We study a system where many identical customers use a single service. Each customer experiences both positive externalities (a positive "network effect") and negative externalities (a "congestion effect") from other customers using the service. Such a model arises frequently in practice: application services on wireless networks are a primary example. We characterize the social optimum, where a social planner determines the usage level of each customer. We also characterize the Nash equilibrium achieved when the usage levels are determined by the customers themselves, in their self-interest. We study the ratio of the welfare in Nash equilibrium to that in the social optimum. We demonstrate that there is an *optimal scale*, i.e., a number of customers at which this ratio is maximized; further, the optimal ratio is unity. We also show that this same optimal scale maximizes the Nash welfare of a single individual. We interpret our results in terms of club formation, and also discuss extensions to a dynamic pricing model with a single monopolistic service provider.

Bio Ramesh Johari is an Assistant Professor at Stanford University, with a full-time appointment in the Department of Management Science and Engineering, and courtesy appointments in the Departments of Computer Science and Electrical Engineering. He received an A.B. in Mathematics from Harvard (1998), a Certificate of Advanced Study in Mathematics from Cambridge (1999), and a Ph.D. in Electrical Engineering and Computer Science from MIT (2004). He is also the recipient of First Place in the INFORMS George E. Nicholson Student Paper Competition (2003), the George M. Sprowls Award for the best doctoral thesis in computer science at MIT (2004), honorable mention in the ACM Doctoral Dissertation Award competition (2005), an Okawa Foundation Research Grant (2005), and an NSF Career Award (2007). His current research interests include incentives and pricing in networked systems.