Public Goods in Networks with Constraints on Sharing

This paper considers incentives to provide goods that are partially excludable along social links. We introduce a model in which each individual in a networked society makes a two-pronged decision: (i) decide how much of the good to provide, and (ii) decide which subset of neighbours to nominate as co-beneficiaries. An outcome specifies an endogenous subnetwork generated by nominations and a public goods game occurring over the realised subnetwork. We show the existence of specialised pure strategy Nash equilibria: those in which some individuals (the Drivers) contribute while the remaining individuals (the Passengers) free ride. We then consider how the set of efficient specialised equilibria vary as the constraints on sharing are relaxed and we show a monotonicity result. Finally, we introduce dynamics and show that only specialised equilibria can be stable against individuals unilaterally changing their provision level

Edge-decompositions of graphs with high minimum degree

A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here a graph $G$ is $F$-divisible if $e(F)$ divides $e(G)$ and the greatest common divisor of the degrees of $F$ divides the greatest common divisor of the degrees of $G$, and $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$. We extend this result to graphs $G$ which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large $K_3$-divisible graph of minimum degree at least $9n/10+o(n)$ has a $K_3$-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large $K_3$-divisible graph with minimum degree at least $3n/4$ has a $K_3$-decomposition. We also obtain the asymptotically correct minimum degree thresholds of $2n/3 +o(n)$ for the existence of a $C_4$-decomposition, and of $n/2+o(n)$ for the existence of a $C_{2\ell}$-decomposition, where $\ell\ge 3$. Our main contribution is a general `iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every $K_3$-divisible graph with minimum degree at least $3n/4+o(n)$ has an approximate $K_3$-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement