We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by [PODC'03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the SUM version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of 3 2 on the approximation ratio. For the MAX version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in 6 5-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the MAX version on general networks in GE. This result implies a locality gap of @math for the metric min-max facility location problem, where n is the number of clients.
Greedy navigability is a central issue in the theory of networks. However, the exogenous nature of network models do not allow for describing how greedy routable-networks emerge in reality. In turn, network formation games focus on the very emergence proess, but the applied shortest-path based cost functions exclude navigational aspects. This paper takes a frst step towards incorporating both emergence (missing in algorithmic network models) and greedy navigability (missing in network formation games) into a single framework, and proposes the Greedy Network Formation Game. Our first contribution is the game definition, where we assume a hidden metric space underneath the network, and, instead of usual shortest path metric, we use the length of greedy paths as the measure of communiation cost between players. Our main finding is that greedy-routable small worlds do not emerge on constant dimensional Eulidean grids. This simply means that the emergence of topologies on which w eunderstood the priniples of greedy forwarding cannot be explained endogenously. We also present a very brief outlook on how the situation hanges in the hyperbolic space.