Four canonical auction formats — English ascending, Dutch descending, first-price sealed-bid and Vickrey second-price — are run here against bidders whose private valuations are drawn from a chosen probability distribution. Running many rounds lets you test the Revenue Equivalence Theorem, the result that all four mechanisms should yield the same expected seller revenue under standard assumptions.
Each round samples valuations from the chosen distribution, computes bids according to that auction's equilibrium bidding rule, and reveals the winner, clearing price and resulting surplus on a live bar chart plus a running revenue-history graph.
Pick English, Dutch, Sealed 1st or Vickrey from the auction tabs, set Number of bidders (3–20), Value distribution (uniform, normal μ=60/σ=20, or right-skewed), Rounds (5–100) and Auction speed, then press Run auction or Batch 100 to simulate many auctions instantly.
In a Vickrey (second-price) auction, bidding your true valuation is a dominant strategy: overbidding risks paying more than the item is worth, and underbidding only risks losing a profitable sale — a result William Vickrey proved in 1961.
The simulator uses the theoretical equivalence between the two formats: with n bidders, the equilibrium bid is value × (n−1)/n, since a rational Dutch bidder stops the clock at exactly the price they would have submitted as a sealed first-price bid.
In the simulation the English auction's price rises in steps of 2 until only one bidder remains, so the price is quantised to the nearest step above the second-highest valuation, whereas Vickrey pays the exact second-highest bid.
Efficiency tracks the percentage of rounds in which the bidder with the truly highest valuation actually won the item — a value below 100% signals that the auction format let a lower-value bidder win.
The skewed option squares a uniform random number and multiplies by 100, which concentrates most valuations near zero while allowing rare high draws — modelling markets where most bidders have low interest but a few are very keen.
Winner's surplus is the winning bidder's valuation minus the price paid; English and Vickrey auctions typically leave the winner the full gap to the second-highest valuation, whereas a sealed first-price winner's surplus is compressed by the equilibrium bid-shading factor (n−1)/n.
Auction theory studies how different rules for selling a good affect the behaviour of self-interested bidders and the revenue earned by the seller. The four classic mechanisms are: the English (ascending-bid) auction, where price rises until only one bidder remains; the Dutch (descending-price) auction, where price falls from a high start and the first bidder to call "mine" wins; the first-price sealed-bid auction, where everyone submits a secret bid simultaneously and the highest pays their bid; and the Vickrey (second-price sealed-bid) auction, where the highest bidder wins but pays only the second-highest bid. These formats form the backbone of everything from eBay listings and government spectrum auctions to Google AdWords and art sales at Christie's.
This simulator generates private valuations for each bidder from a chosen distribution (uniform, normal, or right-skewed), runs hundreds of auction rounds, and tracks average seller revenue, allocation efficiency (did the highest-value bidder win?), and winner surplus. You can also select bidder count, number of rounds, and animation speed to explore the Revenue Equivalence Theorem — the remarkable result that all four formats yield the same expected revenue under standard assumptions.
What is the Revenue Equivalence Theorem?
The Revenue Equivalence Theorem (RET), proved by Vickrey in 1961 and generalised by Myerson, states that any auction mechanism where the highest-value bidder wins and a bidder with zero value pays nothing yields the same expected revenue to the seller, provided bidders have independent private values drawn from the same distribution. In practice, deviations arise from risk aversion, correlated values, budget constraints, or asymmetric distributions, but the RET remains the most powerful benchmark in auction design.
Why is truth-telling the dominant strategy in a Vickrey auction?
In a second-price auction, bidding your true value v is a dominant strategy regardless of what others bid. If you bid above v and win, you might pay more than v (a loss). If you bid below v and lose to a bid between your shaded bid and v, you forgo a profitable opportunity. Neither overbidding nor underbidding can improve your outcome, so truthful bidding weakly dominates any other strategy. This property, called incentive compatibility, makes Vickrey auctions exceptionally useful in mechanism design.
What is the optimal bidding strategy in a first-price sealed-bid auction?
With n risk-neutral bidders whose values are drawn independently and uniformly from [0, V], the Bayesian Nash equilibrium bid for a bidder with value v is b(v) = v(n−1)/n. For example, with 5 bidders, a player with value 80 should bid 64. As n increases the shading factor (n−1)/n approaches 1, so bids converge towards true values and the auction more closely resembles a Vickrey outcome.
In theory, English and Vickrey auctions are strategically equivalent for independent private values: both yield the second-highest valuation as the final price, and both are incentive compatible. However, the English auction reveals information during bidding — once all but one bidder drops out, the winner knows the second-highest value precisely. In practice, the English format is preferred for complex goods because bidders can update their beliefs during the ascent, reducing the winner's curse in common-value settings.
The winner's curse arises in common-value auctions (e.g., oil field rights, where everyone values the same underlying asset) when the winning bidder discovers they overpaid. The winner is typically the most optimistic bidder, whose private signal overestimates the true value. Rational bidders shade their bids below their signal to compensate; naive bidders who ignore the curse systematically overpay. It was first documented empirically in US offshore oil lease auctions in the 1970s.
In a common-value or correlated-value setting, the Dutch auction concludes before bidders can update on rivals' behaviour, limiting information leakage that might drive the price up. Dutch auctions are also much faster — the Dutch flower market at Aalsmeer sells 20 million flowers per day using descending-clock auctions. However, for independent private values, the RET predicts identical revenue, so speed and simplicity are the primary practical reasons to prefer Dutch.
In an all-pay auction every bidder pays their bid regardless of whether they win — only one bidder receives the prize. This models lobbying (every firm spends resources to influence legislation, but only one firm's preferred policy wins), R&D races (every firm invests, but only the first to patent profits), and war of attrition games. Under standard conditions the all-pay auction also satisfies the RET, generating the same expected revenue as the other formats.
With more bidders, competition intensifies, pushing the winning bid closer to the highest private value. Under uniform valuations the expected revenue equals (n−1)/(n+1) × Vmax, which rises monotonically with n. Doubling bidder count from 4 to 8 increases expected revenue from 3/5 × V to 7/9 × V, a roughly 26% gain. This explains why public-sector procurement agencies work hard to attract multiple competing tenders.
A reserve price is a minimum acceptable bid below which the seller keeps the good. Myerson's optimal auction theory shows that with n bidders drawing values from a distribution F, the optimal reserve r* satisfies r* − (1−F(r*))/f(r*) = 0 — setting marginal revenue to zero at the threshold. For uniform values on [0, 100] with any n, the optimal reserve is exactly 50. A well-set reserve raises expected revenue by excluding low-value bids and extracting more from the remaining bidders, at the cost of occasional unsold items.
English ascending auctions: eBay (proxy bidding), art houses (Christie's, Sotheby's), antiques. Dutch descending auctions: Aalsmeer flower market, some US Treasury bill auctions. First-price sealed-bid: government procurement contracts, construction tenders, timber lease auctions. Vickrey/second-price: Google AdWords quality-adjusted ranking, some charity auctions, academic spectrum-allocation research. Combinatorial clock auctions (extensions of English format) are used by Ofcom in the UK for mobile spectrum allocation.
An auction is allocatively efficient when the good is won by the bidder with the highest private value — the person who values it most gets it. English and Vickrey auctions are always efficient under independent private values, because dominant strategies align bids with true values. First-price and Dutch auctions can be inefficient in practice if bidders shade asymmetrically. Efficiency matters because misallocation represents lost social surplus: a painting sold to a speculator who values it at 60 when a collector values it at 90 wastes 30 units of welfare.