This paper considers incentives to provide goods that are partially excludable along social links. Individuals face a capacity constraint in that, conditional upon providing, they may nominate only a subset of neighbours as co-beneficiaries. Our model has two typically incompatible ingredients: (i) a graphical game (individuals decide how much of the good to provide), and (ii) graph formation (individuals decide which subset of neighbours to nominate as co-beneficiaries). For any capacity constraints and any graph, we show the existence of specialised pure strategy Nash equilibria - those in which some individuals (the Drivers, D) contribute while the remaining individuals (the Passengers, P) free ride. The proof is constructive and corresponds to showing, for a given capacity, the existence of a new kind of spanning bipartite subgraph, a DP-subgraph, with partite sets D and P. We consider how the number of Drivers in equilibrium changes as the capacity constraints are relaxed and show a weak monotonicity result. Finally, we introduce dynamics and show that only specialised equilibria are stable against individuals unilaterally changing their provision level.
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