Revenue-optimal public-project auction ($x(v)=1_{\{{\sum }_i{\varphi }_i(v_i)\ge 0\}}$)

In a public-project single-parameter mechanism, all players receive the same binary allocation. For independent regular priors and voluntary participation with zero outside utility, maximizing virtual surplus means providing the project exactly when ${\sum }_i{\varphi }_i(v_i)\ge 0$. The allocation is monotone in each value, so critical-value payments implement it with dominant-strategy incentive compatibility and ex post individual rationality. For independent uniform values on $[0,1]$, the condition is ${\sum }_i{v}_i\ge n/2$, with winning payment $\max \{0,n/2-{\sum }_{j\ne i}{v}_j\}$.