution tilts it towards winning: the equilibrium bid rises from $\tfrac{n-1}{n}$ of value to $\tfrac{n-1}{n-1+\alpha}$. With three bidders and $\alpha = 0.2$ — strongly averse to risk — they bid 91% of their values, and the seller’s average revenue rises from 0.500 to 0.682. Revenue equivalence holds for bidders indifferent to risk, and only for them. Charles Holt in 1980 and Eric Maskin and John Riley in 1984 worked this out in general: with risk-averse bidders, the first-price auction earns more.

Other departures break it in other directions. If values are correlated — an oil field worth roughly the same to every driller, each with a private estimate — the second-price and ascending auctions earn more, because the price paid is tied to the other bidders’ information. If bidders differ — one known to value the object more — the first-price auction may give the object to the wrong bidder, and the comparison can go either way. The theorem’s hypotheses are each doing work.

A reserve price, and the best one #

The theorem compares rules that always sell. A seller can also refuse to sell below a reserve price, and that is where real gains are.

Every curve peaks at $r = \tfrac12$, whatever the number of bidders. A reserve is a threshold set before any value is seen, like the single threshold that half of what an oracle takes showed is enough to secure half of the best possible reward, and it works here for a related reason: it converts competition the seller cannot count on into a price it can. With a single bidder the reserve is the whole story: a posted price of one half sells half the time and averages 0.25, the best a take-it-or-leave-it offer can do. With two bidders the reserve raises the average from $\tfrac13$ to $\tfrac{5}{12}$; with five, from 0.667 to 0.672 — the more bidders, the less the reserve matters, because competition already pushes the second-highest value above one half most of the time.

That the best reserve does not depend on the number of bidders is the striking part, and it is Myerson’s other result of 1981. The best auction of all, among every possible rule, is a standard auction with a reserve set where a quantity he called the virtual value crosses nought — for values uniform on $[0,1]$, at one half. The reserve is fixed by the distribution of values alone. It also means the seller deliberately keeps the object sometimes when a buyer would pay something for it: the revenue-maximising auction is not the efficient one, and the gap is the price of the seller’s private gain.

Where the theory is used #

Vickrey’s second-price auction was a curiosity for decades, then became the model for the auctions that sell most of the world’s online advertising: each search triggers an auction among advertisers, decided in milliseconds, and the generalized second-price auction used for years by the major search engines charges each winner roughly the next bid down. More recently several large advertising exchanges switched to first-price rules, and bidders responded by shading — exactly the adjustment the theorem predicts, with exactly the averages it predicts as long as its hypotheses hold.

Spectrum auctions, where governments sell radio frequencies, are where the departures matter: values are correlated, bidders differ, and licences are wanted in combinations, which breaks the single-object model altogether. The design of those auctions, which raised hundreds of billions, drew directly on the ways equivalence fails. Those are combinatorial problems of the kind prices that nobody can break away from met for many items at once, where a single price per item may not exist and the clean equivalence of this essay has no counterpart.

What the figures cannot show #

Every figure assumes values uniform between nought and one. The equivalence itself holds for any common distribution of independent values, but the numbers — the shading fraction $\tfrac{n-1}{n}$, the average $\tfrac{n-1}{n+1}$, the reserve of one half — are properties of the uniform case, and with other distributions the equilibrium bid is a more complicated function of value and the best reserve moves. The simulations check the formulas only for the uniform case.

The figures also show equilibrium behaviour, which is a prediction, not an observation. Laboratory experiments with first-price auctions consistently find bids above the risk-neutral equilibrium, which is one reason risk aversion is studied so closely; in second-price auctions they find frequent overbidding, which no standard model predicts and which the theorem’s elegant argument for truthful bidding does not prevent. The theory says what the bids should be; the people in the room do not always agree.

Still open: many objects, and the bidders in the room #

For a single object with independent private values, the theory is complete: equivalence, its failures, and the optimal auction are all known. For many objects sold together the picture is far from complete. When bidders want combinations — two adjacent spectrum licences worth more together than apart — the natural generalisation of Vickrey’s auction exists, but it can give low or even zero revenue in reasonable cases and is open to manipulation by bidders who split into several identities. Which auction for many objects is best for the seller, or how close simple rules come to the best, has precise answers only in special cases, and approximately optimal designs are an active field.

The behavioural question is open too. Why bidders in second-price auctions bid above their values, when doing so can only hurt them, has several explanations — a misunderstanding of the rule, a dislike of losing, spite towards other bidders — and no consensus. Since many real systems rely on the argument that truthful bidding is best, the gap between the argument and observed behaviour has practical stakes.