The revenue equivalence theorem says that the four standard auction formats produce the same expected revenue for a seller when five specific conditions hold: bidders are risk-neutral, their valuations are private and independent, they are drawn from the same distribution, the lowest-type bidder expects zero surplus, and payments depend only on the bids submitted. William Vickrey proved the result for specific format pairs in 1961, and Roger Myerson generalized it in 1981. The point of the theorem is less that auctions are interchangeable and more that it tells you exactly which assumption to blame when they aren’t.
What the Theorem Actually Claims
Myerson’s version is the sharpest: a seller’s expected revenue depends only on who wins in each situation and on the expected surplus of the weakest bidder, not on the specific payment rule the auction uses.1Princeton University. Optimal Auction Design Two mechanisms that award the item to the same bidder in every scenario, and give the lowest-type bidder the same starting surplus, must produce identical expected revenue. Vickrey’s earlier pairwise equivalence results fall out as special cases.2Peter Cramton’s Market Design Papers. Counterspeculation, Auctions, and Competitive Sealed Tenders
Vickrey received the Nobel Memorial Prize in Economic Sciences in 1996 for his broader work on incentives under asymmetric information, of which the 1961 auction paper was a foundation.3NobelPrize.org. William Vickrey – Facts
The Four Formats the Theorem Covers
The claim applies across four canonical auction types:
- English auction: the price rises openly, bidders drop out as it passes their valuation, and the last one standing pays just above the second-highest valuation.
- Dutch auction: the price ticks down from a high starting point until someone accepts, and the winner pays that stopping price.
- First-price sealed-bid: everyone submits a hidden bid; the highest wins and pays what they wrote.4EconGraphs. First-Price Auctions
- Second-price sealed-bid, also called the Vickrey auction: everyone submits a hidden bid; the highest wins but pays the second-highest bid.5Stanford University. Vickrey-Clarke-Groves Mechanisms
Two pairs are strategically identical before you even reach revenue. A Dutch auction and a first-price sealed-bid present a bidder with the same decision problem: commit to a price without knowing anyone else’s choice, and pay that price if you win.6Cornell Mathematics. Dutch and First-Price Sealed-Bid Auctions An English auction and a second-price sealed-bid share a dominant strategy under independent private values: stay in, or bid, up to your true valuation, and pay a price set by the second-highest value.
Why Different Formats Yield the Same Expected Revenue
In first-price formats, bidders shade their bids below their true values to leave themselves a profit margin. In second-price formats, bidders bid their true values but pay less than what they bid. These two effects cancel out exactly.
A worked case makes the offset concrete. With valuations drawn uniformly between 0 and 1 and n bidders, equilibrium bidding in a first-price auction has each bidder submit (n − 1)/n times their true value. Expected revenue in both formats works out to (n − 1)/(n + 1).7Brown University. Order Statistics and Revenue Equivalence With two bidders that’s about a third of the maximum possible value; with ten bidders it’s roughly 82%. The number of bidders drives revenue. The format does not.
The Five Conditions
The theorem rests on five assumptions, and each one is worth stating precisely because each corresponds to a way the result can fail in practice.
- Independent private values. Each bidder knows what the item is worth to them, and that valuation doesn’t depend on anyone else’s. A collector bidding on a painting for their living room fits. An oil company bidding on a drilling lease does not, because the resource under the ground has one true value that everyone is trying to estimate.
- Risk neutrality. Bidders care about expected profit, not variance. A risk-neutral bidder is indifferent between a guaranteed $50 and a coin flip paying $100 or nothing.1Princeton University. Optimal Auction Design
- Symmetry. All bidders draw valuations from the same probability distribution. No one has a structural edge like better information or lower costs.
- The lowest-type bidder earns zero surplus. A bidder at the bottom of the distribution expects nothing from participating. This boundary condition is what pins the revenue number down; without it, two mechanisms with the same allocation rule could still produce different revenues by handing more surplus to weak bidders.1Princeton University. Optimal Auction Design
- Payments depend only on bids. The auction can’t condition payment on outside factors, side payments, or fees unrelated to the bidding.
When all five hold, the seller’s expected revenue is fixed by the allocation rule alone. Switch formats and the bids, payment rule, and strategy all change; the expected revenue does not.
When It Breaks, and Which Way
Real auctions rarely satisfy all five conditions. The useful thing about the theorem is that each violation points revenue in a known direction.
Risk-Averse Bidders
When bidders dislike the variance of losing, first-price auctions generate higher expected revenue than second-price or English auctions. A risk-averse bidder in a first-price auction shades less aggressively, accepting a thinner expected margin in exchange for a better chance of winning, and that pushes the winning bid up. In a second-price auction, the dominant strategy is still to bid your true value regardless of risk attitude, so nothing changes on that side. The formats diverge, and sealed first-price wins.
Correlated or Common Values
When valuations are correlated rather than independent, the linkage principle established by Milgrom and Weber in 1982 gives a clean ranking: English auctions beat second-price sealed-bid, which beat first-price sealed-bid.8Peter Cramton’s Market Design Papers. A Theory of Auctions and Competitive Bidding The mechanism is information. Watching a rival stay in an English auction is a signal that the item is probably worth more, and that signal reduces uncertainty for everyone still bidding. Less uncertainty means less cautious bidding. Sealed formats suppress that information flow, so bidders protect themselves with lower offers.
In pure common-value settings, where the item has one true worth nobody knows for certain, the winner’s curse compounds this. The highest bidder is statistically the one who overestimated the item’s value the most, and rational bidders shade downward to protect against that. Open ascending formats let bidders update from each other’s behavior, which softens the curse and lifts revenue relative to sealed bids.
Asymmetric Bidders
When bidders draw from different distributions, revenue equivalence fails outright. Vickrey flagged this in his 1961 paper, noting that the shift to a second-price format “may be to the advantage of the seller” in some asymmetric cases “but that in other cases it may be substantially to his disadvantage.”2Peter Cramton’s Market Design Papers. Counterspeculation, Auctions, and Competitive Sealed Tenders Later work generally finds that first-price auctions outperform second-price with asymmetric bidders, because a weaker bidder bids more aggressively relative to their value to stay in contention.
Reserve Prices and What Actually Moves Revenue
The theorem doesn’t say a seller has nothing to do. Setting a reserve price below which the item won’t sell can raise expected revenue even for a seller who values the item at zero. The reserve forces bidders to clear a floor, extracting more from high-value participants at the cost of occasionally failing to sell.
Myerson showed the optimal reserve depends on the valuation distribution, not on the number of bidders. Two participants or twenty, the optimal reserve is the same in a symmetric independent private values setting.1Princeton University. Optimal Auction Design The optimal reserve is positive even when the seller’s own valuation is zero, because the reserve works as price discrimination: sacrifice low-value sales to capture more from high-value ones.
Bulow and Klemperer proved in 1996 that attracting one additional bidder to an auction with no reserve generates more expected revenue than running an optimally reserved auction with one fewer bidder. If you can only invest effort in one lever, more participation beats better mechanism design.
Why the Theorem Is Still Useful
Given how fragile its assumptions are, the theorem’s ongoing value is diagnostic. When two formats produce different revenues in a real market, the theorem tells a seller or regulator where to look: which of the five conditions is broken, and in which direction the break shifts revenue. Risk-averse bidders push toward first-price. Correlated values push toward open ascending. Asymmetry usually favors first-price. The theorem also seeded mechanism design as a field, and Myerson’s proof strategy of working backward from the allocation rule to pin down payments is now standard well beyond auctions.