Local farsightedness in network formation

We propose the concept of local-$k$ farsighted consistent network for analyzing network formation games where players only consider a limited number of feasible networks. A network $g$ is said to be local-$k$ farsightedly consistent if, for any network $g\prime$ within the distance-$k$ neighborhood of $g$, either $g$ is not defeated by $g\prime$, or $g$ defeats $g\prime$. We show that if the utility function is (componentwise) egalitarian or satisfies reversibility or excludes externalities across components, then local-$k$ farsightedness is more likely to be a good proxy for what would happen when players have full knowledge of all feasible networks.