This paper examines directed networks in which the payoff of a player depends on the total number links formed by her and the other players. After showing that these networks with global spillovers may not always have Nash equilibria in pure strategies, we introduce two additional properties for the payoff function. The first called increasing (or decreasing) difference property states that player i’s payoff increases (decreases) as the number of links between the other n - 1 players increases. The second condition called the strict smaller midpoint property imposes a monotonicity restriction on the payoff function. We show that pure strategy Nash networks always exist under both conditions. The paper then characterizes these Nash equilibria showing that symmetric networks play a crucial role.
