Auctions sell spectrum licenses, artwork, ad slots and government bonds. Their rules are games, and the ideas from Best Responses and Nash Equilibrium and Payoff Matrices and Dominant Strategies predict how people bid.
Key idea
In a second-price sealed-bid auction (highest bid wins, pays the second-highest bid), bidding your true value is a dominant strategy. In a first-price auction (winner pays her own bid), bidders shade their bids; with $n$ bidders whose values are independent and uniform, the equilibrium bid is $b(v) = \dfrac{n - 1}{n}v$. With values uniform on $[0, V]$, both formats give the seller the same expected revenue, $\dfrac{n - 1}{n + 1}V$ (revenue equivalence).
Why it works
In a second-price auction your bid decides only whether you win; the price is set by someone else's bid $B$. Bidding your value $v$ wins exactly when $B < v$, the cases where winning is profitable. Bidding less can only lose some of those profitable wins; bidding more can only add wins at prices above $v$. In a first-price auction your bid is your price, so you trade a lower chance of winning for a bigger margin, and you shade more when there are fewer rivals.
Worked example: Second price
Sealed bids: Ana \$120, Ben \$90, Chloe \$75. Ana's value is \$130.
- Ana wins and pays the second-highest bid, \$90.
- Her surplus: $130 - 90 = 40$. Had she bid \$100 or \$200, she would still win and still pay \$90.
Answer: Ana pays \$90.
Worked example: First price
Four bidders, values uniform on [0, 100]. What does a bidder with value 80 bid, and what is the seller's expected revenue?
- $b = \tfrac{3}{4}(80) = 60$.
- Expected revenue: $\dfrac{4 - 1}{4 + 1}(100) = 60$, the same as in a second-price auction, where the price is the expected second-highest value.
Answer: Bid 60; expected revenue 60.
Common mistake
Shading your bid in a second-price auction "to keep some surplus." Your surplus there is your value minus the second bid, which your own bid doesn't change. Shading only matters when it makes you lose an auction you would have won at a price below your value. Check: compare the outcome of bidding your value with any other bid for every possible highest rival bid; truthful bidding is never worse.
William Vickrey analyzed the second-price auction in 1961. Revenue equivalence needs strong assumptions (independent private values, risk-neutral bidders, symmetric bidders); when they fail, for example with risk-averse bidders, first-price auctions tend to raise more.
Try different formats in the Auctions chapter of Game Theory.