Expected hammer price as the crowd grows, if rival values are spread anywhere up to yours: with N bidders the price lands near the second-highest value â about (Nâ1)/(N+1) of the top. Two bidders leave bargains; twenty leave almost none.
The one theorem you can feel: in an ascending auction, bidding your honest maximum and stopping there is a dominant strategy â no cleverness by anyone else can make deviating from it better for you. Everything that goes wrong in auction rooms is a failure to hold that line: auction fever, sunk-cost pull ("I've come this far"), and rivalry all push people past the number they wrote down in the calm. The price you pay is set by the runner-up, not by your maximum â so the maximum costs nothing to hold honestly and everything to abandon.
Traditional English auction strategy â your walk-away maximum bid, surplus at the next raise, expected hammer price by crowd size, and why bidding your true value and stopping is the dominant strategy.
Expected profit for every price you might stop the clock at. Stop early and you win often but earn little; hold out and the prize grows while your chances collapse. The glowing dot is the mathematical sweet spot â your value times (Nâ1)/N.
Why Dutch bidding is a nerve game and traditional bidding isn't: in the ascending room your best move never depends on reading anyone â bid to your value, stop. On the falling clock, the moment you'd happily press the button, waiting one more tick earns you more if nobody else presses first â so the optimal strategy shades below your true value, and by exactly how much depends on the crowd: with N bidders, stop at value à (Nâ1)/N. More rivals, less patience. Nervous about risk? Economists proved risk-averse bidders should press earlier than this â the formula assumes ice in your veins. One honest boundary: this is the classic textbook model (symmetric bidders, valuations spread evenly, everyone rational); real rooms bend it, but the shading logic survives.
Dutch auction bidding strategy â the optimal stopping price at value times (Nâ1)/N, win probability and expected surplus, the full profit curve, and why more bidders mean less patience.
Expected revenue against crowd size â one curve, because under the classic assumptions all four formats earn the seller exactly the same: $100 Ã (Nâ1)/(N+1). The formats differ in speed, drama, and cheat-resistance, never in expectation.
The Revenue Equivalence Theorem (Vickrey 1961, generalized by Myerson) is auction theory's crown jewel: with risk-neutral bidders holding independent private values from the same distribution, every standard auction format delivers the seller the same expected revenue. The falling Dutch clock and the shouting English room, the sealed envelope and the pay-the-second-price curiosity â all one number in expectation. William Vickrey's share of the 1996 Nobel rests on it. The fine print matters, though: risk aversion, correlated values, and collusion all break the tie, which is why real-world sellers still argue about formats â see the Winner's Curse tab for the biggest breaker.
The four classic auction formats compared â English, Dutch, first-price sealed and Vickrey â optimal strategy in each, and the Revenue Equivalence Theorem showing identical expected revenue.
The winner's expected overestimate as bidders multiply â spread à (Nâ1)/(N+1). With two bidders the winner is a little optimistic; with thirty, the winner is almost certainly the one whose estimate was most wrong. Beating many rivals is itself evidence you overshot.
Born in the oil patch: three Atlantic Richfield engineers â Capen, Clapp and Campbell â noticed in 1971 that companies winning offshore drilling leases kept earning dismal returns, and diagnosed the disease: when everyone estimates the same unknown value, the auction is won by the biggest overestimate. The cure is to bid as if you've already won â that is, as if you've just learned every rival's estimate was below yours â and shade accordingly. The model here (estimates spread evenly Âąs around the truth) is the honest teaching version; real corrections need real geology. The curse stalks IPOs, corporate takeovers, spectrum auctions, and every eBay category where resale value is common knowledge waiting to happen.
Winner's curse calculator â expected overpayment when bidding your estimate in common-value auctions, the break-even correction by crowd size and estimate spread, and the oil-lease origin story.
Why the formats coexist: rough typical paces on a log scale. The Dutch clock exists because tulips wilt â descending price needs exactly one button-press to close, which is how Aalsmeer moves tens of thousands of lots before breakfast.
The honesty box: the paces in the chart are rough, order-of-magnitude figures for context â real auctions vary enormously. The stories are the documented part: the flower clock, the 2004 IPO, the 1971 oil paper, the Praetorian Guard's terrible afternoon. Formats survive where their trade-offs fit: speed for perishables, transparency for art, sealed bids for contracts, and second-price mechanics wherever honest proxy bidding makes life simpler.
Auctions in the real world â the Aalsmeer flower clock, Google's 2004 Dutch-auction IPO, eBay's proxy bidding as a Vickrey auction, spectrum auctions and the 2020 Nobel, candle auctions, and the day the Roman Empire went under the hammer.