Group size and collective action in a binary contribution game

We consider the private provision of a public good with non-refundable binary contributions. A fixed amount of the good is provided if and only if the number of contributors reaches an exogenous threshold. The threshold, the group size, and the identical cost of contributing to the public good are common knowledge. Our main result shows that the maximal probability of reaching the threshold (and thereby obtaining the public good) which can be supported in a symmetric equilibrium of this participation game is decreasing in group size. This generalizes a well-known result for the volunteer’s dilemma – in which the threshold is one – to arbitrary thresholds and thereby confirms a conjecture by Olson for the class of participation games under consideration. Further results characterize the limit when group size goes to infinity and provide conditions under which the expected number of contributors is decreasing or increasing in group size