Efficient provision of a discrete public good

A discrete public good is characterized by a threshold production function: the good is provided if a certain number of contributors is reached. I consider a Bayesian framework where an individual's cost associated with his participation in the provision of a public good is private information. An efficient allocation mechanism is constructed when the outside option-available to individuals refusing to participate in the mechanism-is zero utility, and also for uniform uncertainty when the outside option is either voluntary contribution or forced contribution.