Welfare-Preserving $\varepsilon$-BIC to BIC Transformation with Negligible Revenue Loss

In this paper, we investigate the problem of transforming an $\varepsilon$-BIC mechanism into an exactly BIC mechanism without loss of social welfare, working in a general mechanism design setting. We can transform any $\varepsilon$-BIC mechanism to a BIC mechanism with no loss of social welfare and with additive and negligible revenue loss. We show that the revenue loss bound is tight given the requirement to maintain social welfare. This is the first $\varepsilon$-BIC to BIC transformation that preserves welfare and provides negligible revenue loss. 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, CaiZhao17, Rubinstein18}. The revenue loss achieved by our transformation is incomparable to these earlier approaches and can sometimes be significantly less. We also analyze $\varepsilon$-expected ex-post IC ($\varepsilon$-EEIC) mechanisms~\citep{DuettingFJLLP12}, and provide a welfare-preserving transformation with the same revenue loss guarantee for the special case of uniform type distributions. We give applications of our methods to both linear-programming based and machine-learning based methods of automated mechanism design. We also show the impossibility of welfare-preserving, $\varepsilon$-EEIC to BIC transformations with negligible loss of revenue for non-uniform distributions.