Truthful Multi-unit Procurements with Budgets

TL;DR

This paper investigates truthful procurement mechanisms with budgets, establishing bounds for approximation ratios in various valuation settings, including bounded knapsack and sub-additive valuations, highlighting gaps between optimization and mechanism design.

Contribution

It proves a lower bound for truthful mechanisms in bounded knapsack, and provides near-optimal mechanisms for concave additive and sub-additive valuations.

Findings

No universal truthful mechanism can approximate bounded knapsack within ln n.

A polynomial-time mechanism achieves a 4(1+ln n)-approximation for concave additive valuations.

A truthful mechanism attains an O(log^2 n / log log n)-approximation for sub-additive valuations.

Abstract

We study procurement games where each seller supplies multiple units of his item, with a cost per unit known only to him. The buyer can purchase any number of units from each seller, values different combinations of the items differently, and has a budget for his total payment. For a special class of procurement games, the {\em bounded knapsack} problem, we show that no universally truthful budget-feasible mechanism can approximate the optimal value of the buyer within $\ln n$, where $n$ is the total number of units of all items available. We then construct a polynomial-time mechanism that gives a $4(1+\ln n)$-approximation for procurement games with {\em concave additive valuations}, which include bounded knapsack as a special case. Our mechanism is thus optimal up to a constant factor. Moreover, for the bounded knapsack problem, given the well-known FPTAS, our results imply there is a provable gap between the optimization domain and the mechanism design domain. Finally, for procurement games with {\em sub-additive valuations}, we construct a universally truthful budget-feasible mechanism that gives an $O(\frac{\log^2 n}{\log \log n})$-approximation in polynomial time with a demand oracle.