  We study Bayesian mechanism design problems in settings where agents have
budgets. Specifically, an agent's utility for an outcome is given by his value
for the outcome minus any payment he makes to the mechanism, as long as the
payment is below his budget, and is negative infinity otherwise. This
discontinuity in the utility function presents a significant challenge in the
design of good mechanisms, and classical "unconstrained" mechanisms fail to
work in settings with budgets. The goal of this paper is to develop general
reductions from budget-constrained Bayesian MD to unconstrained Bayesian MD
with small loss in performance. We consider this question in the context of the
two most well-studied objectives in mechanism design---social welfare and
revenue---and present constant factor approximations in a number of settings.
Some of our results extend to settings where budgets are private and agents
need to be incentivized to reveal them truthfully.
