  In this paper, we consider the problem of network design on network games. We
study the conditions on the adjacency matrix of the underlying network to
design a game such that the Nash equilibrium coincides with the social optimum.
We provide the examples for linear quadratic games that satisfy this condition.
Furthermore, we identify conditions on properties of adjacency matrix that
provide a unique solution using variational inequality formulation, and verify
the robustness and continuity of the social cost under perturbations of the
network. Finally we comment on individual rationality and extension of our
results to large random networked games.
