Auction Formats
| Format | Winner pays | Dominant strategy |
|---|---|---|
| Second-price (Vickrey) | Second-highest bid | Bid true value $v_i$ |
| First-price | Own bid | Shade bid below $v_i$ |
| English (ascending) | Dropout price | Stay until price $= v_i$ |
| Dutch (descending) | Stopping price | Stop at optimal shade |
In the Vickrey auction, truthful bidding is a weakly dominant strategy: winning at the second price gives $v_i - b_{(2)} \geq 0$ regardless of others’ bids.
Revenue Equivalence Theorem
Under symmetric independent private values drawn from a common distribution $F$ on $[0,\bar{v}]$, any auction that (i) allocates to the highest type and (ii) gives a type-0 bidder zero expected surplus yields the same expected revenue to the seller. For $n$ bidders with values i.i.d. $F$, the equilibrium expected revenue equals:
\[\mathbb{E}[R] = n \int_0^{\bar{v}} v \, f(v) F(v)^{n-1} dv - \int_0^{\bar{v}} F(v)^{n-1} dv \cdot \text{(boundary term)}\]The first-price equilibrium bid for uniform $F$ on $[0,1]$ is $b^*(v) = \frac{n-1}{n} v$.
Myerson’s Optimal Auction
Myerson (1981) showed the revenue-maximising auction uses virtual valuations $\psi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$. The optimal rule allocates to the bidder with the highest non-negative virtual valuation and sets a reserve price $r^$ solving $\psi(r^) = 0$. For uniform $F$ on $[0,1]$, the optimal reserve is $r^* = 1/2$ regardless of the number of bidders.