Almost Budget Balanced Mechanisms with Scalar Bids For Allocation of a Divisible Good

TL;DR

This paper introduces a mechanism for allocating divisible goods among strategic agents using scalar bids, aiming for near budget balance by returning rebates, and provides a convex optimization approach for designing these rebates.

Contribution

It proposes a novel rebate-based mechanism within the SSVCG framework that achieves near budget balance while maintaining efficiency and voluntary participation.

Findings

Mechanism achieves near budget balance with rebates.

Convex optimization used to design rebate functions.

Finite sample method approximates the solution effectively.

Abstract

This paper is about allocation of an infinitely divisible good to several rational and strategic agents. The allocation is done by a social planner who has limited information because the agents' valuation functions are taken to be private information known only to the respective agents. We allow only a scalar signal, called a bid, from each agent to the social planner. Yang and Hajek [Jour. on Selected Areas in Comm., 2007] as well as Johari and Tsitsiklis [Jour. of Oper. Res., 2009] proposed a scalar strategy Vickrey-Clarke-Groves (SSVCG) mechanism with efficient Nash equilibria. We consider a setting where the social planner desires minimal budget surplus. Example situations include fair sharing of Internet resources and auctioning of certain public goods where revenue maximization is not a consideration. Under the SSVCG framework, we propose a mechanism that is efficient and comes close to budget balance by returning much of the payments back to the agents in the form of rebates. We identify a design criterion for {\em almost budget balance}, impose feasibility and voluntary participation constraints, simplify the constraints, and arrive at a convex optimization problem to identify the parameters of the rebate functions. The convex optimization problem has a linear objective function and a continuum of linear constraints. We propose a solution method that involves a finite number of constraints, and identify the number of samples sufficient for a good approximation.