A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good
IEEE Transactions on Automation Science and Engineering
We address the problem of allocating a single divisible good to a number of agents. The agents have concave valuation functions parameterized by a scalar type. The agents report only the type. The goal is to find allocatively efficient, strategy proof, nearly budget balanced mechanisms within the Groves class. Near budget balance is attained by returning as much of the received payments as rebates to agents. Two performance criteria are of interest: the maximum ratio of budget surplus to
... nt surplus, and the expected budget surplus, within the class of linear rebate functions. The goal is to minimize them. Assuming that the valuation functions are known, we show that both problems reduce to convex optimization problems, where the convex constraint sets are characterized by a continuum of half-plane constraints parameterized by the vector of reported types. We then propose a randomized relaxation of these problems by sampling constraints. The relaxed problem is a linear programming problem (LP). We then identify the number of samples needed for "near-feasibility" of the relaxed constraint set. Under some conditions on the valuation function, we show that value of the approximate LP is close to the optimal value. Simulation results show significant improvements of our proposed method over the Vickrey-Clarke-Groves (VCG) mechanism without rebates. In the special case of indivisible goods, the mechanisms in this paper fall back to those proposed by Moulin, by Guo and Conitzer, and by Gujar and Narahari, without any need for randomization. Extension of the proposed mechanisms to situations when the valuation functions are not known to the central planner are also discussed. Note to Practitioners-Our results will be useful in all resource allocation problems that involve gathering of information privately held by strategic users, where the utilities are any concave function of the allocations, and where the resource planner is not interested in maximizing revenue, but in efficient sharing of the resource. Such situations arise quite often in fair sharing of internet resources, fair sharing of funds across departments within the same parent organization, auctioning of public goods, etc. We study methods to achieve near budget balance by first Manuscript has been with the Department of Electrical Engineering, IIT Madras, where he is currently an Associate Professor. His current research interests are in resource allocation, adaptive transmission, code design, and information theory for multiterminal wireless communication systems. Rajesh Sundaresan (S'96-M'00-SM'06) received the B.Tech. degree in electronics and communication from the Indian Institute of Technology, Madras, the M.A. and Ph.D. degrees in electrical engineering from Princeton University, Princeton, NJ, in 1996 and 1999, respectively. From 1999 to 2005, he worked at Qualcomm Inc., Campbell, CA, on the design of communication algorithms for WCDMA and HSDPA modems. Since 2005, he has been with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore. His interests are in the areas of wireless communication networks and information theory.