Public Goods in Networks: Comparative Statics Results

We consider public goods games played on a potentially non-symmetric network and provide comparative statics results on individual and aggregate contributions, as well as on the effect of transfers between players. We show that, contrary to the case of the complete and symmetric network, a positive shock on a player can have adverse consequences. First, it could actually decrease this player's contribution, unless the interaction matrix is a P-matrix. Second, a positive shock on a contributing player increases aggregate contributions, but a positive shock on a non-contributing player will decrease aggregate contributions, even if the player who benefited from the positive shock increases his own contribution. In each case we provide simple conditions to determine whether a positive shock will have positive or negative consequences on contributions, by looking at the unconstrained solution of an alternative, associated game. The sign of the coordinates of this solution determines the effect of a shock. With this in hand, we further show that the aggregate neutrality result of Andreoni [1990] regarding transfers between players generally does not hold on non-symmetric networks and provide conditions for it to hold. Finally, as an application of previous results, we consider introducing agents that follow Kantian moral principles and show that, depending on their position in the network, the presence of Kantian agents can, counter-intuitively, lead to a decrease in aggregate contributions.