Designing Weighted and Directed Networks Under Complementarities

Abstract

We study the problem of a planner designing a weighted and directed network to achieve her objectives subject to an organizational resource constraint. The network determines the complementarities between agents and, hence, their equilibrium effort. The planner's objective function can be convex to capture efficiency objectives or strictly concave to capture egalitarian concerns. We show that all optimal networks are generalized nested split graphs (GNSGs) that exhibit a `link-dominance' ordering among agents. The concept of GNSGs generalizes the previous notion of a nested split graph defined among unweighted and undirected networks to weighted and directed networks. Under a wide range of conditions, optimal networks must be hierarchical so that some agent is more influential and exerts strictly higher effort than others. This situation occurs even if agents are ex ante identical and the planner has egalitarian concerns. In a noncooperative network formation game, we show that all decentralized equilibrium networks are inefficient GNSGs.