Budget Feasible Mechanisms for Dealers

We consider the problem of designing budget feasible mechanisms for a dealer, who aims to maximize revenue by buying items from a seller market and selling them to a buyer market that consists of unit-demand buyers. Different from the related literature, the dealer’s “value” for a set of items that he purchased from the seller market is not directly given as a number but it is defined to be the maximum revenue the dealer can obtain from selling the items to the buyers. We aim to design mechanisms that are dominant-strategy truthful for the sellers to report their costs and envy-free for the buyers to purchase their most preferred items (given their prices) in the final outcome, such that the total payment to the sellers does not exceed the dealer’s budget and the dealer’s revenue is (approximately) maximized. First, to understand the structure of the optimal mechanisms, we show that the maximum (envy-free) revenue obtainable by the dealer as a function of the set of purchased items is monotone and subadditive. Thus, existing results on subadditive optimization problems are potentially applicable in solving the mechanism design problem for the dealer. However, a crucial assumption adopted by all previous studies on subadditive functions is that the mechanism or algorithm has access to the value oracle and/or the demand oracle. In the dealer’s problem, instead, we show that (1) the demand oracle can be efficiently simulated by the value oracle and (2) both have efficient O(logn)-approximation algorithms, where n is the number of buyers. This is particularly interesting given the literature, since, in general, the demand oracle can always efficiently simulate the value oracle, and there are cases where the demand oracle is strictly more powerful. Our results show that, for the dealer’s problem, the two oracles are as powerful as each other. Finally, we construct a polynomial-time budget feasible mechanism for the dealer that doesn’t use any oracle and provides an O((log n)(log m))-approximation of the optimal revenue, where m is the number of sellers.