  In this paper, we provide a transform from an $\varepsilon$-BIC mechanism
into an exactly BIC mechanism without any loss of social welfare and with
additive and negligible revenue loss. This is the first $\varepsilon$-BIC to
BIC transformation that preserves welfare and provides negligible revenue loss.
The revenue loss bound is tight given the requirement to maintain social
welfare. Previous $\varepsilon$-BIC to BIC transformations preserve social
welfare but have no revenue guarantee~\citep{BeiHuang11}, or suffer welfare
loss while incurring a revenue loss with both a multiplicative and an additive
term, e.g.,~\citet{DasWeinberg12, Rubinstein18, Cai19}. The revenue loss
achieved by our transformation is incomparable to these earlier approaches and
can be significantly less. \newnew{Our approach is different from the previous
replica-surrogate matching methods and we directly make use of a directed and
weighted type graph (induced by the types' regret), one for each agent. The
transformation runs a \emph{fractional rotation step} and a \emph{payment
reducing step} iteratively to make the mechanism Bayesian incentive
compatible.} We also analyze $\varepsilon$-expected ex-post IC
($\varepsilon$-EEIC) mechanisms~\citep{DuettingFJLLP12}. We provide a
welfare-preserving transformation in this setting with the same revenue loss
guarantee for uniform type distributions and give an impossibility result for
non-uniform distributions. We apply the transform to linear-programming based
and machine-learning based methods of automated mechanism design.
