A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good

Abstract

Abstract-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 efficient 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 collecting payments according to the celebrated VCG mechanism, and then returning as much of the collected money as rebates. Our focus on linear rebate functions allows for easy implementation. The resulting convex optimization problem is solved via relaxation to a randomized linear programming problem, for which several efficient solvers exist. This relaxation is enabled by constraint sampling. Keeping practitioners in mind, we identify the number of samples that assures a desired level of "near-feasibility" with the desired confidence level. Our methodology will occasionally require subsidy from outside the system. We however demonstrate via simulation that, if the mechanism is repeated several times over independent instances, then past surplus can support the sub...

Almost budget balanced mechanisms with scalar bids for allocation of a divisible good

“…We therefore adopt Moulin's proposal of minimizing (9). Chorppath et al [5] studied exactly this proposal in the divisible goods setting, but in the simpler restricted VCG setting of footnote 3. While the Moulin proposal is defensible for that setting, it has the drawback in our SSVCG setting that σ(θ) is not known to the social planner.…”

“…where (15) is the same as (4), and (16) is the same as (6). This choice puts us in the optimization framework of Chorppath et al [5], except that we have not yet shown how to resolve the issue of dependence of (16) on the private information v i . We now highlight two important differences between our work and that of Yang and Hajek [25].…”

“…Chorppath et al [5], following Calafiore and Campi [3], studied a random sampling of constraints and provided a bound on the number of samples needed to satisfy (29) with high probability. Here we take a deterministic approach.…”

“…Voluntary participation of agents, i.e., agents being better off by participating in the mechanism, is easily verified. Furthermore, knowledge of the valuation functions could be exploited in defining a criterion for almost budget balance, as is done in Moulin [20] and Chorppath et al [5]. The extension of the almost budget balance notion to the SSVCG setting, however, is not straightforward.…”

Adversarial Behavior in Network Games

“…Consider an auction mechanism in the context of a wireless network and uplink power control setting [22,25] where due to the interference coupling the resource sharing is inherently competitive. Let the user utilities be taken as U i (γ i (Q(x)) and the individual power levels,…”