🔨 Auction Simulator — English, Dutch & Vickrey Auctions

The revenue equivalence theorem, proved independently by several economists including William Vickrey, states that under standard assumptions — risk-neutral bidders, independent private valuations, and a rule that the highest-value bidder wins — the English, Dutch, and Vickrey (sealed second-price) auctions, as well as the sealed first-price auction, all yield the same expected revenue to the seller and the same expected payoff to bidders. This is a striking result because the four formats look and feel completely different, yet on average they raise identical amounts of money, which explains why sellers can choose whichever format is most practical without sacrificing expected revenue.

In a sealed second-price auction, a bidder's payment never depends on their own bid, only on the second-highest bid submitted by someone else. Bidding above your true valuation only risks winning at a price higher than the item is worth to you, while bidding below it only risks losing a profitable sale to a lower rival bid. Because neither deviation can ever improve your outcome and can only hurt it, bidding your true valuation weakly dominates every other strategy regardless of what other bidders do — a property economists call incentive compatibility.

Auction theory as a formal field began with Canadian-American economist William Vickrey's 1961 paper "Counterspeculation, Auctions, and Competitive Sealed Tenders," which introduced the second-price auction and proved its truthfulness property; he shared the 1996 Nobel Memorial Prize in Economic Sciences for this and related work on incentives under asymmetric information. Paul Milgrom and Robert Wilson later won the 2020 Nobel Prize for developing new auction formats and applying them to real-world problems such as the FCC's multi-billion-dollar radio spectrum auctions, cementing auction theory as one of economics' most applied subfields.

English auctions are the classic format at art, antiques, livestock, and estate sales, and are approximated by eBay's proxy bidding system. Dutch auctions are used at the Aalsmeer flower market in the Netherlands, in some IPO share allocations, and in certain cryptocurrency token sales. Sealed second-price-like mechanisms underlie U.S. Treasury bond auctions, government spectrum licence sales, and the generalized second-price auctions that power online search advertising at companies like Google, where advertisers bid for ad placement but the pricing rules borrow directly from Vickrey's insight.

The winner's curse occurs in auctions with a common, uncertain value — such as an oil-drilling lease or a company acquisition — where every bidder estimates the same unknown true value with some error. The bidder with the highest bid is statistically likely to be the one who overestimated the value the most, so winning the auction is itself bad news about how much you overpaid. Rational bidders correct for this by shading their bids below their naive estimate, and the effect grows stronger as the number of competing bidders increases, since the maximum of many noisy estimates is increasingly likely to be an overestimate.