  We study mechanism design for nonexcludable and excludable binary public
project problems. We aim to maximize the expected number of consumers and the
expected social welfare. For the nonexcludable public project model, we
identify a sufficient condition on the prior distribution for the conservative
equal costs mechanism to be the optimal strategy-proof and individually
rational mechanism. For general distributions, we propose a dynamic program
that solves for the optimal mechanism. For the excludable public project model,
we identify a similar sufficient condition for the serial cost sharing
mechanism to be optimal for $2$ and $3$ agents. We derive a numerical upper
bound. Experiments show that for several common distributions, the serial cost
sharing mechanism is close to optimality.
  The serial cost sharing mechanism is not optimal in general. We design better
performing mechanisms via neural networks. Our approach involves several
technical innovations that can be applied to mechanism design in general. We
interpret the mechanisms as price-oriented rationing-free (PORF) mechanisms,
which enables us to move the mechanism's complex (e.g., iterative) decision
making off the network, to a separate program. We feed the prior distribution's
analytical form into the cost function to provide quality gradients for
training. We use supervision to manual mechanisms as a systematic way for
initialization. Our approach of "supervision and then gradient descent" is
effective for improving manual mechanisms' performances. It is also effective
for fixing constraint violations for heuristic-based mechanisms that are
infeasible.
