Price Based Protocols For Fair Resource Allocation: Convergence Time Analysis and Extension to Leontief Utilities Ashish Goel * Stanford University Hamid Nazerzadeh † Stanford University Abstract We analyze several distributed, continuous time protocols for a fair allocation of bandwidths to flows in a network (or resources to agents). Our protocols converge to an alloca- tion which is a logarithmic approximation, simultaneously, to all canonical social welfare functions (i.e. functions which are symmetric, concave, and non-decreasing). These proto- cols can be started in an arbitrary state. While a similar protocol was known before, it only applied to the simple bandwidth allocation problem, and its stability and con- vergence time was not understood. In contrast, our proto- cols also apply to the more general case of Leontief utilities, where each user may place a different requirement on each resource. Further, we prove that our protocols converge in polynomial time. The best convergence time we prove is O(n log ncmaxamax c min a min ), where n is the number of agents in the network, cmax and cmin are the maximum and minimum ca- pacity of the links, and amax,amin are the largest and small- est Leontief coefficients, respectively. This time is achieved by a simple MIMD (multiplicative increase, multiplicative decrease) protocol which had not been studied before in this setting. We also identify combinatorial properties of these protocols that may be useful in proving stronger convergence bounds. The final allocations by our protocols are supported by usage-sensitive dual prices which are fair in the sense that they shield light users of a resource from the impact of heavy users. Thus our protocols can also be thought of as efficient distributed schemes for computing fair prices. 1 Introduction In many problems where a centralized planner needs to distribute a set of resources among a group of individu- als, the individual utility functions are known but there is no well defined notion of how to combine individual utilities into a single aggregate utility function (i.e. a * Department of Management Science and Engineering, and (by courtesy) Computer Science, Stanford University, CA. Research supported in part by NSF ITR grant 0428868, an NSF CAREER award. Email: ashishg@stanford.edu † Department of Management Science and Engineering, Stan- ford University, CA. Research supported in part by NSF ITR grant 0428868. Email: hamidnz@stanford.edu social welfare function). For instance, in most real-life settings, increasing efficiency (i.e. the sum of individ- ual utilities) is an important goal. At the same time, achieving fairness in the allocations to different individ- uals is also desirable. It has recently been shown that it is possible to simultaneously approximate a large class of social welfare functions [19, 14, 20, 2]. Efficient dis- tributed algorithms have been presented for cases where the individual utility functions can be modeled as the amount of flow allocated to that individual in a multi- commodity like problem, and price based protocols have been presented when the flow for an individual in this multi-commodity problem must use a single route fixed in advance [7]. In this paper, we extend the efficient distributed al- gorithm mentioned above to the case where the individ- uals have Leontief utilities. The distributed algorithm must be initialized to a special state and then proceed in a synchronized fashion. We then present two simple price based protocols which achieve the same equilib- rium as the distributed algorithm. These protocols can start with an arbitrary allocation, are stable around the equilibrium point, and converge quickly. The stabil- ity and convergence of price based protocols was left as an open problem even for the simple case of multi- commodity flows. Our price based protocols are simple and efficient enough to be implemented in a TCP-like fashion in computer networks. We believe that our re- sults will also be interesting in designing exchange mar- kets which lead to an approximately fair outcome, under all reasonable definitions of fairness. In order to describe our results in more detail and put them in context, we will first need to define and motivate the problem and also briefly survey recent related work. Problem description The problem of allocation of m resources among n agents with Leontief utilities is described by two positive matrices, denoted C and A. Matrix C =(c 1 , ··· ,c m ) indicates the amount of available resources. Let Y be an allocation of resources among the agents, i.e. y ij unit of