Private provision of a discrete public good with uncertain cost

It is known that a discrete public good is efficiently provided in the subset of 'undominated equilibria' (those not Pareto dominated within the set of Nash equilibria). We make the cost of the discrete public good uncertain at the time the contribution game is played. This can lead to strikingly different results. Often, the public good is underprovided in any Nash equilibrium and there is a unique undominated equilibrium. These results hold for some distributions when there is arbitrarily little uncertainty and always when there is enough uncertainty.