On the Efficiency of Markets with Two-Sided Proportional Allocation Mechanisms

We analyze the performance of resource allocation mechanisms for markets in which there is competition amongst both consumers and suppliers (namely, two-sided markets). Specifically, we examine a natural generalization of both Kelly’s proportional allocation mechanism for demand-competitive markets [9] and Johari and Tsitsiklis’ proportional allocation mechanism for supply-competitive markets [7]. We first consider the case of a market for one divisible resource. Assuming that marginal costs are convex, we derive a tight bound on the price of anarchy of about 0.5887. This worst case bound is achieved when the demand-side of the market is highly competitive and the supply-side consists of a duopoly. As more firms enter the market, the price of anarchy improves to 0.64. In contrast, on the demand side, the price of anarchy improves when the number of consumers decreases, reaching a maximum of 0.7321 in a monopsony setting. When the marginal cost functions are concave, the above bound smoothly degrades to zero as the marginal costs tend to constants. For monomial cost functions of the form C(x) = cx 1 d , we show that the price of anarchy is Ω( 1 d2 ). We complement these guarantees by identifying a large class of two-sided single-parameter market-clearing mechanisms among which the proportional allocation mechanism uniquely achieves the optimal price of anarchy. We also prove that our worst case bounds extend to general multiresource markets, and in particular to bandwidth markets over arbitrary networks.