A Note on Existence of Nash Networks in One-way Flow Models”

The importance of networks is pervasive and now well documented. Among the growing litterature on the formation of networks, a few papers have explored the impact of heterogeneity, although this is a distinctive feature. Galeotti (2006, [3]) is one exception. The author introduces agents heterogeneity in one way flow models of network formation and characterize the architectures of Nash networks. However Nash networks do not alway exist. Heterogeneity in cost of forming links play a major role in this non existence. In recent paper Billand, Bravard and Sarangi (BBS, 2008, [1]) show that Nash networks do not always exist under heterogeneity of costs by pairs. Then in Proposition 3 (pg. 505), they provide a sufficient condition for the existence of Nash networks. Subsequently, by means of a counterexample Derks and Tennekes (2008, [2]) showed that the condition given in Proposition 3 is not sufficient for the existence of Nash networks. In this comment we provide an additional condition which ensures the existence of Nash networks. We also show that although this is a fairly strong condition, it still allows us to have models with non-trivial cost heterogeneity.