Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 40 / 5

A buyer accepts a posted price precisely when their valuation exceeds it; equality has probability zero. The independent uniform valuations therefore give probability $p^2$ that neither accepts, for $0\le p\le 1$. The seller's expected revenue isThe interior critical point is $p=1/\sqrt{3}$, with negative second derivative; the endpoint revenues are zero. Prices above one make no sale and nonpositive prices cannot improve on this positive optimum. HenceWhich willing buyer is served if both accept does not affect this revenue.

In an ideal continuous-price English auction, staying active until one's private valuation is optimal: dropping earlier can forgo positive surplus, while staying later risks paying more than the car is worth. The highest-valued buyer therefore wins, paying the price at which the other buyer drops out, namely $\min (V_1,V_2)$.

For $0\le t\le 1$, independence gives $\mathrm{Pr}(\min (V_1,V_2)>t)=(1-t{)}^2$. Integrating this tail probability gives“Pays his bid” in this ascending auction does not mean that the winning buyer pays his full valuation; the competing buyer's dropout determines the winning price.

With reserve price $r\in [0,1]$, an English auction makes no sale if both valuations are below $r$. If exactly one exceeds it, the price is $r$; if both exceed it, the price is their minimum. ThusThe first term accounts for one qualifying buyer, and the integral uses the density $2(1-t)$ of the smaller valuation. Differentiation gives $R_c^{\prime }(r)=2r(1-2r)$. It increases up to $r=1/2$ and decreases afterwards, while the endpoints give $1/3$ and zero. ThereforeThe strict sale condition at the reserve changes only a probability-zero event.

The resulting revenue comparison for two uniform private values isThus the optimally reserved ascending auction is best, followed by the optimal posted price, followed by the ascending auction without a reserve. These values are in the normalized monetary units used for the valuations.

The revenue equivalence theorem for symmetric independent private values applies to mechanisms with the same allocation probabilities at every type and the same expected utility for the lowest type, under the usual incentive and risk-neutrality assumptions. The reserved and unreserved auctions have different allocation rules. Without a reserve, a bidder of valuation $v$ wins with probability $G(v)=v$. With reserve $r$, that probability is zero for $v<r$ and $v$ for $v>r$. In particular the car is withheld when even the highest valuation is below $r$.

The interim payment identity makes the difference explicit. With lowest-type utility zero, the ordinary auction has expected payment $m(v)=v^2/2$. The reserved auction hasSumming the two bidders' ex ante payments gives $2{\int }_r^1{m}_r(v)dv=1/3+r^2-4r^3/3$, exactly the revenue calculated above. Hence there is no conflict with revenue equivalence: its equal-allocation hypothesis does not hold.