= Revenue comparison for two uniform private values
For two independent uniform valuations on $[0,1]$, optimal <posted price> revenue is $2/(3\sqrt3)$, attained at $p=1/\sqrt3$. An <English auction> without reserve earns the expected minimum, $1/3$. With reserve $r\in[0,1]$, revenue is $r\Pr(\text{exactly one valuation exceeds }r)$ 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>.
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