  Very recently, Hartline and Lucier studied single-parameter mechanism design
problems in the Bayesian setting. They proposed a black-box reduction that
converted Bayesian approximation algorithms into Bayesian-Incentive-Compatible
(BIC) mechanisms while preserving social welfare. It remains a major open
question if one can find similar reduction in the more important
multi-parameter setting. In this paper, we give positive answer to this
question when the prior distribution has finite and small support. We propose a
black-box reduction for designing BIC multi-parameter mechanisms. The reduction
converts any algorithm into an eps-BIC mechanism with only marginal loss in
social welfare. As a result, for combinatorial auctions with sub-additive
agents we get an eps-BIC mechanism that achieves constant approximation.
