Generalized Efficiency Bounds in Distributed Resource Allocation

Game theory is emerging as a popular tool for distributed control of multiagent systems. To take advantage of these game theoretic tools, the interactions of the autonomous agents must be designed within a game-theoretic environment. A central component of this game-theoretic design is the assignment of a local utility function to each agent. One promising approach to utility design is assigning each agent a utility function according to the agent's Shapley value. This method frequently results in games that possess many desirable features, such as the existence of pure Nash equilibria with near-optimal efficiency. In this paper, we explore the relationship between the Shapley value utility design and the resulting efficiency of both pure Nash equilibria and coarse correlated equilibria. To study this relationship, we introduce a simple class of resource allocation problems. Within this class, we derive an explicit relationship between the structure of the resource allocation problem and the efficiency of the resulting equilibria. Lastly, we derive a bicriteria bound for this class of resource allocation problems-a bound on the value of the optimal allocation relative to the value of an equilibrium allocation with additional agents.

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