Group dominant networks and convexity

We study group dominant networks under convexity alone and in combination with either strategic complementarity or substitutability. Group dominant networks consists of one group of completely connected agents and a set of isolated agents. They arise frequently in the networks’ literature, most commonly in collaborative situations. We first provide two conditions that help us select from different equilibrium group dominant networks. They allow us to provide a characterization of equilibrium networks for both strategic complementarity and substitutability cases including identifying conditions for uniqueness. Next we observe that under strategic substitutability the equilibrium set may not be a convex interval, allowing for the possibility of ‘holes’ in it. We provide conditions to eliminate these holes and show that this approach can also help to simplify the process of identifying group dominant equilibria.