Investment and competition with positive externalities in open networks

A key feature of many information services is that they exhibit positive externalities or network effects, i.e., the value of the service increases with the number of users. Here, we consider an “open” network, in which a given service may be offered by multiple providers and the positive externality depends on the total number of customers served by all providers. Providers compete for customers and may also invest in their networks to improve the externality experienced by their own customers. We model this as a two-stage investment and competition game, where after investing, firms compete via Bertrand competition. In both the socially optimal and the monopoly outcomes, under mild conditions, the firm either does not invest or invests at the maximum level. In addition, a monopolist always underinvests. Interestingly, we find that with competition, there can be at most one firm investing.