Distributed resource allocation through utility design - Part I: optimizing the performance certificates via the price of anarchy

Game theory has emerged as a novel approach for the coordination of multiagent systems. A fundamental component of this approach is the design of a local utility function for each agent so that their selfish maximization achieves the global objective. In this paper we propose a novel framework to characterize and optimize the worst case performance (price of anarchy) of any resulting equilibrium as a function of the chosen utilities, thus providing a performance certificate for a large class of algorithms. More specifically, we consider a class of resource allocation problems, where each agent selects a subset of the resources with the goal of maximizing a welfare function. First, we show that any smoothness argument is inconclusive for the design problems considered. Motivated by this, we introduce a new approach providing a tight expression for the price of anarchy (PoA) as a function of the chosen utility functions. Leveraging this result, we show how to design the utilities so as to maximize the PoA through a tractable linear program. In Part II we specialize the results to submodular and supermodular welfare functions, discuss complexity issues and provide two applications.