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The expectation of the seller's total receipts over the buyers' valuation types and any randomization in the sales mechanism. In an auction it is the sum of the bidders' expected payments; with a posted price it is price times sale probability. It does not subtract selling costs or the seller's opportunity cost of retaining the item, so it is revenue rather than profit.

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.

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.

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.