= Uniform lowest-price auction equilibrium
{title2=$b(v)=(n-1)v$}
For $n\geq2$ risk-neutral bidders with independent <uniform distributions> on $[0,1]$, the <lowest-price auction> has a symmetric <Bayesian Nash equilibrium> $b(v)=(n-1)v$. A deviation bidding $(n-1)z$, with $0\leq z\leq1$, wins with probability $z^{n-1}$ and has expected payment $(n-1)z^n/n$. Its expected profit is $vz^{n-1}-(n-1)z^n/n$, whose derivative is $(n-1)z^{n-2}(v-z)$, proving optimality at $z=v$. Bids above $n-1$ give the same outcome as $z=1$. The seller receives $(n-1)$ times the minimum of $n$ independent uniform valuations, with expected revenue $(n-1)/(n+1)$.
// Target: mathematical-optimization.bigb
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