Truthful Multi-unit Procurements with Budgets

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 lnn, where n is the total number of units of all items available. We then construct a polynomial-time mechanism that gives a 4(1+ lnn)-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( log 2 n log logn )-approximation in polynomial time with a demand oracle.