➩ Objective: \(\displaystyle\max_{x, p} f\left(x, p\right)\)

➩ Incentive Compatibility: \(R\left(x\left(t\right); t\right) - p\left(t\right) \geq R\left(x\left(\tau\right); t\right) - p\left(\tau\right)\) for every \(t, \tau \in T\).

➩ Individual Rationality: \(R\left(x\left(t\right); t\right) - p\left(t\right) \geq 0\) for every \(t \in T\).

➩ Replace IC by FOC + SOC.

➩ Differentiate FOC and substitute into SOC to get \(\dfrac{\partial R}{\partial x \partial t} x'\left(t\right) \geq 0\).

➩ Assuming monotone allocation \(x'\left(t\right) \geq 0\), we have SCC.

➩ Given SCC, implementability iff \(x'\left(t\right) \geq 0\): differentiate \(R\left(x\left(\tau\right); t\right) - p\left(\tau\right)\) with respect to \(\tau\) and substitute in FOC of IC.

➩ Differentiate \(R\left(x\left(t\right); t\right) - p\left(t\right)\) with respect to \(t\), assume the derivative is larger than 0.

➩ Use FTC to write \(R\left(x\left(t\right); t\right) - p\left(t\right)\) as an integral to compute \(p\left(t\right)\).

➩ Integrate \(p\left(t\right)\) over \(t\) and do integration by parts to get \(E\left[R\left(x; t\right) - \dfrac{\partial R\left(x; t\right)}{\partial t} \dfrac{1 - F\left(t\right)}{f\left(t\right)}\right]\).

➩ When \(R\left(x; t\right) = x t\), the virtual surplus for auctions \((t - \dfrac{1 - F\left(t\right)}{f\left(t\right)})\).