Nonhierarchical competition between species has been proposed as a potential mechanism for biodiversity maintenance, but theoretical and empirical research has thus far concentrated on systems composed of relatively few species. Here we develop a theory of biodiversity based on a network representation of competition for systems with large numbers of competitors. All species pairs are connected by an arrow from the inferior to the superior. Using game theory, we show how the equilibrium density of all species can be derived from the structure of the network. We show that when species are limited by multiple factors, the coexistence of a large number of species is the most probable outcome and that habitat heterogeneity interacts with network structure to favor diversity.