Network Formation under Heterogeneous Costs: The Multiple Group Model

This discussion paper resulted in a publication in the International Game Theory Review (2007). Vol. 9(4), pp. 599-635. It is widely recognized that the shape of networks influences both individual and aggregate behavior. This raises the question which types of networks are likely to arise. Our paper fits into the literature that addresses this question. The paper continues on the work of Galeotti, Goyal and Kamphorst (2003). In their paper they investigate a model of network formation, where players are divided into groups. The costs of a link between any pair of players are increasing in the distance between the groups that these players belong to. They apply the solution concept of minimal curb sets to the case of two groups. In this paper we give a full characterization of the networks induced by a minimal curb set for any number of groups. To do so, we show that in our multiple group model each minimal curb set is a so-called super-tight curb set, that is a minimal curb set satisfying the condition that in each state of the set every player has the same best reply. From the proof it follows that every recurrent class of an unperturbed best reply dynamics is a minimal (super-tight) curb set and reversely. From this result we obtain the characterization of the types of networks that may arise in a minimal curb set. We show that such networks may have features that can not occur in networks with at most two groups. Nevertheless, local centrality and center-sponsorship are still important characteristics of the networks in minimal curb sets.