Before revision: In this paper, we consider infinite number of non atomic self-interested agentswith private valuation of a divisible good. We design a pricing mechanism that is easy to implement, is individually rational, weakly budget balanced and incentive compatible. In this mechanism, agentssend reports of their types , based on which, the designer solves a constrained optimization problem through Lagrange's mechanism.
After revision: Network utility maximization (NUM) is a general framework for designing distributed optimization algorithms for large-scale networks. An economic challenge arises in the presence of strategic agents' private information. Existing studies proposed (economic) mechanisms but largely neglected the issue of large-scale implementation. Specifically, they require certain modifications to the deployed algorithms, which may bring the significant cost. To tackle this challenge, we present the large-scale Vickery-Clark-Grove (VCG) Mechanism for NUM, with a simpler payment rule characterized by the shadow prices. The Large-Scale VCG Mechanism maximizes the network utility and achieves individual rationality and budget balance. With infinitely many agents, agents' truthful reports of their types , based on which, the designer solves a constrained optimization problem through Lagrange's mechanism.