Truthful multi-unit procurements with budgets

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 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 polynomialtime mechanism that gives a 4(1 +ln n)-approximation for procurement games with 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 sub-additive valuations, we construct a universally truthful budget-feasible mechanism that gives an O(log2 n/log log n)-approximation in polynomial time with a demand oracle.