Auction Theory

Auctions as Games

An auction is a game where bidders compete to buy an item. Each bidder has a private valuation — the most they're willing to pay — and must decide how much to bid. Different auction formats produce different strategic incentives.

The four standard formats are: English (ascending price), Dutch (descending price), first-price sealed-bid(highest bid wins, pays their bid), and Vickrey (highest bid wins, pays the second-highest bid).

Revenue Equivalence

One of the most surprising results in auction theory: under standard assumptions (risk-neutral bidders, independent private values), all four formats generate the same expected revenue for the seller. Run thousands of trials and watch the revenue distributions converge.

Strategic Differences

• Vickrey (second-price): Truth-telling is a dominant strategy. You should always bid your true valuation, because you only pay the second-highest bid if you win.
• First-price: Bidders shade their bids below their valuations. The optimal bid in a Bayesian Nash equilibrium is (n-1)/n of your valuation with n bidders.
• English: Strategically equivalent to Vickrey — the winner pays just above the second-highest valuation.
• Dutch: Strategically equivalent to first-price — you must decide when to stop the clock, just like choosing a sealed bid.