The importance of budget in efficient utility design

Networked control systems can often be viewed through a game theoretic lens with each agent responding to a local utility function. In this paper, we focus on the question of how to design agent utility functions to ensure that the resulting system-wide behavior is desirable. While recent research has identified all utility design methodologies that ensure the existence of a (pure) Nash equilibrium in networked resource allocation problems, it remains on open question as to what utility design optimizes the efficiency of the resulting equilibria. Our first result focuses on the recently introduced concept of smoothness and provides a characterization of how a budget anomaly, which we define as a measure of the difference between the sum of the agents' payoffs and the value of the global objective, impacts the efficiency of the resulting Nash equilibria. Using this characterization, we then propose a new methodology for utility design that is accompanied by automatic efficiency guarantees. Lastly, we illustrate the effectiveness of this methodology on a class of resource allocation problems.