Non-linear anonymous pricing in combinatorial auctions

In combinatorial auctions the pricing problem is of main concern since it is the means by which the auctioneer signals the result of the auction to the participants. In order for the auction to be regarded as fair among the various participants the price signals should be such that a participant that has won a subset of items knows why his bid was a winning bid and that agents that have not acquired any item easily can detect why they lost. The problem in the combinatorial auction setting is that the winner determination problem is a hard integer programming problem and hence in general there does not exist a linear pricing scheme supporting the optimal allocation. This means that single item prices, that support the optimal allocation can in general not be found. In this article we present an alternative. From integer programming duality theory we know that there exist non-linear anonymous price functions that support the optimal allocation. In this paper we will provide a means to obtain a simple form of a non-linear anonymous price system that supports the optimal allocation. Our method relies on the fact that we separate the solution of the winner determination problem and the pricing problem. This separation yields a non-linear price function of a much simpler form compared to when the two problems are solved simultaneously. The pure pricing problem is formulated as a mixed-integer program. If solving this program is too demanding computationally a heuristic can be used which essentially requires us to solve a sequence of linear programming relaxations of a new mixed-integer programming formulation of the pricing problem. The procedure is computationally tested using instances of the combinatorial auctions test suite.