Externalities effects in a network game.

Question

We have a simple game on network and it is given a payoff function: $$u_i(s) = s_i (-c_i + \sum_{j \in N_i(G)} s_j)$$ where $s_i \in \{0,1,2\}, s_i $ is a strategy of player $i$ and $c_i$ is a $i$ player's cost. $N_i$ is a set of neighbours of $i$. Is it a game with positive/negative externalities effect? Is this game complements or/and substitutes strategic? Why?

Answer 1

A game features strategic complementarity if \begin{equation} \frac{\partial ^2}{\partial s_i\,\partial s_j}u_i(s)>0\quad\text{for any $j\ne i$}. \end{equation} In your case, the cross partial derivative evaluates to $1>0$. Therefore it is a game of strategic complementarity.