An Equilibrium Concept for Multi-Network Formation Games

A multi-network, or multi-layer network, is a set of nodes that can be connected via multiple types of links. In this paper, I first define a notion of pairwise stability for multi-networks that is analogous to pairwise stability of a single network. When the utility of each agent derived from the multi-layer network is additively separable across networks, then pairwise stability of all individual networks implies pairwise stability of the multi-network. This notion of pairwise stability needs to be refined when there exists spillovers across network types, in other words, when being connected in one way affects the utility derived from other types of connections. The traditional notion of pairwise stability does not allow for the simultaneous addition or removal of multiple link types. As a result, a multi-layer network may be pairwise stable, even if two agents could rearrange the links between them and achieve a Pareto improvement. I define a new notion of network stability, multi-link pairwise stability, which captures this potential for gains between pairs of agents. Finally, I illustrate the implications of link type complementarity for efficiency and stability in a symmetric connections model. The insight arising from this analysis is that link type complementarity generates multi-layer network configurations that are efficient and stable when the same configurations of individual networks without complementarities are not