The Winner’s Curse

In a common-value auction the item is worth the same to everybody — an oil tract holds however much oil it holds, a spectrum licence generates whatever revenue it generates — but nobody knows what that value is, so each bidder acts on a private, noisy estimate. Suppose every bidder’s estimate is unbiased: on average it is exactly right. Bid your estimate and you should break even, surely? You will not. You only get the item when your estimate was the highest of the n drawn, and the highest of n unbiased estimates is not unbiased — it is optimistic by a predictable amount that grows with n. An estimator that is unbiased unconditionally becomes biased conditional on victory, and the auction hands you the item exactly in the states of the world where you were most wrong. Paul Milgrom and Robert Weber state it in one sentence in the paper that founded the modern theory: “even if all bidders make unbiased estimates, the winner will find that he had overestimated (on average) the value of the rights he has won at auction” (A Theory of Auctions and Competitive Bidding, Econometrica 50(5), September 1982). This is the winner’s curse: as John Kagel and Dan Levin put it, “you win, you lose money, and you curse” (Common Value Auctions and the Winner’s Curse: Lessons from the Economics Laboratory, 2001). The fix is conceptually strange and mechanically simple: bid as though you already know you have won. Klemperer states the same discipline for an ascending auction — a bidder “quits exactly at its value conditional on her winning” (Auction Theory: A Guide to the Literature, Appendix). This note derives the correction, measures the curse and its repair in C, and reviews the empirical evidence with the scepticism it deserves.

This is the common-value branch of stage P4 · Auctions in Games and Strategic Systems in C MOC. It is the note where the independence assumption that carries Revenue Equivalence, The Vickrey Auction and Reserve Prices and Optimal Auctions is dropped, and almost every conclusion from those notes changes.

1. Two Kinds of Uncertainty

Kagel and Levin draw the line in one sentence: “Private-value auctions, where bidders know the value of the item to themselves with certainty but there is uncertainty regarding other bidders’ values. Common-value auctions, where the value of the item is the same to everyone but different bidders have different estimates about the underlying value.”

The two settings are not variations on a theme; they are different problems with different pathologies.

	Private values (IPV)	Common values
What you don’t know	what others value it at	what the item is worth
What you do know	your own value, exactly	a noisy signal of a shared truth
Learning a rival’s information	changes nothing about your value	changes your value estimate
Winning tells you	you had the highest value	your signal was the most optimistic
The pathology	shading in first-price (First-Price and Second-Price Auctions)	the winner’s curse
Efficiency question	who values it most?	none — everyone values it equally
Second-price truthfulness	dominant strategy	not a dominant strategy

That last row is the one that trips people. In a common-value auction the second-price auction is not strategyproof, because your willingness to pay depends on information you do not have and can only infer from other people’s behaviour. See The Vickrey Auction for the private-value result whose hypotheses this violates.

Kagel and Levin are careful that the dichotomy is an idealization: “Most (non-laboratory) auctions have both private value and common value elements. […] Bidders for an oil painting may purchase for their own pleasure, a private-value element, but they may also bid for investment and eventual resale, reflecting the common-value element.” Anything you might resell has a common-value component. So does anything whose cost of exploitation is uncertain — which is why construction procurement is a textbook case.

They also note that in a pure common-value auction, “there are no efficiency issues […] as all bidders place equal value on the item.” Allocative efficiency, the objective that dominates the private-value literature, is vacuous here. What is left is revenue, information aggregation, and whether the winner survives.

flowchart TB
    T["True common value x0<br/>(unknown to everyone)"]
    S1["bidder 1 signal<br/>x1 = x0 + noise"]
    S2["bidder 2 signal<br/>x2 = x0 + noise"]
    S3["bidder 3 signal<br/>x3 = x0 + noise"]
    SN["…bidder n signal<br/>xn = x0 + noise"]
    W["The auction selects<br/>the MAXIMUM of the signals"]
    B["max of n unbiased draws<br/>is biased UPWARD"]
    C["Winning is evidence that<br/>your noise term was large<br/>and positive"]
    T --> S1 & S2 & S3 & SN --> W --> B --> C

What this shows and what to take from it: the auction is a selection mechanism running on your estimation error. Each signal is individually unbiased, but the auction does not sample a signal at random — it deliberately picks the largest one, and the largest of n unbiased draws is systematically too high. The insight to carry: the curse is not a statement about bidders being irrational or bad at estimating. It survives perfectly calibrated, perfectly unbiased estimation. It is a property of the conditioning event, not of the estimator.

2. The Mathematics: How Much Is Winning Worth Knowing?

2.1 The model

Follow the Kagel–Levin experimental design exactly, because it is the one with published measurements to check against. A common value x0 is drawn from a known uniform distribution on [x_lo, x_hi]. Bidder i receives a private signal

x_i  ~  Uniform[ x0 − ε ,  x0 + ε ]

with ε known to everyone. Reading the symbols: x0 is the true value the item will turn out to have; ε is the precision of everyone’s information — small ε means sharp estimates, large ε means fog. Each signal is unbiased: E[x_i | x0] = x0. Nobody is systematically optimistic.

Kagel and Levin restrict analysis to “region 2”, the interval x_lo + ε ≤ x ≤ x_hi − ε, “where the bulk of the observations lie.” The reason is a boundary effect: a bidder holding a signal below x_lo + ε learns something extra from the support of the prior (if your signal is x_lo + 1 you know x0 cannot be as low as x_lo + 1 − ε, so you can sharpen your estimate). Inside region 2 no such endpoint information exists and the algebra is clean. Every measurement below restricts to region 2 and says so.

2.2 The bias of the winning signal

x_i = x0 + e_i where e_i ~ U[−ε, ε] independently. The highest signal exceeds the truth by max_i e_i, and the expected maximum of n i.i.d. draws from U[−ε, ε] is

E[ max(e_1, …, e_n) ]  =  −ε + 2ε · n/(n+1)  =  ε · (n−1)/(n+1)

Derivation: the maximum of n i.i.d. U[0,1] draws has mean n/(n+1) (its CDF is z^n, so E = ∫₀¹ (1 − z^n) dz = 1 − 1/(n+1)). Rescale U[0,1] → U[−ε, ε] by z ↦ −ε + 2ε z and the mean maps to −ε + 2ε·n/(n+1) = ε(n−1)/(n+1).

So the winning bidder’s signal is too high by ε(n−1)/(n+1) on average. That expression is worth staring at:

at n = 1 it is 0 — with no competition there is no selection and no curse;
at n = 2 it is ε/3;
at n = 10 it is 0.818 ε;
as n → ∞ it tends to ε, the entire width of the noise.

Adding bidders makes the curse worse, monotonically and without limit (up to ε). This is the single most counter-intuitive comparative static in auction theory, because in a private-value auction more bidders is unambiguously good news for the seller and irrelevant to a truthful bidder’s strategy. Here more bidders means you must bid less, and Kagel and Levin identify getting this backwards as diagnostic: “Higher individual bids in response to increased numbers of rivals is often considered to be the hallmark characteristic of a winner’s curse.”

xychart-beta
    title "Bias of the winning signal, eps(n-1)/(n+1), with eps = 30"
    x-axis "number of bidders n" [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
    y-axis "how much the winning signal overstates the truth" 0 --> 30
    bar [0.0, 10.0, 15.0, 18.0, 20.0, 21.4, 22.5, 23.3, 24.0, 24.5]
    line [30, 30, 30, 30, 30, 30, 30, 30, 30, 30]

What this shows and what to take from it: the bars are the expected overstatement of the winning signal; the flat line is ε, the full width of the noise, which the bias approaches but never reaches. The shape matters: the curse arrives fast — half of its asymptotic size is already present at n = 2, and three-quarters by n = 7. The takeaway is that “there are only a few bidders, so the effect must be small” is wrong. Two rivals is already most of the problem.

2.3 The conditional expectation — what you should have bid

The quantity you actually want is E[x0 | X = x_1:n]: the expected true value given that your signal x was the highest of the n drawn. Kagel and Levin state it (their equation 2) as

E[ x0 | X = x_1:n ]  =  x  −  ((n−1)/(n+1)) · ε

for signals in region 2. Deriving it independently confirms the formula and shows where it comes from. Given the highest signal equals x, the true value must lie in [x − ε, x + ε]. Writing u = x0 − x, the likelihood of observing “max signal = x” given x0 is proportional to the density of one signal at x times the probability the other n−1 land below x, which is ((x − x0 + ε)/(2ε))^{n−1} = ((ε − u)/(2ε))^{n−1}. With a locally flat prior on x0 (region 2), the posterior density of u is proportional to (ε − u)^{n−1} on [−ε, ε]. Substituting w = ε − u:

E[u]  =  ∫₀^{2ε} (ε − w) w^{n−1} dw  /  ∫₀^{2ε} w^{n−1} dw
      =  [ ε(2ε)^n/n − (2ε)^{n+1}/(n+1) ]  /  [ (2ε)^n/n ]
      =  ε  −  2εn/(n+1)
      =  −ε (n−1)/(n+1)

which reproduces Kagel and Levin exactly. Note that this is the same number as §2.2’s unconditional bias of the winning signal — a coincidence of the uniform model, not a general identity, and one that makes this design unusually good for teaching.

The correction, in one sentence: subtract ε(n−1)/(n+1) from your signal before you even start thinking about strategy.

A bidder who bids exactly E[x0 | X = x_1:n] earns zero expected profit. That is the break-even bid, not the optimal one, and the distinction matters.

2.4 The bid factor decomposes into two independent pieces

Kagel and Levin’s risk-neutral Nash equilibrium (RNNE) bid function for the first-price auction — originally due to Wilson — is

b(x)  =  x  −  ε  +  h(x),        h(x) = (2ε/(n+1)) · exp( −(n/(2ε)) · (x − (x_lo + ε)) )

Provenance of the equilibrium — Wilson 1969 and 1977, verified bibliographically

Klemperer’s survey credits Robert Wilson with having “introduced the (pure) common-value model and developed the first closed-form equilibrium analysis of the winner’s curse”, and Kagel and Levin write that “Wilson (1977) was the first to develop the Nash equilibrium solution for first-price common-value auctions, and Milgrom and Weber (1982) provide significant extensions and generalizations.” Both Wilson papers were located as JSTOR scans on a Princeton course mirror and their identities confirmed: Wilson (1969), “Competitive Bidding With Disparate Information”, Management Science 15(7), March 1969 (text layer readable; it is a two-bidder model, presented as a “Communications to the Editor” note), and Wilson (1977), “A Bidding Model of Perfect Competition”, Review of Economic Studies 44(3), October 1977, pp. 511–518. The 1977 scan has no text layer at all — pdftotext returns 9 characters — so its cover page was read by rendering it with pdftoppm -png -r 110; “could not extract text” is not the same as “could not read.” The body of Wilson (1977) was not read for this note, only its bibliographic identity confirmed; the b(x) formula above is taken from Kagel and Levin’s survey, which reproduces it as their equation (1). #uncertain

h(x) decays exponentially and is negligible except near the bottom of region 2, so b(x) ≈ x − ε: shade by the full noise width. The total shading of ε splits cleanly:

component	size	why
adverse-selection correction	`ε (n−1)/(n+1)`	to stop losing money — undo the bias winning induces
strategic shading	`2ε / (n+1)`	to make money — the ordinary first-price margin
total	`ε`

The two sum to exactly ε, since (n−1)/(n+1) + 2/(n+1) = 1. Kagel and Levin give the same split in percentage terms — “with n = 4 the bid factor is 60% of ε and with n = 7 it is 75%” — and (4−1)/(4+1) = 60%, (7−1)/(7+1) = 75%, so the derivation checks against the source.

Their explanation of why the strategic piece exists is worth quoting because it is the part people skip: “if just correcting for the adverse selection effect, the winner would earn zero expected profits, which is not a very attractive outcome. As such, a bidder would find it profitable to lower her bid from this hypothetical benchmark since zero expected surplus is lost by doing so even if this causes her not to win the item, and strictly positive expected surplus is awarded should she win the item with the lower price.”

flowchart LR
    SIG["your signal<br/>x"]
    ADV["− eps(n−1)/(n+1)<br/>ADVERSE SELECTION<br/>(survival)"]
    ZERO["break-even bid<br/>E(x0 given you won)<br/>expected profit = 0"]
    STR["− 2eps/(n+1)<br/>STRATEGIC MARGIN<br/>(profit)"]
    RNNE["equilibrium bid<br/>b(x) ≈ x − eps<br/>expected profit = 2eps/(n+1)"]
    SIG --> ADV --> ZERO --> STR --> RNNE

What this shows and what to take from it: the two shadings do completely different jobs and fail differently. Skip the strategic margin and you break even. Skip the adverse-selection correction and you go bankrupt — and the amount you must subtract grows with the number of rivals, which is exactly when a naive bidder’s instinct says to bid more. The measured consequence is §4.

3. “Bid As Though You Already Won”

The correction has a formulation that turns it from an algebraic adjustment into a discipline you can apply without any algebra at all. Klemperer’s exposition of the ascending common-value auction states it as the equilibrium condition:

“when the player with the second-highest signal quits, she knows (assuming equilibrium behaviour) that the remaining signal is (weakly) higher than hers. So she is sure the actual value of the object to her cannot be less than the price at which she is quitting […] This illustrates the winner’s curse. The point is that what is relevant to her is not the expected value of the object, but rather its expected value conditional on her winning it. Only when she wins the object does she care about its value, so she quits exactly at its value conditional on her winning.”

That last clause is the whole technique. The naive bidder asks “what do I think this is worth?”. The corrected bidder asks “suppose I am about to win at this price — what would I then believe it is worth?”, and bids that. The counterfactual is legitimate because the only states of the world in which your bid has any consequence at all are the states in which you win. Your bid does nothing when you lose; conditioning on the event that matters is not paranoia, it is the only correct thing to condition on.

Klemperer notes the same principle running through the analysis of public information: “All bidders evaluate the additional public information assuming that their signal is the highest since, in equilibrium, they only win in this case.”

Once framed this way, the winner’s curse joins a family of selection effects the vault already documents in other clothes:

Adverse selection. Kagel and Levin’s reference list opens with Akerlof’s The Market for Lemons (1970) for exactly this reason: the used car you are offered is offered because the seller knows something. Winning is the same signal.
Berkson’s paradox / collider bias. Conditioning on a collider (here: “I won”) induces dependence between your estimation error and everything else.
Regression to the mean. The best-performing thing you selected on will do worse next period, because you selected partly on noise.
The swing voter’s curse. Kagel and Levin cite Feddersen and Pesendorfer (1998, 1999): condition on your vote being pivotal and it tells you what everyone else must have voted.

If you already have intuition for any of those, you have intuition for the winner’s curse; what is peculiar about the auction is that the mechanism manufactures the selection deliberately and hands you a bill for it.

Terminological trap — economists use "winner's curse" for two different things uncertain

Kagel and Levin flag this explicitly and it causes real confusion when reading across the literature: “many economists, particularly theorists, when discussing the winner’s curse use the term to refer to the difference between the expected value of the item conditional on the event of winning and the naive expectation. Further, their use of the term typically refers to players who fully account for this winner’s curse, rather than those who fall prey to it.” So in a theory paper “the winner’s curse” is usually the quantity ε(n−1)/(n+1) that rational bidders correct for; in an experimental or applied paper it is usually the error of failing to correct. This note uses the experimental sense (the error) throughout and says “the adverse-selection correction” for the quantity. When reading a paper, check which sense is in force before concluding the authors disagree with each other.

4. Measuring the Curse in C

Assertion is cheap. Here is the curse, and its repair, measured.

4.1 The four bidding rules

#define LO 50.0                    /* common value x0 ~ U[LO, HI]           */
#define HI 950.0
static double EPS = 30.0;          /* signal x_i ~ U[x0-EPS, x0+EPS]        */
 
static double b_naive (double x,int n){ return x; }                      /* bid your estimate  */
static double b_shade (double x,int n){ return x - EPS/2.0; }            /* "shade a bit"      */
static double b_cond  (double x,int n){ return x - EPS*(n-1.0)/(n+1.0); }/* break-even bid     */
static double b_rnne  (double x,int n){                                  /* Nash equilibrium   */
  double h = (2.0*EPS/(n+1.0)) * exp(-(n/(2.0*EPS)) * (x - (LO+EPS)));
  return x - EPS + h;
}

Rule 1 is the naive bidder: unbiased estimate in, bid out. Rule 2 is the plausible-sounding heuristic “there’s uncertainty, I’ll knock off half the noise” — included because it is what a careful but uninformed person actually does, and because it makes the point that shading is not the same as shading correctly. Rule 3 is the break-even bid from §2.3. Rule 4 is the Wilson/Kagel–Levin RNNE.

Note that rules 3 and 4 take n as an argument and rules 1 and 2 do not. A correct common-value strategy cannot be written without knowing how many rivals you face. In a private-value second-price auction it can — truthful bidding is dominant regardless of n. That difference is the whole practical difficulty of the subject: the correction you must apply depends on a number you frequently do not know (§6.3).

The measurement harness is the same splitmix64 generator, per-seed means, and standard deviation across seeds described in Reserve Prices and Optimal Auctions §5. All runs below restrict x0 to [LO + 2ε, HI − 2ε] so every signal lands in region 2; that retains 780/900 ≈ 86.7% of the prior’s support.

4.2 The bias is exactly what the algebra says

./curse part 1 — measured E[max signal − x0] against the prediction ε(n−1)/(n+1), 20 seeds × 300,000 trials, ε = 30:

`n`	measured	sd across seeds	predicted `ε(n−1)/(n+1)`
1	−0.0006	0.0359	0.0000
2	9.9964	0.0240	10.0000
3	15.0018	0.0231	15.0000
4	17.9929	0.0201	18.0000
5	19.9963	0.0149	20.0000
6	21.4237	0.0152	21.4286
7	22.4993	0.0119	22.5000
8	23.3297	0.0107	23.3333
10	24.5423	0.0094	24.5455

Every row is within one to two seed-standard-deviations of the prediction, and the n = 1 row confirms there is no bias without competition. The signal is unbiased; the winning signal is not, and by n = 10 it is optimistic by 82% of the entire noise width.

4.3 Profit by strategy: the curse and its repair

./curse part 2 — first-price auction, all n bidders using the same rule, winner’s expected profit x0 − b₁, 20 seeds × 300,000 trials:

`n`	naive `b = x`	shade `b = x − ε/2`	break-even `b = x − ε(n−1)/(n+1)`	RNNE
2	−10.001 ± 0.025	+4.999 ± 0.025	−0.001 ± 0.025	+19.771 ± 0.025
3	−15.001 ± 0.017	−0.001 ± 0.017	−0.001 ± 0.017	+14.950 ± 0.017
4	−18.001 ± 0.017	−3.001 ± 0.017	−0.001 ± 0.017	+11.986 ± 0.017
5	−19.997 ± 0.015	−4.997 ± 0.015	+0.003 ± 0.015	+9.999 ± 0.015
6	−21.422 ± 0.011	−6.422 ± 0.011	+0.007 ± 0.011	+8.577 ± 0.011
7	−22.496 ± 0.010	−7.496 ± 0.010	+0.004 ± 0.010	+7.504 ± 0.010
8	−23.332 ± 0.009	−8.332 ± 0.009	+0.002 ± 0.009	+6.668 ± 0.009
10	−24.546 ± 0.009	−9.546 ± 0.009	−0.001 ± 0.009	+5.454 ± 0.009

Four things fall out, all of them checkable against closed forms:

Naive profit is exactly −ε(n−1)/(n+1). The naive column is the negative of §4.2’s bias column, to four decimals. The loss is the bias, nothing more and nothing less. It is negative at every n ≥ 2 and gets monotonically worse.
The break-even bid breaks even, at every n, to within ±0.007 on a 900-wide value scale. The formula does what it claims.
RNNE profit is exactly 2ε/(n+1) — the strategic component of §2.4. Check: n = 5 → 60/6 = 10.0 (measured 9.999); n = 10 → 60/11 = 5.454 (measured 5.454); n = 2 → 20.0 (measured 19.771, the gap being the exponential h(x) term, which is only non-negligible at small n near the bottom of region 2).
The plausible heuristic is right for exactly one value of n. “Shade by half the noise” breaks even at n = 3 — because ε(3−1)/(3+1) = ε/2 — and loses money at every larger n. This is the most practically dangerous column in the table: a rule that was calibrated when the market had three bidders quietly becomes a money-loser as the market thickens, with no signal that anything changed.

xychart-beta
    title "First-price common-value auction: winner's profit by strategy (eps = 30)"
    x-axis "number of bidders n" [2, 3, 4, 5, 6, 7, 8, 9, 10]
    y-axis "winner's expected profit" -30 --> 25
    line [-10.0, -15.0, -18.0, -20.0, -21.4, -22.5, -23.3, -24.0, -24.5]
    line [5.0, 0.0, -3.0, -5.0, -6.4, -7.5, -8.3, -9.0, -9.5]
    line [19.8, 14.9, 12.0, 10.0, 8.6, 7.5, 6.7, 6.0, 5.5]

What this shows and what to take from it: three trajectories. The bottom line is the naive bidder, diving further below zero with every rival added. The middle line is the half-shading heuristic, crossing zero at n = 3 and never coming back. The top line is the equilibrium bidder, always positive but earning less as competition rises — the ordinary effect of competition, which is what a healthy market looks like. The takeaway is the divergence: the naive bidder and the equilibrium bidder move in opposite directions as n grows, so a field that looks profitable at n = 3 can be catastrophic at n = 8 with no change in anyone’s behaviour.

4.4 The correction protects you; it does not enrich you

The obvious follow-up: what happens to a single corrected bidder in a field of naive ones? ./curse part 3 — one RNNE bidder against n−1 naive bidders bidding b = x:

`n`	P(the corrected bidder wins)	corrected bidder’s profit when winning	naive bidders’ average profit when winning
2	12.67%	+9.94	−2.90
3	4.19%	+7.51	−11.09
4	1.56%	+6.01	−15.43
5	0.62%	+5.00	−18.18
6	0.26%	+4.34	−20.08
7	0.11%	+3.73	−21.46
8	0.05%	+3.35	−22.51
10	0.01%	+2.84	−24.00

The corrected bidder almost never wins. At eight bidders they take the item once in two thousand auctions. When they do, they make money; the rest of the time they watch a naive rival overpay. This is the operationally important fact and it is the one that makes the correction hard to sustain inside an organisation: the correct strategy looks, from the outside, exactly like not competing. Kagel and Levin report the mechanism by which this gets overridden, from a geologist friend in Houston:

“there was such a broad range of legitimate value estimates for most tracts that when the bidding department started reducing bids relative to value estimates to the point that they were winning very few auctions, the geologists simply raised their estimates. (Geologists love to drill and failure to win tracts means they can’t drill.)”

That anecdote is the winner’s curse as an organisational failure rather than a statistical one, and it belongs alongside the statistics. A bid discipline that produces a 0.05% win rate will not survive contact with a sales target.

5. Second-Price and Ascending Auctions: Less Bad, Not Cured

5.1 Deriving the second-price equilibrium

Truthful bidding is not dominant here, so what is the equilibrium? The Milgrom–Weber condition for a symmetric second-price common-value auction is: bid the value of the item conditional on your signal being x and the highest rival signal also being exactly x — the “tie at the top” event, which is precisely the event in which your bid is pivotal.

For this uniform model, conditioning on own signal x, highest rival signal x, and the remaining n−2 signals below x, the posterior density of u = x0 − x is proportional to (ε − u)^{n−2} on [−ε, ε], and the same substitution as §2.3 gives E[u] = ε(2−n)/n. So

b(x)  =  x  −  ε (n−2)/n

Sanity: at n = 2 this is b(x) = x — with one rival, “I win and we tie” means both signals equal x, whose posterior mean is x, so bidding your signal is correct. At n = 10 it is x − 0.8 ε.

Uncertain — this second-price bid function is derived here, not taken from a source uncertain

Verify: b(x) = x − ε(n−2)/n as the symmetric equilibrium of the second-price auction in the Kagel–Levin uniform signal design, region 2. Reason: none of the sources fetched for this note states this particular closed form. Kagel and Levin give the first-price RNNE and cite Kagel, Levin and Harstad (1995) for the second-price comparative statics without reproducing the bid function; Klemperer’s appendix gives the ascending-auction equilibrium in a different (mangled-in-extraction) notation. The derivation above and the numerical equilibrium check in §5.3 are my own. To resolve: check Kagel, Levin and Harstad (1995) or Milgrom and Weber (1982) for the explicit uniform-signal second-price bid function.

5.2 Measured

./curse2 part A — second-price common-value auction, 20 seeds × 300,000 trials:

`n`	naive `b = x`, winner profit	equilibrium `b = x − ε(n−2)/n`, winner profit	shading applied
2	+9.997 ± 0.022	+9.997 ± 0.022	0.00
3	−0.001 ± 0.021	+9.999 ± 0.021	10.00
4	−6.000 ± 0.022	+9.000 ± 0.022	15.00
5	−9.999 ± 0.021	+8.001 ± 0.021	18.00
6	−12.859 ± 0.017	+7.141 ± 0.017	20.00
7	−14.997 ± 0.014	+6.431 ± 0.014	21.43
8	−16.665 ± 0.016	+5.835 ± 0.016	22.50
10	−19.088 ± 0.014	+4.912 ± 0.014	24.00

Compare with the first-price table in §4.3. The naive bidder’s loss at n = 10 is −19.1 here versus −24.5 there — the curse is genuinely milder in a second-price auction, but it is not eliminated. The reason is mechanical: the winner pays the second bid, which was submitted by someone with the second-most-optimistic signal, and the second-highest of n draws is less biased than the highest. That buys a real but partial reprieve, and only up to n = 2, where naive bidding happens to be correct.

Milgrom and Weber give the deeper reason, which applies to the ascending (English) auction rather than the sealed second-price one, and which is the single most-quoted practical result in this literature:

“when bidders are uncertain about their value estimates, the English and second-price auctions are not equivalent: the English auction generally leads to larger expected prices. One explanation of this inequality is that when bidders are uncertain about their valuations, they can acquire useful information by scrutinizing the bidding behavior of their competitors during the course of an English auction. That extra information weakens the winner’s curse and leads to more aggressive bidding in the English auction, which accounts for the higher expected price.”

They then rank the four standard formats by expected price — “The English auction generates the highest prices followed by the second-price auction and, finally, the Dutch and first-price auctions” — and connect it to practice: “This may explain the observation that ‘an estimated 75 per cent, or even more, of all auctions in the world are conducted on an ascending-bid basis’.”

This is a genuine break from the private-value world. Under independent private values all four formats raise the same expected revenue (Revenue Equivalence) and the ascending auction is strategically equivalent to the second-price one (English and Dutch Auctions). Introduce a common-value component and the equivalences collapse in a specific direction: information revealed during the auction is worth money to the seller, because a bidder who watches rivals drop out has less residual uncertainty and therefore needs less adverse-selection insurance. Klemperer calls the general statement the Linkage Principle: “expected revenue is raised by linking the winner’s payment to information that is affiliated with the winner’s information.”

sequenceDiagram
    participant B4 as bidder 4 (lowest signal)
    participant B3 as bidder 3
    participant B2 as bidder 2
    participant B1 as bidder 1 (highest signal)
    Note over B1,B4: price rises continuously from zero
    B4->>B1: quits at p4
    Note over B1,B3: everyone infers b4's signal from p4<br/>and revises their estimate of x0
    B3->>B1: quits at p3
    Note over B1,B2: two rivals' signals now inferred;<br/>residual uncertainty about x0 has fallen
    B2->>B1: quits at p2
    Note over B1: wins at p2, but with far less<br/>estimation risk than a sealed bid would carry

What this shows and what to take from it: the ascending auction is an information-revelation protocol that happens to also allocate an item. Every drop-out is a public message about a rival’s private signal, so by the time the last rival quits, the winner is conditioning on n − 1 signals rather than one. The insight: the winner’s curse is caused by conditioning on too little, and the English auction attacks it by supplying more to condition on. This is why Milgrom and Weber rank it first for revenue and why spectrum regulators specify it — the format is chosen for its side effect on bidders’ beliefs, not for its allocation rule.

5.3 Attacking the claimed equilibrium

The MOC’s rule for P7 — “a strategyproofness claim you have tried and failed to break is worth more than one you have proved on paper and mistyped” — applies here. ./dev puts n−1 bidders on b(x) = x − ε(n−2)/n and lets one deviate to b(x) + δ, 10 seeds × 4,000,000 trials, n = 5:

`δ`	deviator’s expected profit
−15	+0.8542 ± 0.0037
−9	+1.2417 ± 0.0036
−3	+1.5475 ± 0.0037
0	+1.5998 ± 0.0035
+3	+1.6001 ± 0.0040
+15	+1.5929 ± 0.0054
+30	+1.4252 ± 0.0059
+45	+0.7462 ± 0.0060
+60	+0.0001 ± 0.0060

No profitable deviation was found: δ = 0 is the maximum, downward deviations lose strictly and immediately, and large upward deviations collapse to zero profit at δ = 60 = 2ε (bid two noise-widths above equilibrium and you win everything at prices that exactly consume your surplus). The interesting feature is that the top of the curve is flat on the upward side — δ = 0, +3 and +15 are indistinguishable at this sample size. That flatness is not sampling noise but a property of the equilibrium: the bid is defined as the price at which you break even conditional on winning at exactly that price, so the marginal auctions you gain by bidding slightly higher earn approximately zero. It is a first-order-flat optimum, which is worth knowing because it means a small miscalibration upward costs almost nothing while a small miscalibration downward costs real money — the opposite asymmetry to the reserve-price case in Reserve Prices and Optimal Auctions §8.7.

6. The Jar of Coins

The classroom demonstration, and the first published experiment. Kagel and Levin: “Bazerman and Samuelson (1983) conducted the first experiment demonstrating a winner’s curse. Using M.B.A. students at Boston University, the experiment was conducted in class, with students participating in four first-price sealed-bid auctions. Bidders formed their own estimates of the value of each of four commodities — jars containing 800 pennies, 160 nickels, 200 large paper clips each worth four cents, and 400 small paper clips each worth two cents. Unknown to subjects, each jar had a value of $8.00. […] The number of bidders varied between 4 and 26.”

The reported outcome, which is the reason everyone teaches this:

quantity	value
true value of every jar	$8.00
average estimate across all four commodities	** $5.13 * * (i . e .$ 2.87 too low)
average winning bid	$10.01
average loss to the winner	−$2.01
share of auctions where the winning bid lost money	over half

Kagel and Levin draw the crucial inference: “As the authors note, this underestimation should reduce the likelihood and magnitude of the winner’s curse.” The population was systematically pessimistic by nearly three dollars and still overpaid by two. Whatever produced the $10.01 winning bid, it was not optimism — it was the maximum operator.

6.1 Reproducing the shape

./jar — n bidders each form an estimate N(5.13, σ) (the mean fixed at Bazerman and Samuelson’s measured value) and bid it; first-price:

`σ`	`n = 4`	`n = 10`	`n = 15`	`n = 26`	winner profit at `n = 15`	% of auctions the winner loses
2.0	7.19	8.21	8.60	9.09	−0.60	69.2%
2.5	7.70	8.98	9.47	10.08	−1.47	86.6%
3.0	8.22	9.74	10.34	11.07	−2.34	93.8%
4.0	9.25	11.28	12.07	13.05	−4.07	98.3%
5.0	10.28	12.82	13.81	15.04	−5.81	98.3%

A dispersion of about σ ≈ 2.8–3.0 in the estimates reproduces both the observed winning bid ($10.34 modelled vs $10.01 observed at a mid-range n = 15) and the observed loss (−$2.34 vs −$2.01). That is a plausible spread for coin-jar estimates — a 90% interval of roughly ± $4.70 o nan$ 8 jar.

Uncertain — this is a calibration, not a replication uncertain

Verify: nothing about Bazerman and Samuelson’s data is being tested here. The model assumes normal, independent, identically-distributed estimates and bidders who bid their estimate with no shading at all; the real experiment had a mixed n from 4 to 26, subjects who did shade somewhat with uncertainty (“a $1.00 increase in the 90% confidence interval reduced bids by 3¢” — a tiny effect), and, critically, winning bids that Kagel and Levin say “were sensitive to a handful of grossly inflated bids,” i.e. a fat right tail that a Gaussian cannot produce. All this simulation establishes is that a plausible dispersion parameter puts the model in the observed range, which is weak evidence that the selection effect alone can account for the result without invoking any additional bidder irrationality. It is not a fit and no parameter was estimated from their data. To resolve: read Bazerman & Samuelson (1983), Journal of Conflict Resolution 27:618–634, for the bid distributions.

Two features of the table are robust to the modelling choices and worth taking away. The loss grows with n, again. And the fraction of auctions in which the winner loses money exceeds 90% at realistic dispersions — the “over half” that Bazerman and Samuelson observed is, if anything, mild.

6.2 Running it yourself

The demonstration works in a room because the two errors compound. People are bad at estimating jar contents (large σ), and they are unaware of the selection effect (no correction). To run it well: use a jar, take sealed bids, announce n before bidding, and — this is the part usually omitted — collect each bidder’s point estimate separately from their bid. The teaching moment is not that the winner overpaid. It is that the average estimate was fine, sometimes even conservative, and the winner still overpaid, so no individual can be blamed for bad estimation. Then re-run with the same jar and the same people; Kagel and Levin’s laboratory finding is that a single round of feedback is nowhere near enough (§ Production Notes).

Failure Modes and Gotchas

1. Everything scales with `n`, and `n` is the number you least reliably know

The correction ε(n−1)/(n+1) requires n. In a sealed-bid auction you frequently do not know how many rivals showed up, and getting it wrong is asymmetric: guess n too low and you under-correct and lose money; guess too high and you merely lose the auction.

This is not hypothetical — it is the leading alternative explanation for the best-known field evidence. Kagel and Levin: “Hendricks, Porter, and Boudreau (1987) found that in early OCS lease sales, average profits were negative in auctions with seven or more bidders. […] one possible explanation for this outcome is the increased severity of the adverse selection problem associated with more bidders. However, they note that the data could also be explained by bidder uncertainty regarding the number of firms competing on a given tract (their preferred explanation). That is, since most tracts received less than six bids, it seems likely that firms would expect this number or less. As a result, although firms might have fully accounted for the adverse selection effect based on the expected number of firms bidding on a tract, they would nevertheless be incorrect for tracts that attracted above average numbers of bidders, and overbid on those tracts.”

Note that the authors of the empirical paper preferred the non-curse explanation. That is the correct default posture toward field evidence here, and §Production Notes returns to it.

2. The heuristic that was calibrated at one `n` silently rots

§4.3’s second column: “shade by half the noise” is exactly right at n = 3 and loses money at every larger n. A bidding rule tuned by trial and error in a thin market becomes a loss-maker as the market thickens, with no error signal — the losses look like bad luck for a long time, because per-auction variance dwarfs the bias.

3. Bidding more when rivals appear is the diagnostic symptom

Kagel and Levin: “Higher individual bids in response to increased numbers of rivals is often considered to be the hallmark characteristic of a winner’s curse.” Their measurement of it is direct: comparing 3–4-bidder against 6–7-bidder auctions with the same moderately experienced subjects, “actual profit decreased substantially more than profit opportunities as measured by the RNNE criteria. This implies that subjects were bidding more aggressively, rather than less aggressively, as the number of rivals increased, contrary to the RNNE prediction.”

If your bidding system responds to “more competitors” by raising bids in a common-value setting, it has the sign wrong. In a private-value setting that response is correct. The two look identical from inside a bidding tool.

4. Public information can reduce revenue when the curse is present

Theory says releasing unbiased public information raises the seller’s expected revenue (the Linkage Principle). Kagel and Levin measured the opposite in the large-group condition: in 3–4-bidder auctions, “public information resulted in statistically significant increases in revenue that averaged 38% of the RNNE model’s prediction. However, in auctions with larger numbers of bidders (6–7), public information reduced average sellers’ revenue by $1.79 p er a u c t i o n, co m p a r e d t o t h e R N N E m o d e l^{'} s p r e d i c t i o n o f anin cr e a seo f$ 1.78.”

The mechanism is instructive: “If bidders suffer from a winner’s curse, the high bidder consistently overestimates the item’s value, so that announcing x_L is likely to result in a downward revision of the most optimistic bidders’ estimate.” Public information cures the curse, and curing the curse costs the seller the money the curse was transferring. A seller who benefits from an information asymmetry is a seller who is being paid by bidders’ mistakes, and improving disclosure will show up as a revenue regression.

5. Learning does not transfer between contexts

The most sobering laboratory finding. Kagel and Levin’s subjects who had adjusted well in 3–4-bidder auctions relapsed in 6–7-bidder auctions: “This result also suggests that the underlying learning processes is context specific rather than involving some sort of ‘theory absorption’ that readily generalizes to new environments.” Bidders learn the level of shading that stopped hurting, not the principle. Move them and they are naive again.

They extend this pessimism to the field: “there are dynamics of interactions within organizations that may retard adjustment to the winner’s curse. These include, (i) payments of large salaries to petroleum geologists to estimate likely reserves, and then having to recognize that these estimates still have a very large variance and are not very precise, (ii) transfers of personnel within the firm and between firms prior to receiving feedback about the profitability of bids, and (iii) gaming that goes on within organizations.”

Point (ii) is the one that generalises furthest. In any domain where the feedback loop is longer than the tenure of the person making the decision — oil leases, spectrum, acquisitions, multi-year contracts, hiring — nobody in the loop ever personally experiences the consequence of the bias.

flowchart TB
    A["bid team applies the correct<br/>adverse-selection correction"]
    B["win rate collapses<br/>(0.05% at n = 8, measured §4.4)"]
    C["'we never win anything'<br/>pressure from management"]
    D["geologists raise their estimates<br/>/ analysts raise their multiples"]
    E["bids rise back above<br/>E(value given we won)"]
    F["losses appear years later,<br/>after the people involved have moved on"]
    A --> B --> C --> D --> E --> F
    F -. "no feedback reaches the decision" .-> D

What this shows and what to take from it: the winner’s curse is not only a statistics problem; it is a control-loop problem. The correct strategy produces an observable that looks like failure (a near-zero win rate) on a timescale of weeks, and a consequence that looks like success (profitable wins) on a timescale of years — and Kagel and Levin note that personnel move “prior to receiving feedback about the profitability of bids.” The loop that should correct the bias is broken by latency, so the visible short-term signal wins. Any organisation bidding on long-payback assets should assume this loop exists until it can show otherwise.

6. Second-price truthfulness does not survive here — do not import it

The Vickrey Auction’s dominant-strategy result is a private-value theorem. In a common-value auction, bidding your estimate in a second-price auction loses money from n = 3 onward (§5.2, measured at −19.1 by n = 10). Any system that reasons “it’s a second-price auction so truthful bidding is optimal” without checking whether values are private has imported a theorem outside its hypotheses. Advertising auctions, where the value of a click is partly a shared estimate of user quality, are exactly the ambiguous case.

7. Limited liability was proposed as an artefact explanation, and refuted

Worth knowing because it is the sharpest attack the laboratory result has faced. Hansen and Lott (1991) argued that Kagel and Levin’s subjects overbid rationally because they could not lose more than their starting cash balance — a free option. Two independent refutations followed, both reported by Kagel and Levin:

Recomputation: “for almost all bidders cash balances were always large enough so that it never paid to deviate from the Nash equilibrium bidding strategy in a one-shot auction,” and a sign check — “limited-liability arguments imply more aggressive bidding in auctions with fewer rather than larger numbers of bidders, just the opposite of what the data shows.”
New experiments where the mechanism cannot operate. Lind and Plott (1991) ran simultaneous guaranteed-profit private-value auctions to keep cash flowing in, and had subjects agree to work off negative balances “doing work-study type duties (photocopying, running departmental errands, etc.) at the prevailing market wage rate.” Cox, Dinkin and Smith (1998) reinitialised balances each period so bankruptcy was impossible; they “find no significant differences in individual bid patterns in the unlimited liability treatment, contrary to HL’s argument.”

Kagel and Levin also note that Holt and Sherman (1994) rule out the other obvious alternative, “that the winner’s curse is a result of an added thrill, or extra utility, from winning.”

8. But bidding is still closer to Nash than to full naivety, which needs explaining

The honest counterweight, in the authors’ own words: “We are still left with the puzzle, first expressed by Lind and Plott, that although many experiments report a clear winner’s curse (negative profits), comparing between the symmetric RNNE and totally naive bidding models offered in the literature, bidding is closer to the RNNE. One promising explanation for this phenomenon appears to be that bidders are cursed to different degrees. That is, agents may make partial, but incomplete, adjustments for the adverse selection effect […] with the perfectly rational and perfectly naive bidding models being polar cases.” They cite Eyster and Rabin (2000) for a formal model of partial cursedness.

So “bidders are naive” is too strong; “bidders under-correct” is the supported claim. §4.3’s naive column is the worst case, not the typical one.

Alternatives and When to Choose Them

Two different questions hide here — what should a bidder do, and what should a seller do — and they have opposite answers.

For the bidder:

Strategy	Requires	Expected profit	When
Bid your estimate	nothing	`−ε(n−1)/(n+1)` — negative at every `n ≥ 2`	never in a common-value auction
Fixed percentage haircut	a tuning session	right at one `n`, wrong at all others (§4.3)	only if `n` is genuinely fixed and you re-tune when it changes
Break-even bid `E[v \| I won]`	`n`, `ε`	exactly zero	as a floor, not a strategy — never bid above this
RNNE / full equilibrium	`n`, `ε`, the format, symmetry	`2ε/(n+1)` in first-price	when you can model the environment; the theoretical benchmark
Bid a lower bound (`x − ε`)	`ε` only	risk-free but rarely wins	Kagel and Levin note `max[x − ε, x_lo]` “is a risk-free strategy that fully protects a bidder from negative earnings” — the conservative default when `n` is unknown
Don’t participate	—	zero	when you cannot estimate `ε` or `n`, and rivals are better informed. Losing an auction costs nothing; winning one can cost everything
Buy better information (reduce `ε`)	money	scales the whole problem down	the only lever that improves both terms of the bid factor at once

The last row deserves emphasis because it is the one an engineering organisation can actually act on. Every quantity in this note is proportional to ε. Halving your estimation noise halves the curse, halves the required correction, and halves the equilibrium profit-per-auction while doubling the number of auctions you can safely enter. Compare with the private-value world, where better information about your own value is worth nothing because you already know it.

For the seller:

Format	Effect on the curse	Effect on revenue
First-price sealed bid	worst — no information revealed	lowest of the four, under affiliation
Second-price sealed bid	milder (winner pays the second signal’s bid)	middle
English / ascending	mildest — drop-out prices reveal rivals’ signals continuously	highest (Milgrom–Weber)
Release public information	reduces the curse	raises revenue in equilibrium; reduces it when the curse is strong (§Failure Modes 4)

The seller’s dilemma is genuine and not merely theoretical: a naive bidding population is a transfer to the seller, and every step that protects bidders — ascending format, published reserve data, disclosed geology — reduces that transfer in the short run. The counter-argument is participation: bidders who lose money leave, and by Bulow–Klemperer a lost bidder costs the seller more than almost anything else. Kagel and Levin report that in the inexperienced-bidder auctions, “40% of all subjects starting these auctions went bankrupt” — a market that eats 40% of its participants does not have many rounds left.

See English and Dutch Auctions for the format mechanics, Revenue Equivalence for the private-value benchmark these rankings violate, and Ad Auctions and GSP for a live market with a partial common-value component.

Production Notes

The field evidence, and why to hold it loosely

The winner’s curse entered economics from industry, not academia. Kagel and Levin: “The winner’s curse story begins with Capen, Clapp, and Campbell (1971), three petroleum engineers who claimed that oil companies had suffered unexpectedly low rates of return in the 1960’s and 1970’s on OCS [outer continental shelf] lease sales ‘year after year.’” Klemperer describes the same paper as “a well-known, more popular account of the winner’s curse in practice that was immensely important in influencing bidding practice.” Milgrom and Weber, writing in Econometrica eleven years later, still cite it as the motivating fact.

Claims have since been made in many markets. The list, with the sourcing Kagel and Levin give:

Market	Claim source
Offshore oil leases (OCS)	Capen, Clapp & Campbell 1971; Lorenz & Dougherty 1983
Book publication rights	Dessauer 1981
Professional baseball free agency	Cassing & Douglas 1980; Blecherman & Camerer 1998
Corporate takeover battles	Roll 1986
Real estate auctions	Ashenfelter & Genesove 1992

Uncertain — every field claim in this table is second-hand and none was independently verified uncertain

Verify: the specific claims of Capen/Clapp/Campbell (1971), Cassing & Douglas (1980), Blecherman & Camerer (1998), Roll (1986), Dessauer (1981), Ashenfelter & Genesove (1992), and Hendricks/Porter/Boudreau (1987). Reason: none of these papers was fetched. Every statement about them in this note comes from Kagel and Levin’s 2001 survey, which was read in full. Kagel and Levin themselves flag the whole category: “It is exceedingly difficult to support claims of a winner’s curse using field data because of reliability problems with the data and because alternative explanations for overbidding are often available.” They give the Hendricks et al. case in detail (see Failure Modes 1), where the empirical authors preferred a non-curse explanation for their own finding. To resolve: read the primary field papers, but expect the ambiguity to survive.

There is also a cartel-theory alternative that undercuts the origin story itself, and Kagel and Levin raise it against their own subject: “An alternative and simpler explanation as to why oil companies might claim that they fell prey to a winner’s curse lies in cartel theory, as responsiveness to the winner’s curse claim could serve as a coordination device to get rivals to reduce their bids in future sales.” Three petroleum engineers publishing “we are all bidding too much” in the Journal of Petroleum Technology is, read cynically, a public request for everyone to bid less. That does not make them wrong; it means the paper is not evidence.

The honest summary of the empirical position: the winner’s curse is very well established in the laboratory, across many designs, with the obvious artefact explanations tested and refuted. In the field it is well-motivated, widely believed, and hard to demonstrate, because the counterfactual (what would profits have been under correct bidding?) is unobservable and because uncertainty about n produces the same signature.

The laboratory record, which is strong

Kagel and Levin’s own numbers, from auctions with inexperienced bidders who had already been through 2–3 practice rounds with full feedback:

Measure	Value
Average profit over the first nine auctions	*− $2.57 * (R N N E p r e d i c t i o n : * * +$ 1.90**)
Auctions with positive profit	17%
Individual bids above `E[x0 \| X = x_1:n]`	59%
High bids above `E[x0 \| X = x_1:n]`	82%
Subjects who went bankrupt	40%

Their verdict: “the winner’s curse is a genuinely pervasive problem for inexperienced bidders. It is remarkably robust being reported under a variety of treatment conditions […] and for different subject populations, including professional bidders from the commercial construction industry (Dyer, Kagel and Levin, 1989).”

With moderately experienced bidders it improves but does not vanish, and the improvement is conditional on n: “For small groups (auctions with 3–4 bidders), the general pattern was one of positive profits averaging $4.32 * * p er a u c t i o n [\dots] b u t s t i l l w e l l b e l o w t h e R N N E p r e d i c t i o n o f * *$ 7.48. In contrast, for these same bidders, bidding in larger groups (auctions with 6–7 bidders) profits averaged − $0.54 * * p er a u c t i o n, co m p a r e d t o t h e R N N E p r e d i c t i o n o f * *$ 4.82.”

Read those two rows together: experienced bidders capture 58% of available profit at n = 3–4 and are underwater at n = 6–7. Experience is worth something and it is not worth enough.

Spectrum auctions: the curse as a design constraint

The most consequential production use of this theory is the design of radio spectrum auctions. Kagel and Levin: “Experimental studies of auction markets have played a significant role in the design and execution of the recent wave of spectrum (air wave rights) auctions carried out in this country and abroad. […] a central design element has been to use ascending-price auctions (with price feedback for bidders) to both minimize the presence of the winner’s curse and to generate increased revenue in the absence of a winner’s curse.”

That is Milgrom and Weber’s ranking (§5.2) turned into a procurement specification. They quote The Economist (15 April 2000, p. 36) on Britain’s 3G sale: “An ascending auction ought to remove another common problem with auctions, the ‘winner’s curse.’ […] the winner’s curse is much likelier in sealed-bid auctions, where bidders lack an important piece of information about the value of the asset: the valuations of other, perhaps better-informed, bidders. In an ascending auction, however, that information is clearly revealed.” And McAfee and McMillan (1996, p. 161): “by allowing bidders to respond to each other’s bids, [an ascending-price auction] diminishes the winner’s curse.”

The scale involved, per Kagel and Levin’s footnote: the US FCC “conducted a number of sales to date raising a total of $23.9 bi l l i o n * * an d se l l in g o v er * * 10, 000 l i ce n ses * * b e tw ee n J u l y 1994 an dJ u l y 2000, " an d " inana u c t i o n e n d in g in A p r i l 2000, t h e B r i t i s h g o v er nm e n t r ai se d * * 22.5 bi l l i o n p o u n d s ($ 35.53 billion) from the sale of ‘third generation’ mobile phone licenses.”

Uncertain — the 3G auctions are cited for and against the winner's curse uncertain

Verify: whether the European 3G auctions of 2000 are an instance of the winner’s curse. The Kagel–Levin passage and The Economist quote above argue the ascending format prevented it. But the same auctions are routinely cited as the canonical modern winner’s curse, on the grounds that several winners subsequently wrote down the licences and some operators were financially crippled. Both readings appear in the literature and this note has read only the first. Reason: Klemperer’s post-mortem work on the European 3G auctions could not be retrieved — paulklemperer.org serves a shell with no paper index, and the historical Oxford paths (nuffield.ox.ac.uk/users/klemperer/*.pdf) return 404 for everything except survey.pdf. To resolve: read Klemperer’s “How (Not) to Run Auctions: The European 3G Telecom Auctions” and “What Really Matters in Auction Design”. Note that ex-post write-downs are not by themselves evidence of a winner’s curse: a licence can be correctly valued at auction and then lose value because the world changed, which is a forecasting failure and not a selection effect.

The oldest recorded case, probably

Klemperer’s survey, in a footnote on the history of auctions: “Most famously, the whole Roman empire was sold by ascending auction in A.D. 193 by the Praetorian Guards; the winner, and therefore next Emperor, was Didius Julianus who reigned for just over two months before being overthrown and executed by Septimius Severus (an early and sad case of the winner’s curse).” Whether an emperorship is a common-value good is left as an exercise; the outcome is not in dispute.

Where this shows up in systems work

Procurement and low-bid contracting. Kagel and Levin report the same results “in low bid wins, supply auctions with both student subjects and professional bidders drawn from the commercial construction industry.” A vendor who wins your RFP by a wide margin has probably underestimated the work. The systems reading: a suspiciously cheap bid is a signal about the bidder’s estimate, not a bargain — treat abnormally low bids as a risk flag, which is exactly what public procurement rules that require justification of “abnormally low tenders” are doing.
Hiring and free agency. The candidate you are most excited about is the one your process’s noise favoured most. Same estimator, same selection, same correction.
Capacity and cost estimation under bidding. Any internal market where teams bid for shared capacity (Multi-Tenancy and Fairness in LLM Serving, Rate Limiting as a Mechanism) inherits the structure the moment the resource’s true value is shared and uncertain.
Adversarial selection generally. Kagel and Levin list the curse’s extensions: “bilateral bargaining games […] blind bid auctions […] markets where quality is endogenously determined […] and voting behavior (the swing voter’s curse).” Their reference list opens with Akerlof’s Market for Lemons for a reason.

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The Winner's Curse