Lowest-price auction

The highest bidder wins a single item but pays the lowest submitted bid, while losers pay nothing. This differs from a lowest unique bid auction. With at least two bidders using increasing strategies, the winning bidder pays the smallest opponent bid.

// Target: mathematical-optimization.bigb

For $n\ge 2$ 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 $0\le z\le 1$, wins with probability $z^{n-1}$ and has expected payment $(n-1)z^n/n$. Its expected profit is $v{z}^{n-1}-(n-1)z^n/n$, 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