Hierarchical Network Formation Games
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Classical {\em network-formation games\/} (NFGs) are played on directed graphs, and are used in network design and analysis. Edges in the network are associated with costs and players have reachability objectives, which they try to fulfill at a minimal cost. When several players use the same edge, they share its cost. The theoretical and practical aspects of NFGs have been extensively studied and are well understood. All studies of NFGs, however, consider an {\em explicit\/} representation of the network. In practice, networks are often built in a {\em hierarchical\/} manner. Technically, some of the vertices in the network are {\em boxes}, associated with nested sub-networks, where a sub-network may be ``called" by several boxes in the network. This makes hierarchical networks exponentially more succinct than traditional ``flat'' networks.
We introduce {\em hierarchical network formation games\/} (HNFGs) and study theoretical and practical aspects of the hierarchical setting. Different applications call for different cost-sharing mechanisms, which define how edge-formation costs are shared by their users. Indeed, in some applications, cost sharing should refer to the flat expansion of the network and in some it should take into account the hierarchical structure of the network.
We study properties of HNFGs like stability and equilibrium inefficiency in the different mechanisms. We also study computational aspects of HNFGs, where the principal question is whether their exponential succinctness with respect to NFGs leads to an exponential increase in the complexity of reasoning about them.
This question is analogous to research done in the formal-verification community about the ability to model-check hierarchical systems in their succinct presentation.
We show that the picture is diverse and depends on the mechanism applied.