Private information and auctions

Every model so far has assumed each player knows the other's payoffs exactly, and a bidder at an auction does not know what the object is worth to the person beside them.

Giving each player a secret

The difficulty is easy to state and looks fatal. If you do not know my payoffs, you cannot compute my best response, so you cannot compute yours. Worse, you have beliefs about my payoffs, and I have beliefs about your beliefs, and there is no obvious place for that regress to stop.

John Harsanyi's solution, in a three-part paper published in Management Science across 1967 and 1968, was to convert the unknown into a move by nature. Each player has a type $θ_{i}$ , a private variable holding everything they know and others do not: their valuation, their cost, their patience. Nature draws the whole profile of types from a probability distribution that is common knowledge, and then tells each player their own type and nothing else. The regress collapses, because everyone's beliefs about everyone's beliefs are derived by conditioning the same shared prior, rather than being separate objects to be specified.

The move is not free. It replaces "I do not know your payoffs" with "I do not know your payoffs but I know exactly the distribution they were drawn from", which is a substantial assumption and is precisely what the common prior gives up. In an auction for oil rights it is defensible; in a first meeting between two firms from different industries it is a fiction that should be flagged.

A strategy in such a game is no longer a single action. It is a function from types to actions: a rule saying what you do for each valuation you might turn out to have. A Bayes-Nash equilibrium is a profile of such rules in which each type of each player is maximising expected payoff, the expectation taken over the other players' types using the prior and over their actions using their rules. This is the same equilibrium idea as ever, applied to a game in which the strategies are functions.

The auction as a laboratory

An auction is the setting where all of this pays off, for three reasons: the type is one number, the rules are written down, and the outcomes are observed in public with money attached.

Fix the standard independent private values model. There is one indivisible object and $n$ bidders. Bidder $i$ 's value $v_{i}$ is drawn independently from the uniform distribution on $[0, 1]$ , is known to bidder $i$ alone, and is unaffected by what anyone else's value turns out to be. A bidder who wins at a price $p$ gets $v_{i} - p$ and a bidder who loses gets nothing.

Four formats are standard. In an English auction the price rises until one bidder remains. In a Dutch auction the price falls until someone claims the object. In a first-price sealed bid the highest bid wins and pays what it bid. In a second-price sealed bid, proposed by William Vickrey in 1961, the highest bid wins and pays the second-highest bid.

Truth-telling in a second-price auction

The second-price rule looks like a mistake, since the seller deliberately collects less than the winner offered. It buys something in return.

In a second-price auction, bidding your true value weakly dominates every other bid. The argument needs no probabilities at all, which is what makes it so strong. Let $h$ be the highest bid among your rivals, whatever it is.

Consider bidding above your value $v$ . This changes nothing unless $h$ lands between $v$ and your bid. In that case you win and pay $h > v$ , taking a loss, where truthful bidding would have lost the auction and paid nothing. Now consider bidding below $v$ . This changes nothing unless $h$ lands between your bid and $v$ . In that case you lose the auction, where truthful bidding would have won it at a price below your value. Every deviation is either irrelevant or harmful, which is weak dominance exactly as defined in the dominance lesson.

The consequence is that a second-price auction requires no strategic thought whatsoever. A bidder does not need to estimate how many rivals there are, how aggressive they are, or how the values are distributed. This is the property that makes the format attractive to designers, and it is the same argument that runs an English auction, where staying in until the price reaches your value is optimal for the same reason.

Example. Your value for a painting is £70 in a second-price sealed-bid auction. Compare bidding £70, £85 and £55 against a highest rival bid of first £80 and then £60.

Against £80: bidding 70 loses, worth 0; bidding 85 wins at a price of 80, worth $70 - 80 = - £ 10$ ; bidding 55 loses, worth 0. Against £60: bidding 70 wins at 60, worth £10; bidding 85 also wins at 60, worth £10; bidding 55 loses, worth 0. Truthful bidding is never beaten and it strictly beats overbidding in the first case and underbidding in the second.

Now you. Your value is £40 and you bid £50. The highest rival bid turns out to be £45. What happens, and what would truthful bidding have given?

Answer

You win and pay £45 for something worth £40 to you, a loss of £5. Truthful bidding would have lost the auction and paid nothing, which is better. Overbidding only ever changes the outcome in the range where winning is a mistake, and that is the whole content of the dominance argument.

Shading in a first-price auction

Now the format where thinking is required. In a first-price sealed-bid auction, bidding your value guarantees a payoff of zero whether you win or lose, so every bidder shades downwards. How far depends on how many rivals there are, which means the equilibrium has to be solved for.

Look for a symmetric equilibrium in which every bidder uses the same increasing rule, and guess that it is linear: $b (v) = k v$ for some constant $k$ to be found. Take one bidder with value $v$ considering a bid $b$ . They win when every one of the other $n - 1$ bidders has $k v_{j} < b$ , that is $v_{j} < b / k$ . Since values are uniform on $[0, 1]$ , each such event has probability $b / k$ , and independence multiplies them:

expected payoff = (v - b) {(\frac{b}{k})}^{n - 1}

The constant $k^{n - 1}$ does not affect where the maximum is, so maximise $(v - b) b^{n - 1}$ . Differentiating and setting to zero gives $- b^{n - 1} + (v - b) (n - 1) b^{n - 2} = 0$ , and dividing through by $b^{n - 2}$ leaves $- b + (n - 1) (v - b) = 0$ , so $n b = (n - 1) v$ and

b (v) = \frac{n - 1}{n} v

The guess is confirmed, with $k = (n - 1) / n$ . Read the formula. With two bidders you bid half your value. With four you bid three quarters. With ten you bid nine tenths, and as the field grows the shading vanishes, because the chance that shading costs you the object rises while the saving on each pound stays the same.

Example. Values are uniform on $[0, 1]$ and there are four bidders. What should a bidder with a value of 0.8 bid, and what is their expected payoff?

The bid is $0.75 \times 0.8 = 0.6$ . They win when all three rivals have values below 0.8, which has probability ${0.8}^{3} = 0.512$ , and when they win they earn $0.8 - 0.6 = 0.2$ . The expected payoff is $0.2 \times 0.512 = 0.1024$ .

Now you. Same setting with ten bidders. What does the bidder with value 0.8 bid, and what is their expected payoff now?

Answer

The bid is $0.9 \times 0.8 = 0.72$ , and the win probability is ${0.8}^{9} = 0.134$ . The expected payoff is $(0.8 - 0.72) (0.134) = 0.0107$ , a tenth of what it was with four bidders. Competition destroys bidder surplus twice over, by cutting the margin and by cutting the chance of collecting it, and the seller takes both.

Revenue equivalence

Two formats, two completely different bidding rules. Which raises more?

Work out the second-price revenue first. Everyone bids their value, so the price is the second-highest of $n$ draws from the uniform distribution on $[0, 1]$ . The expected value of the $k$ th highest of $n$ such draws is $(n + 1 - k) / (n + 1)$ , so the expected revenue is $(n - 1) / (n + 1)$ .

Now the first-price revenue. The winner is the bidder with the highest value, whose expectation is $n / (n + 1)$ , and they pay $(n - 1) / n$ of it. Multiplying:

\frac{n - 1}{n} \times \frac{n}{n + 1} = \frac{n - 1}{n + 1}

Identical. With two bidders both formats raise $1 / 3$ on average; with five, $2 / 3$ ; with ten, $9 / 11$ . The seller is indifferent, and so, it turns out, is every bidder before learning their value.

This is the revenue equivalence theorem, proved in general by Roger Myerson and by John Riley and William Samuelson, both in 1981. Any auction mechanism in which the object always goes to the bidder with the highest value, and in which a bidder with the lowest possible value expects zero, raises the same expected revenue, whatever its rules look like. The result is a strong warning against auction design by intuition: the choice between formats cannot be justified by revenue under these assumptions, so any real argument for one has to point at an assumption that fails.

Four of them fail regularly and each names a real design decision. Risk aversion breaks the tie in favour of first-price, since a risk-averse bidder bids nearer their value to reduce the chance of losing. Correlated values break it the other way: Paul Milgrom and Robert Weber showed in 1982 that when values are affiliated, the English auction raises most and the first-price sealed bid least, because open bidding reveals information that reduces the winner's fear of overpaying. Collusion is easier in second-price and English formats, where a ring can allocate the object internally and one member bids low with no risk of being outbid by a defector who would have to pay their own bid. And entry matters more than any of it: a format that discourages weak bidders from turning up loses more revenue than any shading effect, which is why sealed bids are often preferred where one bidder is known to be strong.

The winner's curse

Everything above assumes private values: what the painting is worth to you says nothing about what it is worth to me. Change that assumption and the arithmetic turns hostile.

In a common value auction the object is worth the same to everyone, and nobody knows what that is. An offshore oil tract holds a quantity of oil that is what it is; each bidder has a geologist producing an estimate. Suppose the true value is $V$ and each of $n$ bidders receives a signal drawn uniformly from $[V - 10, V + 10]$ , unbiased in the sense that its expected value is $V$ .

The trap is that you do not win at random. You win when your signal is the highest of the $n$ , and the highest of $n$ draws from that distribution has expected value

V - 10 + \frac{20 n}{n + 1} = V + 10 \frac{n - 1}{n + 1}

With ten bidders that is $V + 8.18$ . A bidder who treats their unbiased signal as an estimate of the value, bids close to it and wins has systematically overpaid by more than eight, and the more rivals they beat the worse it is. This is the winner's curse, named by Ed Capen, Bob Clapp and Bill Campbell in a 1971 paper in the Journal of Petroleum Technology, written after observing that returns on Gulf of Mexico leases in the 1950s and 1960s were far below what the geology had promised.

The repair is a conditioning argument, and it belongs on the list of things this course exists to teach. Do not ask what the object is worth given your signal. Ask what it is worth given your signal and given that your signal was the highest, since those are the only cases in which you pay anything. Formally, bid on $E [V | s_{i} = s, s_{i} = \max]$ rather than on $E [V | s_{i} = s]$ . In equilibrium bidders shade for this reason on top of the ordinary first-price shading, and the equilibrium does not lose money.

Real bidders do not do this reliably. Max Bazerman and William Samuelson's 1983 experiment auctioned jars of coins worth $8 t o c l a s s e s o f M B A s t u d e n t s . T h e a v e r a g e b i d w a s$ 5.13, so the group as a whole was cautious. The average winning bid was $10.01, s o t h e w i n n e r s l o s t a n a v e r a g e o f$ 2.01 per jar. Caution in the aggregate is no protection: the auction selects the most optimistic estimate in the room and hands them the bill.

Example. Ten bidders each receive a signal uniform within £10 million of the true value of a tract. You receive £58 million and win. What is a sensible estimate of the tract's value?

The expected overshoot of the highest of ten signals is $10 \times 9 / 11 = £ 8.18$ million, so conditional on your signal being the highest, an unbiased estimate of the value is $58 - 8.18 = £ 49.8$ million. Your bid should be below that, not below £58 million, and the gap between those two numbers is what has bankrupted bidders in real lease sales.

Now you. The same tract and the same signal of £58 million, but only four bidders. What is the estimate now, and why has it moved?

Answer

The overshoot is $10 \times 3 / 5 = £ 6$ million, so the estimate is £52 million. Fewer rivals means winning is weaker evidence that you were the optimist, so the curse is milder. The general and slightly alarming implication is that a bidder should get more cautious as the field gets larger, which is the opposite of the instinct competition produces.

What the theory has actually built

Auction theory is the part of this subject with the clearest record of practical effect, and the record includes both directions.

The New Zealand government ran a second-price sealed-bid auction for radio spectrum in 1990 on exactly the truth-telling logic above, and discovered its political weakness: the gap between winning bids and prices paid is public. One licence attracted a winning bid of NZ $100, 000 a n d s o l d f o r N Z$ 6, and another drew NZ $7 m i l l i o n a n d s o l d f o r N Z$ 5,000, because in each case the runner-up was far behind. The theory was working as designed and the format was abandoned.

The United Kingdom's auction of five third-generation mobile licences in April 2000, designed with advice from Ken Binmore and Paul Klemperer, ran as an ascending auction over 150 rounds and raised £22.5 billion. Germany's auction four months later raised about €50 billion. Other European countries using different rules in the same year and the same industry raised radically less, and Klemperer's account of why puts almost none of the weight on the format's revenue properties in theory and almost all of it on entry and collusion: Switzerland's auction, which allowed joint bidding until the number of bidders had fallen close to the number of licences, raised roughly €20 per head of population against the United Kingdom's €650.

That is the honest summary of what the machinery is worth. The equilibrium calculations are correct and second-order. What decides an auction is how many serious bidders show up and whether they can quietly agree not to compete, and the contribution of theory is mostly that it says clearly which of those questions to ask.

Where this leaves us

Private information turned out not to break the framework. Types, a common prior and Bayes-Nash equilibrium reproduce everything the earlier lessons did, with strategies as functions rather than actions, and they deliver sharp results: truth-telling under a second-price rule, a shading factor of $(n - 1) / n$ , revenue equivalence across formats, and a correction for the curse of winning.

The course now has its full toolkit, and one obligation left. Across fourteen lessons the theory has predicted penalty kicks accurately, predicted ultimatum offers badly, produced a folk theorem that predicts nothing, and produced auction advice that governments paid for and used. The last lesson lays those results side by side, asks what distinguishes the successes from the failures, and looks at the models built to fit the failures: level-k reasoning, quantal response and social preferences. It also confronts the temptation named in the very first lesson, that any behaviour at all can be explained after the fact by rewriting the payoffs.