Revenue Equivalence

Revenue equivalence is the result that under a specific and quite restrictive list of assumptions — symmetric bidders, independent private values, risk neutrality, no budget constraints, an atomless and strictly increasing value distribution — every auction mechanism that (i) always awards the object to the bidder with the highest value and (ii) gives the lowest possible type zero expected surplus yields the same expected revenue to the seller (Vickrey 1961; Myerson 1981, Corollary to Lemma 3; Riley & Samuelson 1981, Proposition 1). First-price, second-price, English, Dutch, all-pay — same money. The theorem is genuinely surprising and genuinely useful, but its real value in practice is inverted: it is a diagnostic, not a design principle. Because the conclusion is so strong, any observed revenue difference between two formats is a proof that at least one assumption fails, and the interesting engineering question is always which one. This note states the assumptions as a checklist, then breaks each one and measures what happens, in C, against exactly-computed benchmarks.

Mental Model: five load-bearing assumptions, five directions to fall

flowchart TB
    RET["<b>Revenue Equivalence</b><br/>all standard formats yield<br/>the same expected revenue"]
    A1["A1 · Risk neutrality"]
    A2["A2 · Independent signals"]
    A3["A3 · Symmetric bidders<br/>(one common distribution)"]
    A4["A4 · No budget constraints"]
    A5["A5 · F atomless and<br/>strictly increasing on its support"]
    A1 --> RET
    A2 --> RET
    A3 --> RET
    A4 --> RET
    A5 --> RET
    A1 -. "break it" .-> B1["<b>First-price wins</b><br/>risk-averse bidders shade less<br/>measured: +11.1% at rho=1/2"]
    A2 -. "break it" .-> B2["<b>English wins</b> (linkage principle)<br/>English &ge; second &ge; first<br/>measured: +1.3% to +6.8%"]
    A3 -. "break it" .-> B3["<b>Either can win</b><br/>shifts/stretches favour first-price<br/>measured: +100% and +14%<br/>low-end mass favours the open auction"]
    A4 -. "break it" .-> B4["<b>First-price wins</b><br/>and efficiency collapses<br/>measured: +27.6% revenue,<br/>efficiency 0.94 &rarr; 0.28"]
    A5 -. "break it" .-> B5["<b>Equivalence simply fails</b><br/>two-type example: revenue is<br/>strictly increasing in the price offered"]
    style RET fill:#1f3a6f,color:#fff

What it shows: the theorem’s five hypotheses and the direction each failure pushes revenue. The insight is that four of the five failures point the same way — toward the sealed first-price/Dutch format — and only affiliation points toward the open ascending format. That asymmetry is why practitioners’ instincts (“English auctions raise the most”) and theorists’ results so often collide: the affiliation effect is the one every textbook emphasises, and it is measured below to be the smallest of the four.

Statement, and What Each Word Is Doing

The cleanest textbook statement is Levin’s:

Theorem (Revenue Equivalence). Suppose $n$ bidders have values $s_{1}, \dots, s_{n}$ identically and independently distributed with cdf $F (\cdot)$ . Then all auction mechanisms that (i) always award the object to the bidder with highest value in equilibrium, and (ii) give a bidder with valuation $\underline{s}$ zero profits, generate the same revenue in expectation. (Levin, Auction Theory, §1.4)

Klemperer’s version adds the fine print that Levin’s suppresses, and the fine print matters:

Revenue Equivalence Theorem (Private-Value Case). Assume each of $n$ risk-neutral potential buyers has a privately-known value independently drawn from a common distribution $F (v)$ that is strictly increasing and atomless on $[\underline{v}, \overline{v}]$ . Suppose that no buyer wants more than one of the $k$ available identical indivisible objects. Then any auction mechanism in which (i) the objects always go to the $k$ buyers with the highest values, and (ii) any bidder with value $\underline{v}$ expects zero surplus, yields the same expected revenue, and results in a buyer with value $v$ making the same expected payment. (Klemperer, Appendix A)

Walking the symbols:

$n$ risk-neutral potential buyers. Risk neutrality means a bidder maximises expected money, so utility is linear: $U = v - p$ if she wins, $0$ otherwise. This is assumption A1.
privately-known value. Each bidder knows her own $v_{i}$ exactly, and $v_{i}$ does not depend on anyone else’s information. This rules out common values and the winner’s curse (see The Winner’s Curse).
independently drawn. The $v_{i}$ are stochastically independent. Myerson is explicit that this is a temporary assumption: “Until §7, we will assume that the value estimates of the $n$ bidders are stochastically independent random variables.” This is A2.
from a common distribution. Common — the same $F$ for every bidder. This is A3, symmetry.
strictly increasing and atomless. $F$ has no flat stretches (no gaps in the support) and no jumps (no value that occurs with positive probability). This is A5, and it is the assumption nobody quotes. Klemperer flags it in a footnote: “To see the necessity of a strictly increasing or atomless distribution, see note 117.”
no buyer wants more than one object. Unit demand. Multi-unit demand is a different subject (see Ad Auctions and GSP).
the objects always go to the buyers with the highest values. The allocation rule is fixed. This is the real content: revenue equivalence says the allocation rule determines the payments up to a constant.
any bidder with value $\underline{v}$ expects zero surplus. The boundary condition. Fixing the payment of one type pins down the payments of all types.
Nowhere in the statement does A4 appear, because budget constraints are outside the model entirely — a bidder is implicitly assumed able to pay any amount she bids.

Why it is true: the envelope argument in six lines

The modern proof is a single application of the envelope theorem, and it is worth carrying because it shows exactly which assumption enters where. Following Levin §1.4:

An auction rule is a pair of functions of the bid profile: $x_{i} (b)$ , the probability $i$ gets the object, and $t_{i} (b)$ , $i$ ‘s payment. Bidder $i$ ‘s expected payoff from signal $s_{i}$ and bid $b_{i}$ is

$U_{i} (s_{i}, b_{i}) = s_{i} E_{b_{- i}} [x_{i} (b_{i}, b_{- i})] - E_{b_{- i}} [t_{i} (b_{i}, b_{- i})] .$

Read that line carefully: $s_{i}$ multiplies the win probability linearly. That linearity is risk neutrality (A1), and it is what makes the derivative below so simple. Now suppose $b_{i} (\cdot)$ is an equilibrium. Because $i$ is best-responding, the envelope theorem gives

$\frac{d}{d s} U_{i} (s)_{s = s_{i}} = E_{b_{- i}} [x_{i} (b_{i} (s_{i}), b_{- i})] = F^{n - 1} (s_{i}),$

where the last equality uses (i), the assumption that the highest value wins, and independence (A2) plus symmetry (A3) to write the probability that all $n - 1$ rivals are below $s_{i}$ as $F^{n - 1} (s_{i})$ . Integrating, and using (ii) to set $U_{i} (\underline{s}) = 0$ :

$U_{i} (s_{i}) = \int_{\underline{s}}^{s_{i}} F^{n - 1} (\tilde{s}) d \tilde{s} .$

Combining with the definition of $U_{i}$ , bidder $i$ ‘s expected payment is

$E [t_{i}] = s_{i} F^{n - 1} (s_{i}) - \int_{\underline{s}}^{s_{i}} F^{n - 1} (\tilde{s}) d \tilde{s} = \int_{\underline{s}}^{s_{i}} \tilde{s} d F^{n - 1} (\tilde{s}),$

by integration by parts. As Levin puts it: “Since $x_{i} (\cdot)$ does not enter into this expression, bidder $i$ ‘s expected equilibrium payment given his signal is the same under all auction rules that satisfy (i) and (ii).” Sum over the $n$ bidders and you get expected revenue $= E [S^{2 : n}]$ , the expectation of the second-highest of $n$ draws — a constant.

flowchart TB
    S1["U_i(s,b) = s * Pr[win] - E[payment]<br/><i>payoff is LINEAR in s</i>"]
    S2["dU_i/ds = Pr[win] = F^(n-1)(s)<br/><i>envelope theorem at the equilibrium bid</i>"]
    S3["U_i(s) = integral of F^(n-1) from the<br/>lowest type up to s<br/><i>using the boundary condition</i>"]
    S4["E[payment | s] = integral of s' dF^(n-1)(s')<br/><b>no dependence on the payment rule</b>"]
    S5["Revenue = n * E[payment]<br/>= E[2nd highest value]"]
    S1 --> S2 --> S3 --> S4 --> S5
    A1(["A1 risk neutrality<br/>makes the payoff linear"]) -.-> S1
    A2(["A2 independence + A3 symmetry<br/>let Pr[win] factor into F^(n-1)"]) -.-> S2
    C1(["condition i:<br/>highest value wins"]) -.-> S2
    C2(["condition ii:<br/>lowest type gets zero"]) -.-> S3
    A5(["A5 atomless F makes b() strictly<br/>increasing, so the integral exists"]) -.-> S4
    style S4 fill:#1f6f43,color:#fff
    style S5 fill:#1f3a6f,color:#fff

What it shows: the five-step envelope proof with each hypothesis attached to the step it supports. The insight is that the green box is the whole theorem — once the expected payment of a type is written as an integral that mentions only the allocation rule, the payment rule has been eliminated. Every failure mode later in this note is an arrow into this diagram being cut: risk aversion breaks step 1, affiliation breaks step 2, asymmetry breaks step 2 and condition (i), budgets break condition (i), atoms break step 4.

Myerson reaches the same place from the mechanism-design side and states it as a corollary rather than a theorem: “The seller’s expected utility from a feasible auction mechanism is completely determined by the probability function $p$ and the numbers $q_{i} (p, x, a_{i})$ for all $i$ … Thus, for example, the seller must get the same expected utility from any two auction mechanisms which have the properties that (1) the object always goes to the bidder with the highest value estimate above $t_{0}$ and (2) every bidder would expect zero utility if his value estimate were at its lowest possible level.” He then notes that “Vickrey’s equivalence results may be viewed as a corollary of our equation (4.12)” (Myerson 1981, §4).

Riley and Samuelson’s independent route is the most intuitive of the three. Their Proposition 1 says expected seller revenue depends on the auction rules only through $v^{*}$ , the reservation value below which it is not worth bidding: “the expected revenue of the seller from quite different auctions can be compared simply by determining the lowest reservation value, $v^{*}$ , for which it is worthwhile bidding” (Riley & Samuelson 1981, §I). Every standard auction without a reserve price has $v^{*} = \underline{v}$ ; hence they all raise the same money; hence Reserve Prices and Optimal Auctions is the lever, because it is the only one that moves $v^{*}$ .

The computer-science statement, in Nisan et al., Algorithmic Game Theory, Theorem 9.46, generalises past auctions entirely: “for every two Bayesian–Nash implementations of the same social choice function $f$ … if for some type $t_{i}^{0}$ of player $i$ the expected payment of player $i$ is the same in the two mechanisms, then it is the same for every value of $t_{i}$ .” Their gloss is the one to remember: “the revenue equivalence theorem tells us that we cannot increase revenue without changing appropriately the allocation rule (social choice function) itself.”

The Assumption Checklist

#	Assumption	Where it enters the proof	Breaking it favours	Measured below
A1	Bidders are risk neutral	The linear $s_{i} \cdot Pr [win]$ term	First-price / Dutch	+11.1% at CRRA $ρ = 1/2$ , $n = 5$
A2	Signals are independent (and values private)	Writing $Pr [win] = F^{n - 1} (s_{i})$ as a product	English (linkage principle)	+1.3% to +6.8%
A3	Bidders are symmetric — one common $F$	The same $F$ for all $i$ ; “highest value wins” in equilibrium	Either, depending on the asymmetry’s shape	+100% (shift), +14% (stretch), −50% (low-end mass)
A4	No budget constraints	Implicit: a bidder can always pay her bid	First-price	+27.6% revenue, at a large efficiency cost
A5	$F$ atomless and strictly increasing	Needed for a strictly increasing equilibrium bid function	Neither — equivalence just fails	Two-type counterexample
—	Unit demand, single object	Throughout	—	Out of scope; see Ad Auctions and GSP
—	Highest value wins in equilibrium	Condition (i)	—	Violated by A3 and A4 breaks
—	Lowest type gets zero surplus	Condition (ii)	—	Violated by subsidies to losers

Baseline: confirming equivalence before breaking it

Everything that follows is measured against a baseline whose exact answer is known. With $n$ bidders and values i.i.d. uniform on $[0, 1]$ :

$E [k -th highest of n] = \frac{n + 1 - k}{n + 1}$ , so second-price revenue $= \frac{n - 1}{n + 1}$ ;
the first-price equilibrium is $b (v) = \frac{n - 1}{n} v$ , so first-price revenue $= \frac{n - 1}{n} \cdot \frac{n}{n + 1} = \frac{n - 1}{n + 1}$ ;
the all-pay equilibrium is $b^{A} (v) = \frac{n - 1}{n} v^{n}$ , and $n \cdot E [b^{A}] = (n - 1) \cdot \frac{1}{n + 1} = \frac{n - 1}{n + 1}$ .

All three agree. These closed forms were cross-checked with Python’s fractions.Fraction so that no floating-point rounding sits between the theory and the test:

from fractions import Fraction as F
def order_stat(n, k):                 # E[k-th highest of n iid U(0,1)]
    return F(n+1-k, n+1)
for n in range(2, 9):
    sp = order_stat(n, 2)             # second-price revenue
    fp = F(n-1, n) * order_stat(n, 1) # first-price: coefficient x E[max]
    ap = F(n-1) * F(1, n+1)           # all-pay: n * ((n-1)/n) * E[v^n]
    assert sp == fp == ap == F(n-1, n+1)

For $n = 5$ every format’s exact expected revenue is $4/6 = 2/3$ . The Monte Carlo harness (200,000 trials × 10 independent seeds, English and Dutch run as actual clocks rather than order statistics — see English and Dutch Auctions) returns:

format	mean	sd across seeds	95% CI	bias vs 2/3	sd per trial
English	0.666600	0.000480	[0.666302, 0.666898]	−0.000066	0.178148
Dutch	0.666479	0.000243	[0.666329, 0.666630]	−0.000187	0.112755
First-price	0.666529	0.000243	[0.666379, 0.666680]	−0.000137	0.112755
Second-price	0.666550	0.000480	[0.666253, 0.666848]	−0.000116	0.178148
All-pay	0.666230	0.001138	[0.665525, 0.666935]	−0.000437	0.449547

Every interval covers $2/3$ . This is what a confirmed theorem looks like, and it is also the first place practitioners over-read it. The last column is not the same across formats and never was: the per-trial spread of the realised price differs by a factor of 1.58 between the ascending and descending formats, and by a factor of 4 between all-pay and first-price. Revenue equivalence is a statement about the first moment only. Vickrey said so explicitly, computing the ratio in his Appendix I as $2 N / (N - 1)$ ; at $N = 5$ that is $2.5$ , and the measured variance ratio is $2.496$ .

A single run is not a result

Each row above is 10 independent seeds of 200,000 trials. Reporting only the pooled mean would hide the fact that the across-seed standard deviation for the all-pay auction (0.001138) is 4.7× that of the first-price auction (0.000243) — the same fact as the per-trial spread, and the reason an all-pay experiment needs roughly 20× the sample size to reach the same precision. If you report one number from one seed, you cannot distinguish a real effect from seed noise, and several of the effects measured below are only a few percent.

Break A1 — Risk-Averse Bidders: First-Price Wins

Klemperer’s derivation is a two-sentence asymmetry:

“In a second-price (or an ascending) auction, risk-aversion has no effect on a bidder’s optimal strategy which remains to bid (or bid up to) her actual value. But in a first-price auction, a slight increase in a player’s bid slightly increases her probability of winning at the cost of slightly reducing the value of winning, so would be desirable for a risk-averse bidder if the current bidding level were optimal for a risk-neutral bidder. So risk-aversion makes bidders bid more aggressively in first-price auctions.” (Klemperer §5)

The primary source is Riley and Samuelson’s Proposition 4, read directly:

“PROPOSITION 4: Suppose assumption IID holds and all buyers share a common utility function displaying risk aversion. Then (i) In the second bid auction, bidders continue to bid their reservation values, that is, $b_{i} = v_{i}$ . (ii) In the high bid auction, as bidders become more risk averse, they make uniformly higher bids. (iii) Consequently, the seller enjoys a greater expected profit under the high bid auction than under the second bid auction.” (Riley & Samuelson 1981, §III)

Their intuition: “with risk aversion the marginal increment in wealth associated with a successful, slightly lower bid is weighted less heavily than the possible loss $(v_{i} - b_{i})$ if, as a result of lowering the bid, the buyer is no longer the high bidder. This leads risk-averse bidders always to shade their bids less than risk-neutral bidders.”

The exact equilibrium, derived

For constant relative risk aversion $u (x) = x^{ρ}$ with $ρ \in (0, 1]$ and values uniform on $[0, 1]$ , the symmetric first-price equilibrium has a clean closed form. A bidder with value $v$ choosing which type $x$ to imitate solves $max_{x} (v - b (x))^{ρ} x^{n - 1}$ . The first-order condition is

$- ρ (v - b (x))^{ρ - 1} b^{'} (x) x^{n - 1} + (n - 1) x^{n - 2} (v - b (x))^{ρ} = 0,$

and imposing $x = v$ at equilibrium gives $ρ b^{'} (v) v = (n - 1) (v - b (v))$ . Trying $b (v) = c v$ yields $ρ c = (n - 1) (1 - c)$ , so

$b (v) = \frac{n - 1}{n - 1 + ρ} v$

At $ρ = 1$ this is the familiar $\frac{n - 1}{n} v$ ; as $ρ ↓ 0$ the coefficient rises to 1 and shading disappears. Expected revenue is $\frac{n - 1}{n - 1 + ρ} \cdot \frac{n}{n + 1}$ , while second-price revenue stays pinned at $\frac{n - 1}{n + 1}$ .

Measured

With $n = 5$ , 400,000 trials × 10 seeds:

$ρ$	bid coefficient	FPA mean	SPA mean	FPA − SPA	sd(FPA, seeds)	exact FPA
1.00 (risk neutral)	0.800000	0.666564	0.666461	+0.000102	0.000121	0.666667
0.80	0.833333	0.694337	0.666461	+0.027876	0.000126	0.694444
0.667	0.857143	0.714175	0.666461	+0.047714	0.000129	0.714286
0.50	0.888889	0.740626	0.666461	+0.074165	0.000134	0.740741
0.333	0.923077	0.769112	0.666461	+0.102650	0.000139	0.769231
0.25	0.941176	0.784192	0.666461	+0.117731	0.000142	0.784314

At $ρ = 1/2$ — a moderate degree of risk aversion — the first-price auction raises 11.1% more ( $20/27$ versus $2/3$ exactly; simulated $0.740626$ versus $0.666461$ ). The gap is 550 across-seed standard deviations wide, so it is not noise. At $ρ = 1$ the difference collapses to $+ 0.0001$ , one standard deviation: the theorem holds where it is supposed to.

Note what does not move: the second-price column is constant to six decimal places across every row. Bidding one’s value is dominant for any monotone utility function, so risk attitude cannot touch it. That invariance is the whole mechanism of the break.

The mirror image: a risk-averse seller prefers the same format

Klemperer works the other side and gets a chain of three:

“Conditional on the winner’s actual information, the price is fixed in the first-price auction, and random but with the same mean in the second-price auction. So also unconditional on the winner’s information, the price is riskier (but with the same mean) in the second-price auction. So a risk-averse seller prefers the first-price auction to the second-price sealed-bid auction and, for a similar reason, prefers the second-price sealed-bid auction to an ascending open auction.”

The measured per-trial standard deviations in the baseline table are exactly this ordering: 0.1128 (first-price/Dutch) < 0.1781 (second-price/English). A risk-averse seller facing risk-neutral bidders and a risk-neutral seller facing risk-averse bidders both prefer the sealed first-price auction — for entirely different reasons, in the same direction.

Break A2 — Affiliated Values: The English Auction Wins (a Little)

If signals are affiliated — informally, a high signal for one bidder makes high signals for others more likely; formally $f (s \land s^{'}) f (s \lor s^{'}) \geq f (s) f (s^{'})$ (Levin §2.1) — the product $F^{n - 1} (s_{i})$ in the envelope argument is no longer the right win probability, and the theorem’s conclusion fails. Milgrom & Weber (1982) replace it with a strict ranking:

Theorem 11: expected price in the English auction $\geq$ that in the second-price auction.
Theorem 15: expected price in the second-price auction $\geq$ that in the first-price auction.

Levin’s summary: “Milgrom and Weber (1982) prove a very general result called the ‘linkage principle’ which basically states that in this general symmetric setting, the more information on which the winner’s payment is based, the higher will be the expected revenue.” Klemperer draws the seller’s corollary: “if the seller has access to any private source of information, her optimal policy is to precommit to revealing it honestly.”

How big is it?

The theory gives a sign, not a magnitude, and the magnitude turns out to be modest. Quint (2016) supplies the quantified example: a binary state $θ \in {0, 1}$ equally likely with common value $v_{i} = θ$ ; signals with density $α s^{α - 1}$ under $θ = 1$ and $α (1 - s)^{α - 1}$ under $θ = 0$ . He compares a full-information button auction (Milgrom–Weber’s “English”: drop-out prices observed) against a no-information one (their “second-price”). Reimplementing his equilibrium bid functions in C and running 300,000 trials × 8 seeds:

$N$	$α$	no-info revenue ×1000	full-info revenue ×1000	lift	Quint’s Table 1/2
3	1.5	388.8	393.8	+1.29%	389 → 394, 1.3%
3	2.0	388.1	400.9	+3.32%	388 → 401, 3.3%
3	2.5	415.9	432.0	+3.85%	416 → 432, 3.8%
3	3.0	443.6	458.6	+3.39%	443 → 458, 3.4%
5	2.0	427.8	453.8	+6.07%	428 → 454, 6.1%
7	2.0	445.5	475.9	+6.82%	446 → 476, 6.8%

An independent C reimplementation matching a published table to three significant figures in every cell. Across-seed standard deviations are 0.3–0.8 in these units.

The largest lift anywhere in Quint’s grid is 6.8%, and he warns that even that may be inflated: “this example has one feature that may exaggerate the magnitude of this effect… Another example I tried… generates a revenue difference of only 0.4% when $N = 3$ .” The effect also vanishes at both ends — uninformative signals leave no information rents to compete away, and near-perfectly-informative signals leave none either.

This is the single most important calibration in the note. The affiliation break is the one every treatment of auction theory emphasises, and it is the smallest of the four measured here. Risk aversion at $ρ = 1/2$ is worth 11%; asymmetry can be worth 100%; budget constraints 27%. Affiliation is worth a few percent, and it is the only one of the four that favours the ascending auction. A designer who chooses an ascending format “because of the linkage principle” while ignoring bidder asymmetry is optimising the small term.

And the extreme case: with correlation, the seller can take everything

There is a much more dramatic failure lurking one step further. Myerson’s §7 shows that if bidders’ private information is correlated (not merely affiliated), “the seller can construct a mechanism that yields for herself the entire social surplus that would be feasible if bidders’ information were fully public,” by offering each bidder a menu of bets among which she must choose to participate (Klemperer’s summary of Myerson §7; Crémer & McLean 1985 generalise it). Klemperer is appropriately sceptical: the result “does seem to rely heavily on assumptions such as risk-neutrality of both the bidders and the seller, common knowledge of the distributions from which bidders’ signals are drawn, the inability of bidders to collude, and the ability of the seller to credibly and costlessly communicate and enforce the auction’s results (often including extracting large payments from losing bidders).” Full-surplus extraction is a theorem about a world nobody lives in, but it does establish that “revenue equivalence fails under correlation” is an understatement.

Break A3 — Asymmetric Bidders: Either Format Can Win

This is the largest and least tidy break. Vickrey already noticed it in 1961, and Maskin & Riley’s Asymmetric Auctions (Review of Economic Studies 67, 2000) is the systematic treatment. Their summary in an earlier paper, quoted by Klemperer, is the rule of thumb: “roughly speaking, the sealed-bid auction generates more revenue than the open [second-price] auction when bidders have distributions with the same shape (but different supports), whereas the open auction dominates when, across bidders, distributions have different shapes but approximately the same support.”

The mechanism, per Klemperer §7.1: “In a first-price auction a bidder whose value is drawn from a weaker distribution bids more aggressively (closer to her actual value) than a bidder from a stronger distribution, [so] a first-price auction also discriminates in favour of selling to the weaker bidder, in contrast to a second-price (or ascending) auction which always sells to the bidder with the higher valuation.” Maskin and Riley’s footnote makes it arithmetic: “In a first-price auction the first-order condition of a bidder with value $v$ considering raising her bid $b$ … sets $(v - b) Δ p - p Δ b = 0$ . Weaker bidders have smaller probabilities of winning, $p$ , and hence smaller profit margins, $v - b$ , when they do win.”

Case 1 — a shift: the “Getty effect”

Maskin and Riley’s Example 1: the weak buyer’s value is uniform on $[0, 1]$ , the strong buyer’s on $[2, 3]$ .

Open / second-price: the strong buyer always wins and pays the weak buyer’s value, so revenue $= E [v_{w}] = 1/2$ .
First-price: the strong buyer’s best response to $b_{w} (v_{w}) = v_{w}$ is to bid $b = 1$ — for $v_{s} \geq 2$ , “the solution is $b = 1$ ”, because the gain from certain victory dominates. Revenue $= 1$ , with zero variance.

The first-price auction raises exactly twice as much. Maskin and Riley name it: “This principle might be termed the ‘Getty effect,’ after the wealthy art museum known for out-bidding its competition.” Simulated over 400,000 trials × 10 seeds:

  first-price  = 1.000000  (sd across seeds 0.000000)
  second-price = 0.500024  (sd across seeds 0.000668)
  first-price premium = +99.99%

The zero variance on the first-price row is not a bug: in this equilibrium the strong buyer bids exactly 1 in every trial and always wins, so the realised price is deterministic. That is itself a striking observation — an asymmetry severe enough can make the first-price auction a posted price.

Case 2 — a stretch: still first-price, by a realistic margin

Example 1 is extreme by construction. Maskin and Riley’s Example 2 is the calibrated version: weak buyer uniform on $[0, \frac{1}{1 + z}]$ , strong on $[0, \frac{1}{1 - z}]$ — same shape, same lower endpoint, wider support for the strong bidder. They derive the asymmetric equilibrium inverse bid functions in closed form:

$b_{w}^{- 1} (b) = \frac{2 b}{1 + z ( 2 b ) ^{2}}, b_{s}^{- 1} (b) = \frac{2 b}{1 - z ( 2 b ) ^{2}} .$

Inverting those numerically and simulating (400,000 trials × 10 seeds):

$z$	first-price revenue	open revenue	difference	sd(FP, seeds)
0.00	0.333377	0.333450	−0.000073	0.000222
0.10	0.334190	0.330504	+0.003686	0.000232
0.20	0.337169	0.324171	+0.012998	0.000234
0.30	0.342040	0.315537	+0.026503	0.000242
0.40	0.349390	0.306133	+0.043258	0.000214

At $z = 0$ both formats return $1/3$ , the exact symmetric answer, and the difference is within one standard deviation of zero — the theorem holds at the symmetric point. As $z$ grows the two curves separate in opposite directions, exactly as Maskin and Riley’s algebra predicts (first-price revenue “rises with $z$ ”; the open auction’s winning-bid cdf “is increasing in $z$ ”, so its revenue falls). At $z = 0.4$ the first-price auction raises 14.1% more.

Case 3 — the other direction: when the open auction wins

The asymmetry has to have the right shape. Maskin and Riley’s Example 3 reallocates probability mass to the bottom of the weak bidder’s distribution: both bidders’ values are concentrated on ${0, 2}$ , the strong bidder always at 2, the weak bidder at 0 or 2 with probability $\frac{1}{2}$ each.

Open auction: revenue $= \frac{1}{2} \times 2 = 1$ .
First-price: the strong buyer can win with probability at least $\frac{1}{2}$ by bidding just above zero, earning at least 1 — so she never bids above 1. The weak buyer can then win for certain at $1 + ε$ . Buyers’ total expected payoff is at least $1 \frac{1}{2}$ ; social surplus is 2; so revenue is at most $\frac{1}{2}$ .

The open auction raises at least twice as much. The mechanism is the strong bidder’s “incentive to ‘low ball’”: she gains nothing from bidding high enough to be sure of winning (she would have to bid 2 and earn nothing), but gains plenty from bidding very low. Maskin and Riley’s Proposition 3.5 generalises this to continuous distributions.

Example 3 breaks two assumptions at once

The two-point distributions in Example 3 have atoms, so they also violate A5. The paper’s Proposition 3.5 is the continuous version, and it is the one to cite for “asymmetry can favour the open auction” without confounding. I did not simulate Example 3, because its first-price equilibrium is in mixed strategies with atoms and computing it correctly is a separate exercise; the argument above is Maskin and Riley’s own and needs no simulation, since it bounds revenue rather than computing it.

Case 4 — Vickrey’s own 1961 example, re-derived and re-measured

Vickrey posed the simplest possible asymmetry: bidder 2’s value is a known constant $a$ ; bidder 1 draws $v \sim U (0, 1)$ . He reports (p. 19) that under the progressive auction “for values of $v_{1}$ greater than $a$ , bidder 1 gets the article at a price of $a$ , while for a $v_{1}$ less than $a$ , bidder 2 gets the article at a price of $v_{1}$ . The average price is thus $a - \frac{1}{2} a^{2}$ . With the Dutch auction the average realized price turns out to be greater than this for $a$ greater than about 0.43, and less for values below about 0.43.”

The English side is a two-line calculation: $(1 - a) \cdot a + a \cdot \frac{a}{2} = a - \frac{a ^{2}}{2}$ . The first-price side requires the mixed equilibrium, which Vickrey solved graphically and which I re-derived from the indifference conditions. Bidder 2 must be indifferent across her support, so $β^{- 1} (b) (a - b) = π_{2}$ is constant. Vickrey’s Figure 3 notes that bidder 1 never wins below $a /2$ , which pins $π_{2} = \frac{a}{2} (a - \frac{a}{2}) = \frac{a ^{2}}{4}$ , giving

$β (v) = a - \frac{a ^{2}}{4 v} (v \geq a /2), b_{m a x} = a - \frac{a ^{2}}{4} .$

Bidder 1’s own first-order condition $G^{'} (b) (v - b) = G (b)$ then determines bidder 2’s mixing cdf; substituting $w = 2 b - a$ and integrating gives $G (b) \propto e^{- a / w} / w$ , normalised so $G (b_{m a x}) = 1$ . Simulating this (60,000 trials × 6 seeds at each of 86 values of $a$ ):

$a$	English (exact)	first-price (MC)	difference
0.20	0.180000	0.172693	−0.007307
0.30	0.255000	0.247181	−0.007819
0.40	0.320000	0.317010	−0.002990
0.45	0.348750	0.350664	+0.001914
0.60	0.420000	0.446476	+0.026476
0.80	0.480000	0.565411	+0.085411
0.95	0.498750	0.648968	+0.150218

simulated sign change (first-price overtakes English) at a = 0.4329
Vickrey (1961, p.19) says "about 0.43"

An independently derived mixed-strategy equilibrium, simulated in C, reproduces a threshold Vickrey computed by hand in 1961 to three significant figures. It also shows the direction: with a strong, predictable rival, the seller wants the sealed first-price format — the same conclusion as the Getty effect, arrived at from a completely different asymmetry.

Break A4 — Budget Constraints: First-Price Wins, and Efficiency Collapses

Klemperer calls this “an important reason why revenue equivalence may fail in practice,” and the argument he gives is a clever use of the theorem against itself:

“Consider the standard model in which revenue equivalence applies and bidders have independent private values $v_{i}$ , but let bidder $i$ have budget constraint $\overset{w}{ˉ}_{i}$ . Then in a second-price auction $i$ bids exactly as if she had value $x_{i} = min (v_{i}, \overset{w}{ˉ}_{i})$ but no budget constraint, so by the revenue equivalence theorem the expected revenue equals that from a first-price auction in which bidders have values $x_{i}$ and no budget constraints… It is intuitive that this is less expected revenue than from a first-price auction in which bidders have values $v_{i} \geq x_{i}$ and budget constraints $\overset{w}{ˉ}_{i} \geq x_{i}$ . So first-price auctions are more profitable than second-price auctions. This and similar results are obtained in Che and Gale (1998).” (Klemperer §13.1)

Read the argument carefully and you see why it is only a sign: the last step is “it is intuitive that.” There is no closed form for the constrained first-price equilibrium, so measuring the magnitude requires actually solving the game.

A numerical Bayes–Nash solver, and why the obvious one fails

The tool is a damped best-response iteration in bid space. The strategy is a real-valued map from types to bids on a grid; the belief is the cdf of one opponent’s bid; each iteration computes the exact best response for every type and moves the strategy a fraction $λ$ toward it.

Two implementation details are load-bearing, and I only found them by watching the solver fail:

Belief-space fictitious play does not work here. Mixing the belief distribution toward the best-response distribution never converges, because the best response to a smoothed belief is a sharp pure strategy and the $L^{1}$ distance between them stays around 1.9 forever. Damping must be applied to bids, not to distributions.
A raw empirical cdf makes the whole upper half of the strategy pool at one bid. If $G$ is built as the empirical cdf of the current profile, $G (x) = 1$ for every $x$ at or above the highest current bid — so no type ever gains by bidding above it, and the strategy freezes. Diagnostically this is unmistakable once you print it: for $n = 5$ the solver returned $b (v) = 0.8 v$ for $v ≲ 0.4$ and then a flat $b = 0.499$ for every higher type, against a true $b (v) = 0.8 v$ reaching $0.8$ . The fix is a kernel-smoothed belief whose bandwidth is annealed to zero, so that bidding just above the current top still pays.

/* Belief: the cdf of ONE opponent's bid, as a kernel-smoothed histogram of the
 * current strategy.  The smoothing is essential: with a raw empirical cdf,
 * G(x)=1 for every x at or above the highest bid in the current profile, so no
 * type ever has an incentive to bid above it and the upper half of the strategy
 * pools at one bid.  Bandwidth h is annealed to 0 over the run.              */
static void build_G(double h){
    memset(hist, 0, sizeof hist);
    double w = 1.0/(double)NT;
    for (int t = 0; t < NT; t++){                 /* histogram of current bids */
        double x = bfun[t]; if (x < 0) x = 0; if (x > Bmax) x = Bmax;
        hist[(int)(x/Bmax*(NG-1)+0.5)] += w;
    }
    int hw = (int)(h/Bmax*(NG-1)); if (hw < 1) hw = 1;
    pre[0] = 0;                                   /* prefix sum => O(1) box kernel */
    for (int i = 0; i < NG; i++) pre[i+1] = pre[i] + hist[i];
    double acc = 0;
    for (int i = 0; i < NG; i++){
        int lo = i-hw, hi2 = i+hw+1;
        if (lo < 0) lo = 0;
        if (hi2 > NG) hi2 = NG;
        acc += (pre[hi2]-pre[lo])/(double)(2*hw+1);   /* smoothed density */
        Gg[i] = acc;                                  /* integrate to a cdf */
    }
    double tot = Gg[NG-1]; if (tot > 0) for (int i = 0; i < NG; i++) Gg[i] /= tot;
}

flowchart LR
    subgraph SP["Second-price, budgets private"]
        R1["Rich bidder<br/>v = 0.9, no cap"]
        P1["Poor bidder<br/>v = 0.8, w = 0.3"]
        R1 --> O1["Rich wins,<br/>pays min(v,w) of the runner-up<br/><b>price = 0.3</b>"]
        P1 --> O1
    end
    subgraph FP["First-price, budgets private"]
        R2["Rich bidder<br/>v = 0.9, no cap"]
        P2["Poor bidder<br/>v = 0.8, w = 0.3"]
        R2 --> O2["Rich must OUTBID a bid<br/>distribution she cannot see,<br/>so shades far less<br/><b>price above 0.3</b>"]
        P2 --> O2
    end
    O1 -.->|"the poor bidder's CAP<br/>becomes the price"| O2
    style O1 fill:#7a3b12,color:#fff
    style O2 fill:#1f6f43,color:#fff

What it shows: why a private budget favours the first-price format. In the second-price auction the runner-up’s cap is handed to the winner as her price — the seller is charged for the poorest surviving bidder’s poverty. In the first-price auction the winner pays her own bid, which she must choose against uncertainty about rivals’ caps as well as their values, so she shades less. The insight to carry: the effect is driven by the budget being private, which the common-cap experiment below confirms — with a publicly known identical cap the gap shrinks from 27.6% to 5.8%.

Validating the solver before trusting it

A numerical equilibrium solver that has not been checked against a closed form is a random number generator. Running it on twelve cases where the answer is known exactly (no budget constraint, $b (v) = \frac{n - 1}{n - 1 + ρ} v$ , revenue $\frac{n - 1}{n - 1 + ρ} \cdot \frac{n}{n + 1}$ ), with 256 type points and 600 iterations at $λ = 0.10$ :

$n$	$ρ$	solver $b (0.25)$	exact $b (0.25)$	solver revenue	exact revenue
2	1.00	0.125052	0.125977	0.332972	0.333333
3	1.00	0.168405	0.167969	0.499729	0.500000
5	1.00	0.200295	0.201563	0.665839	0.666667
8	1.00	0.219876	0.220459	0.776062	0.777778
5	0.50	0.223783	0.223958	0.739819	0.740741
8	0.25	0.243773	0.243265	0.856740	0.858238

Maximum relative revenue error across all twelve cases: 0.2%. The solver independently rediscovers both the risk-neutral shading rule and the CRRA rule derived by hand above. Only now is it worth pointing at a case with no closed form.

Measured: private budgets

Values $v \sim U (0, 1)$ ; budgets $w \sim U (0, W_{m a x})$ drawn independently of value; $n = 5$ . The second-price side needs no solver — the dominant feasible bid is $min (v, w)$ — so its revenue is a computable order statistic. The first-price side is solved numerically.

$W_{m a x}$	first-price revenue	second-price revenue	difference	first-price gain	first-price efficiency
4.00	0.625810	0.616201	+0.009610	+1.56%	0.8691
2.00	0.585306	0.561027	+0.024279	+4.33%	0.7581
1.00	0.502319	0.445864	+0.056455	+12.66%	0.5767
0.80	0.461505	0.393263	+0.068242	+17.35%	0.5076
0.60	0.398155	0.322849	+0.075306	+23.33%	0.4194
0.50	0.353050	0.280523	+0.072527	+25.85%	0.3701
0.40	0.297022	0.233431	+0.063592	+27.24%	0.3237
0.30	0.231762	0.181655	+0.050107	+27.58%	0.2830
0.20	0.159297	0.125348	+0.033948	+27.08%	0.2497
0.10	0.081641	0.064715	+0.016926	+26.15%	0.2232

“Efficiency” is the probability that the first-price auction allocates to the highest-value bidder, estimated over 300,000 draws.

Three findings, only the first of which was predicted:

Che and Gale’s sign is confirmed and the magnitude is large. First-price beats second-price at every budget level tested, and the gap peaks around +27.6% — more than double the risk-aversion effect and four times the affiliation effect. As the constraint relaxes ( $W_{m a x} = 4$ , binding for only a quarter of bidders) the gap shrinks to 1.56% and both formats climb back toward the unconstrained $2/3$ , which is the right limiting behaviour.
The absolute difference is hump-shaped while the percentage is not. In levels the gap peaks near $W_{m a x} \approx 0.6$ at $+ 0.075$ and falls away in both directions; in percentage terms it plateaus near 27% and stays there. Which statistic a designer cares about depends on whether the budget is a fixed institutional cap or something proportional to the prize.
Efficiency is destroyed. At $W_{m a x} = 0.3$ the first-price auction puts the object in the highest-value bidder’s hands only 28% of the time. Revenue equivalence’s condition (i) — “the object always goes to the bidder with the highest value” — is not merely an accounting assumption; when budgets bind it is false, and the theorem’s hypothesis fails at the front door rather than through the envelope argument. A seller reading “first-price raises 27% more” as a recommendation is also choosing a mechanism that misallocates the object nearly three times out of four.

Isolating the cause: it is the private budget, not the budget

A natural follow-up: is the 27% gap caused by bidders being poor, or by the seller not knowing which bidders are poor? Re-running the same solver with a common, commonly-known cap $B$ — every bidder constrained identically, and everyone knows it — separates the two:

$B$	first-price revenue	second-price revenue	difference	first-price gain
1.00 (never binds)	0.665839	0.666667	−0.000827	−0.12%
0.80	0.665821	0.647081	+0.018740	+2.90%
0.70	0.645953	0.610361	+0.035593	+5.83%
0.60	0.581026	0.553344	+0.027682	+5.00%
0.50	0.494155	0.479165	+0.014990	+3.13%
0.40	0.398564	0.392491	+0.006074	+1.55%
0.30	0.299755	0.298056	+0.001699	+0.57%
0.20	0.199979	0.199723	+0.000256	+0.13%

The gap peaks at 5.8% — about a fifth of the 27.6% seen with private budgets. (The $B = 1.00$ row is the solver’s error floor: the constraint never binds there, both formats should return exactly $2/3$ , and the solver’s first-price answer is 0.12% low. Every other number in this table should be read with that ±0.1% band in mind.)

So the bulk of the effect is informational. When budgets are private, a bidder’s cap is a second dimension of hidden type, and the first-price auction — where the bid itself must be paid — lets a rich bidder profitably outbid a poor one at a price the poor one could never have reached, while the second-price auction hands the winner the poor bidder’s cap as the price. When the cap is common knowledge and identical, that channel closes and only the mechanical truncation remains. The lesson generalises: breaking A4 hurts mostly through its interaction with A2 and A3, because a private budget is exactly an extra private, asymmetric dimension of type.

Klemperer notes one further escalation: “auction forms that take payments from losers, such as lotteries and all-pay auctions, can be more profitable still in the presence of budget constraints” (citing Che & Gale 1996). That is condition (ii) failing rather than (i) — losers no longer get zero — and it is the same door the all-pay auction walks through.

Uncertain

Verify: the exact statements and conditions of Che & Gale (1998), “Standard Auctions with Financially Constrained Bidders,” Review of Economic Studies 65(1), pp. 1–21. Reason: the paper is behind the Oxford Academic paywall and no readable mirror was located from this machine (this is a retrieval failure, not a text-extraction failure — I never obtained a PDF to render). Everything above is Klemperer’s §13.1 summary, which I read in full, plus my own simulation. My model (budgets independent of values, uniform) is a Che–Gale-style environment, not necessarily their environment; the sign agrees, and the magnitudes are mine and should not be attributed to them. To resolve: read the RES paper and check whether the revenue gap is characterised analytically there. uncertain

Break A5 — Atoms: The Assumption Nobody Quotes

Klemperer’s Appendix A gives the counterexample directly, and it is small enough to check by hand:

“Assume instead that there are just two types, $v = 0$ and $v = 1$ , and each of two bidders is equally likely to be of either type (independent of the other’s type) and the seller begins by simultaneously offering both bidders a price $p$ ; if just one accepts then the trade is made at price $p$ , if both accept then the unit is allocated by lottery at price $p$ ; if neither accepts then the unit is allocated by lottery at price 0. Then a high type prefers to accept, so [conditions (i) and (ii) hold] for any $p \in (0, \frac{2}{3})$ , but the seller’s expected revenue is strictly increasing in $p$ , so Revenue Equivalence fails.”

The mechanism satisfies both stated conditions — the object goes to a highest-value bidder, and the lowest type expects zero surplus — and yet a one-parameter family of such mechanisms has strictly different revenues. The reason is that with a discrete distribution the equilibrium bid function is not strictly increasing; there are ties with positive probability, and the envelope integral that pins payments down needs the value distribution to have no atoms. Klemperer notes that “Riley (1989a) [gives] revenue equivalence results for discrete distributions,” so the discrete case is not hopeless — it is just a different theorem.

This break matters more in computer science than in economics, because discrete type spaces are the default in algorithmic mechanism design: bids are integers of cents, ad-auction quality scores are quantised, and combinatorial valuations live on finite lattices. Anyone importing “all these formats are revenue-equivalent” into a discrete-bid setting has imported a theorem whose hypothesis they have already violated. The practical symptom is exactly the tie-breaking rule mattering to revenue, which under A5 it could not.

Failure Modes and Gotchas

Reading it as “format doesn’t matter.” The theorem says the mean does not depend on the format, given five assumptions that essentially never all hold. The correct reading is Nisan et al.’s: you cannot raise revenue without changing the allocation rule. That is a statement about where to look for revenue (the allocation rule, i.e. reserve prices and exclusion — see Reserve Prices and Optimal Auctions), not a licence to pick formats at random.

Forgetting condition (ii). “Highest value wins” is the memorable condition; “lowest type gets zero” is the one that quietly fails. Entry subsidies, participation payments, loser rebates, and free options all violate it. Riley and Samuelson’s framing is the most robust check: has the marginal entering type $v^{*}$ changed? If yes, revenue changed.

Applying it to a mechanism whose equilibrium is not efficient. Condition (i) is about the equilibrium allocation, not the rules. Asymmetric first-price auctions and budget-constrained auctions both violate it. If your mechanism sometimes gives the object to the second-highest-value bidder, the theorem does not apply at all — you cannot invoke it and then discuss “the revenue difference,” because there is no baseline.

Mistaking equal means for equal distributions. Measured above: per-trial standard deviations of 0.113 (first-price), 0.178 (second-price), and 0.450 (all-pay), all with the same mean. Revenue risk differs by a factor of 4. Any seller who has to finance against expected proceeds, or any procurement office with an annual budget, is exposed to the second moment.

Comparing a single run of two formats. The affiliation effect is 1–7%; the across-seed sd of a 300,000-trial run of that model is ~0.15% of revenue. A single 10,000-trial run has an sd of roughly 0.8%, which will not separate $α = 1.5$ from $α = 2.0$ . Every table in this note reports the across-seed sd for exactly this reason. A single run is not a result.

Trusting a numerical equilibrium solver you have not validated. Documented above: the naive belief-space iteration failed silently (a plausible-looking bid function that was correct for the bottom 40% of types and constant thereafter, and a revenue of 0.494 against a true 0.667 — a 26% error with no error message). It was caught only by printing $b (v)$ against the closed form. Always run the solver on a case with a known answer first, and check the whole strategy, not one point of it.

Assuming the linkage principle is the dominant effect. Measured: it is the smallest of the four. Cramton’s practitioner survey lists ex-ante asymmetry, risk aversion, collusion, and entry as the considerations that actually decide format choice; affiliation appears among them but is not privileged. If a real auction shows a 30% format difference, affiliation is not the explanation.

Using it to predict rather than to diagnose. The theorem’s assumptions are strong enough that its predictive record is poor. Maskin and Riley open their paper by observing that the observed dominance of certain formats is “inconsistent with theory, since a corollary of the revenue equivalence theorem is that the expected revenue [is format-independent],” and conclude: “These discrepancies suggest that the hypotheses of the revenue-equivalence theorem [fail].” That is the productive use. Revenue equivalence is a null hypothesis.

Alternatives and When to Choose Them

There is no alternative theorem — revenue equivalence is a fact about a model. What there are alternatives to is the modelling frame, and each one buys a different kind of guidance.

Frame	What it gives you	When to reach for it
Revenue equivalence (Vickrey, Myerson, Riley–Samuelson)	A null hypothesis: format is irrelevant, so any observed difference localises an assumption failure	Sanity-checking a simulator; deriving one auction’s equilibrium from another’s (Klemperer’s Appendix A trick); arguing that a proposed format change is not where the money is
Marginal-revenue / optimal auctions (Myerson 1981; Bulow & Roberts 1989)	A constructive answer: allocate to the highest marginal revenue, not the highest value; set the reserve where marginal revenue equals the seller’s cost	When you can change the allocation rule and want the revenue-maximising one. See Reserve Prices and Optimal Auctions
The linkage principle (Milgrom & Weber 1982)	A ranking under affiliation: English ≥ second-price ≥ first-price, plus “commit to honest disclosure”	Common-value settings — mineral rights, spectrum, financial instruments
Asymmetric-auction analysis (Maskin & Riley 2000)	Format rankings that depend on the shape of the asymmetry, plus the participation consequences	Whenever bidders are visibly unequal: incumbents vs. entrants, one deep-pocketed buyer
“What really matters” (Klemperer 2002)	The claim that entry, collusion, and political robustness dominate all of the above	Designing a real auction with real money and real lawyers
Bulow & Klemperer’s bidder-count result	An extra bidder is worth more than an optimal reserve price	Deciding between spending effort on mechanism design and spending it on marketing

Klemperer’s What Really Matters in Auction Design is the corrective a reader should hold against everything above: the fine structure of format choice matters far less than whether bidders show up and whether they collude. The measured effects in this note are 1–28%; a failed auction with two bidders instead of six loses much more than that.

Cramton’s summary of the same point, from the format side: “With neither theory nor experiments providing definitive results, it is easy to conclude that which auction format is best is an empirical matter. How one would go about answering this empirical question is difficult at best, since rarely do we see multiple formats being used in easily comparable settings” (Cramton 1998).

Production Notes

FCC spectrum auctions, and A4 in the wild. The budget-constraint break is not a theoretical curiosity in high-stakes auctions; it is the default. Cramton reports that in the FCC’s C-block auction, “bidding teams typically face budget constraints, which are negotiated with the Board of Directors (or outside investors),” and — crucially — that the format interacts with the constraint: “in an ascending auction, the bidding team can return to the Board with hard and fast ‘evidence’ supporting their optimistic estimates: the high bids of the other bidders confirm how much these items are worth. In this way, budget constraints can be relaxed.” NextWave, the largest C-block winner at $4.2 bi l l i o n (41$ 20 per person “ended with the winners paying twice that.” The sequel is the cautionary half: the FCC “has as of August 1997 collected only about 10% of this money (the initial deposit),” with winners unable to make installment payments. An auction that raises revenue by relaxing budget constraints during bidding raises revenue it may not collect. My simulation’s efficiency column is the static version of the same disease.

Bidding credits as an explicit A3 correction. If asymmetry breaks revenue equivalence and can cost the seller half her revenue, the designer’s response is to manufacture symmetry. Cramton: “Bidding credits of from 10 to 40 percent were used in the FCC spectrum auctions to encourage participation by women, minorities, and small businesses. In the regional narrowband auction, these preferences had the effect of stimulating competition and significantly raising revenues (Ayres and Cramton 1996).” The theoretical justification is Myerson’s: an optimal auction “typically discriminates in favour of selling to bidders whose values are drawn from lower distributions, that is, ‘weaker’ bidders” (Klemperer §7.1). Handicapping is not a fairness concession bolted onto a revenue-maximising auction; under asymmetry it is the revenue-maximising auction. Cramton’s caveat is real: “the government’s use of credits is crude, since rarely does it have good information about the extent of asymmetries among bidders.”

Reading bidders’ format preferences as a diagnostic. Maskin and Riley “show quite generally that strong buyers prefer the second-price auction, whereas weak buyers prefer the first-price auction.” Cramton turns this into an operational trick: “Bidders with high values tend to favour ascending auctions; whereas, bidders with low values prefer sealed bidding. If only a single bidder is insisting on an ascending auction, then it is more likely that there is a large asymmetry favouring this bidder. In this case, a sealed-bid auction probably should be used.” A public consultation on auction rules is, itself, a mechanism that elicits information about the asymmetry — and revenue equivalence is what tells you the elicited information is worth acting on.

eBay and condition (ii). eBay’s proxy-bid auction is a second-price mechanism, so under A1–A5 it is revenue-equivalent to a sealed first-price sale. It is not, and the reason is a sixth assumption nobody lists: the theorem assumes the auction ends. eBay’s hard close creates sniping — 40% of eBay Computers auctions and 59% of Antiques auctions have a last bid in the final five minutes, against ~3% on Amazon’s auto-extending format (Roth & Ockenfels 2002) — and a successful snipe can mean the highest-value bidder does not win, violating condition (i). The termination rule is part of the allocation rule. See English and Dutch Auctions for the full data.

Procurement and the all-pay family. Klemperer notes that the all-pay auction models “queues… legal battles, and markets with consumer switching costs in which firms compete for the prize of selling to new unattached consumers by lowering their prices to their old locked-in customers,” and that the war of attrition “is also a kind of all-pay auction.” Revenue equivalence says a lobbying contest and a sealed-bid sale transfer the same expected amount. It also says — via the measured variance — that they transfer it with wildly different risk, and the all-pay form additionally burns the losers’ expenditure as a social cost rather than a transfer. Equal revenue is not equal welfare.

vault

Explorer

Revenue Equivalence

Revenue Equivalence

Mental Model: five load-bearing assumptions, five directions to fall

Statement, and What Each Word Is Doing

Why it is true: the envelope argument in six lines

The Assumption Checklist

Baseline: confirming equivalence before breaking it

Break A1 — Risk-Averse Bidders: First-Price Wins

The exact equilibrium, derived

Measured

The mirror image: a risk-averse seller prefers the same format

Break A2 — Affiliated Values: The English Auction Wins (a Little)

How big is it?

And the extreme case: with correlation, the seller can take everything

Break A3 — Asymmetric Bidders: Either Format Can Win

Case 1 — a shift: the “Getty effect”

Case 2 — a stretch: still first-price, by a realistic margin

Case 3 — the other direction: when the open auction wins

Case 4 — Vickrey’s own 1961 example, re-derived and re-measured

Break A4 — Budget Constraints: First-Price Wins, and Efficiency Collapses

A numerical Bayes–Nash solver, and why the obvious one fails

Validating the solver before trusting it

Measured: private budgets

Isolating the cause: it is the private budget, not the budget

Break A5 — Atoms: The Assumption Nobody Quotes

Failure Modes and Gotchas

Alternatives and When to Choose Them

Production Notes

See Also

Graph View

Table of Contents

Backlinks