Equilibrium existence and uniqueness in network games with additive preferences

•A network game is analyzed through a complementarity problem.•We consider concave additive preferences that encompass the quasi-linear ones.•Equilibrium uniqueness is established with a P-matrix.•Our results extend previous findings related to network sparsity.

A directed network game of imperfect strategic substitutes with heterogeneous players is analyzed. We consider concave additive separable utility functions that encompass the quasi-linear ones. It is found that pure strategy Nash equilibria verify a non-linear complementarity problem. By requiring appropriate concavity in the utility functions, the existence of an equilibrium point is shown and equilibrium uniqueness is established with a P-matrix. For this reason, it appears that previous findings on network structure and sparsity hold for many more games.