The weighted proportional allocation mechanism (original) (raw)

We consider a weighted proportional allocation of resources that allows providers to discriminate usage of resources by users. This framework is a generalization of well-known proportional allocation accommodating allocation of resources proportional to weighted bids or proportional to submitted bids but with weighted payments. We study a competition game where everyone is selfish: providers choose discrimination weights aiming at maximizing their individual revenues while users choose their bids aiming at maximizing their individual payoffs. We analyze revenue and social welfare of this game. We find that the revenue is lower bounded by k/(k + 1) times the revenue under standard price discrimination scheme, where a set of k users is excluded. For users with linear utility functions, we find that the social welfare is at least 1/(1 + 2/ √ 3) of the maximum social welfare (approx. 46%) and that this bound is tight. We extend the efficiency result to a broad class of utility functions and to multiple competing providers. We also describe an algorithm used by the provider to adjust the user discrimination weights without a prior knowledge of user utility functions and establish convergence to equilibrium points of our game. Our results show that, in many cases, weighted proportional sharing achieves competitive revenue and social welfare, despite the fact that everyone is selfish. The mechanism allows for resource constraints described by general polyhedrons, thus accommodating a variety of resources, including bandwidth of communication networks, systems of computing resources, and sponsored search ad slots. The Framework. In this paper, we consider a class of auctions that allows providers to discriminate among different users. Specifically, we are interested in auctions that are simple in terms of the information provided by users, and are easy to describe to users. We consider two natural instances of weighted proportional allocation: (1) weighted bid auction where the allocation to a user is proportional to the bid submitted by this user weighted with a discrimination weight that is selected by the provider, and the payment by the user is equal to his own bid, and (2) weighted payment auction where the allocation to a user is proportional the bid of this user and the payment is equal to the weighted bid, where the weight is selected by the provider. The weighted bid auction is a novel proposal while weighted payment auction was recently proposed by Ma et al [18]. As in the network pricing literature [20], we consider these allocation problems in the full information setting. The justification for this setting is the fact that in practice allocation auctions are run repeatedly and thus, providers can learn about the behavior and private information of users. As discussed in the beginning of this section, even in this setting there are several advantages of using proportional sharing-like auctions over fixed price schemes. Namely, both of the auctions that we consider are akin and natural generalizations of well-known proportional allocation (e.g. [13, 11, 8], see related work discussed later in this section). Thus, this class of mechanisms inherits many natural properties of the traditional proportional sharing rule making it easy and robust to implement in practice. First, these mechanisms are simple for bidders, they only need to know the total of others' bids. Second the allocation is a natural and continuous function of the bid vector, and, therefore, it can be robustly implemented in a distributed way (as will be shown later in our paper). From an engineering point of view, this is an important feature of practical allocation rules. For example, when users demands are inelastic (users' utilities are close to linear) proportional sharing-like mechanisms are much preferred to fixed price schemes. Another important reason that motivates us to study these weighted proportional rules is the fact that in settings where providers' goal is to maximize revenue, the weighted proportional sharing is preferred over the traditional proportional sharing. As will be shown later, while weighted proportional sharing always generates near-optimal revenue, the revenue of traditional proportional sharing provides no such guarantee, and in fact, can be arbitrarily bad. We study these allocation rules in general convex environments that capture many special cases of resource allocation problems such as the network bandwidth sharing, sponsored search, and scheduling of resources in cloud computing (see Figure 1 for an illustration). We provide a deeper discussion of these applications in Appendix A. We consider a provider that offers a resource to a set of users U = {1, 2,. .. , n} where n ≥ 1 (for the case of multiple providers, the auctions as described in the following are applied by each individual provider). We denote with x = (x 1 , x 2 ,. .. , x n) and q = (q 1 , q 2 ,. .. , q n) the vector of allocations and payments by users, respectively. The resource owned by the provider is allowed to be an arbitrary infinitely divisible resource with constraints specified by a convex set, say P ∈ IR n +. An allocation vector x is said to be feasible if x ∈ P. The provider discriminates users by assigning a discrimination weight C i ≥ 0, for every user i. Each user i, submits a bid w i ≥ 0. Our weighted bid auction determines the allocation and payment for each user as follows: