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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.

A reserve price is the seller's minimum acceptable price. In an English auction it changes both the sale rule and the payment rule: the highest valuation must exceed the reserve and the winner pays at least the reserve. Mechanisms with different reserves generally have different allocation probabilities, so revenue equivalence need not equate their expected revenues.

For two independent uniform valuations on $[0,1]$, optimal posted price revenue is $2/(3\sqrt{3})$, attained at $p=1/\sqrt{3}$. An English auction without reserve earns the expected minimum, $1/3$. With reserve $r\in [0,1]$, revenue is $r\mathrm{Pr}(\text{exactly one valuation exceeds }r)$ plus the expected smaller valuation when both exceed $r$, namely $1/3+r^2-4r^3/3$. It is maximized at $r=1/2$, giving $5/12$. Different reserve-dependent allocation probabilities explain why this differs from the no-reserve revenue without violating revenue equivalence.