When does everyone genuinely contribute in the private provision of a local public good? We first introduce a monotonic condition to characterize the relationship between the structure of the network that underlie the noncooperative game of private provision of local public goods on the one hand, and the preferences of the agents on the other, showing that the monotonic condition is a sufficient and necessary condition of existence of a distributed Nash equilibrium (DNE) in which each agent exerts a positive amount of effort to provision of the public good (Theorem 1). We then study the number of equilibria, and, by using the monotonic condition, characterize the condition under which the DNE set is a singleton, a continuum, or null (Theorem 2). As it turns out, the structure of the network and the agents' preferences jointly shape the effort profile in the provision of local public goods. © 2012 Wiley Periodicals, Inc.