The price of anarchy of serial, average and incremental cost sharing

We compute the price of anarchy (PoA) of three familiar demand games, i.e., the smallest ratio of the equilibrium to efficient surplus, over all convex preferences quasi-linear in money. For any convex cost, the PoA is at least $$\frac{1}{n}$$ in the average and serial games, where n is the number of users. It is zero in the incremental game for piecewise linear cost functions. With quadratic costs, the PoA of the serial game is $$\theta (\frac{1}{\log n})$$ , and $$\theta (\frac{1}{n})$$ for the average and incremental games. This generalizes if the marginal cost is convex or concave, and its elasticity is bounded.