Single-parameter mechanism

In a single-parameter mechanism, player $i$ has one private scalar value $v_i$, receives an allocation amount $x_i$, and has quasilinear utility $v_i{x}_i-p_i$. The allocation-payment relation is governed by incentive compatibility and the interim payment identity.

For an absolutely continuous valuation distribution with density $f(v)>0$, its virtual valuation is $v-(1-F(v))/f(v)$. The virtual-surplus revenue identity converts expected incentive-compatible payments into expected allocations weighted by virtual valuations. A regular prior makes this quantity nondecreasing.

Virtual surplus weights allocation amounts by the corresponding virtual valuations. Under the hypotheses of the virtual-surplus revenue identity, maximizing it subject to implementability and feasibility yields a revenue-optimal mechanism when lowest-type utilities can be normalized to zero.

For an independent private values model, let $X_i(v)$ and $P_i(v)$ be interim allocation and payment. The interim payment identity gives $P_i(v)=v{X}_i(v)-U_i({\underline{v}}_i)-{\int }_{{\underline{v}}_i}^v{X}_i(t)dt$. Applying Fubini's theorem to reverse the order of integration yields $\mathbb{E}{\int }_{{\underline{v}}_i}^{V_i}{X}_i(t)dt=\int X_i(t)(1-F_i(t))dt$. Subtracting this term from $\int v{X}_i(v)f_i(v)dv$ proves the identity. Interim individual rationality bounds the lowest-type utilities below by zero.

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 $\max \{0,n/2-{\sum }_{j\ne i}{v}_j\}$.

A valuation distribution is regular when its virtual valuation is a nondecreasing function of value. This is an auction-theory condition, not the unrelated regularity notions used elsewhere in mathematics.