Competitive Equilibrium Relaxations in General Auctions

Abstract: The goal of an auction is to determine commodity prices such that all participants are perfectly happy. Such a solution is called a competitive equilibrium and does not exist in general. For this reason we are interested in solutions which are similar to a competitive equilibrium. The article introduces two relaxations of a competitive equilibrium for general auctions. Both relaxations determine one price per commodity by solving a difficult non-convex optimization problem. The first model is a mathematical program with equilibrium constraints (MPEC), which ensures that each participant is either perfectly happy or his bid is rejected. An exact algorithm and a heuristic are provided for this model. The second model is a relaxation of the first one and only ensures that no participant incurs a loss. In an optimal solution to the second model, no participant can be made better off without making another one worse off.