English Auction: With 5 bidders and uniform values, expected revenue ≈ 67% of max value
With 5 bidders drawing values uniformly from [0, 100], the expected revenue in any standard auction is approximately (n-1)/(n+1) × 100 = 66.7. The Revenue Equivalence Theorem tells us that English, Dutch, first-price sealed, and Vickrey auctions all produce the same expected revenue under standard assumptions. The simulation confirms this with minor random variation.
The Science of Selling
Auctions have allocated resources for millennia — from Babylonian wife auctions to modern spectrum sales worth billions of dollars. Auction theory provides mathematical models of strategic bidding behavior, explaining why different auction formats exist and when each is optimal. The field earned William Vickrey (1996) and Roger Myerson (2007) Nobel Prizes in Economics.
Four Classic Auction Formats
The simulator implements four standard formats. The English auction (ascending price) is the most familiar — bidders openly raise prices until only one remains. The Dutch auction (descending price) starts high and drops until someone claims the item. First-price sealed-bid auctions have bidders submit private bids simultaneously; the highest bidder wins and pays their bid. Vickrey auctions (second-price sealed-bid) also use sealed bids but charge the winner only the second-highest price, incentivizing truthful bidding.
Revenue Equivalence
One of the most beautiful results in economics is the Revenue Equivalence Theorem: under standard assumptions (independent private values, risk-neutral bidders), all four auction formats generate identical expected revenue. Watch the simulation confirm this — despite radically different rules and strategies, the average selling prices converge across formats. More bidders means more competition and higher revenue, approaching the true maximum value.
Beyond Standard Theory
Revenue equivalence breaks down with risk aversion, asymmetric bidders, or correlated values. Risk-averse bidders bid more aggressively in first-price auctions, making them more profitable for sellers. Common-value settings (like oil lease auctions) introduce the winner's curse — the winner is the bidder who most overestimated the value. Adjust the parameters to explore how bidder count, value distribution, and format interact to shape outcomes.
FAQ
What is the Revenue Equivalence Theorem?
The Revenue Equivalence Theorem (Myerson, 1981; Riley & Samuelson, 1981) states that under standard conditions (independent private values, risk-neutral bidders, symmetric distributions), all standard auction formats yield the same expected revenue to the seller. This means English, Dutch, first-price sealed, and second-price sealed auctions are all equally profitable on average.
What is a Vickrey auction?
A Vickrey (second-price sealed-bid) auction is one where bidders submit sealed bids, the highest bidder wins, but pays the second-highest bid. William Vickrey proved in 1961 that the dominant strategy is to bid your true value, making it truthful and efficient. This mechanism is used by Google and Facebook for ad auctions.
How do bidders strategize in first-price auctions?
In first-price sealed-bid auctions, rational bidders shade their bids below their true values. With n bidders drawing values uniformly from [0,1], the optimal bid is v·(n-1)/n. More bidders means less shading (more aggressive bidding), as competition for winning increases.
What makes an auction efficient?
An auction is efficient when the item goes to the bidder who values it most. English auctions and Vickrey auctions are always efficient (in the independent private values setting) because bidders reveal or bid their true values. First-price auctions are also efficient in symmetric settings, as all bidders shade equally, preserving the ranking.