Interim payment identity ($m(\theta )=\theta G(\theta )-{\int }_0^{\theta }G(t)dt-u(0)$)

For a risk-neutral single-parameter bidder with incentive-compatible type reports, interim utility satisfies $u^{\prime }(\theta )=G(\theta )$ wherever the winning probability is continuous. Consequently expected payment is determined by allocation probabilities and the utility of the lowest type. The common normalization $u(0)=0$ must be justified, not obtained from allocation alone.

Two auctions with the same interim allocation probabilities and the same lowest-type utilities have the same interim expected payments under the usual risk-neutral single-parameter incentive conditions. Equal realized payments are not required. This conclusion follows directly from the interim payment identity.