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Resource allocation in protected and shared bands: uniqueness and efficiency of Nash equilibria

Motivated by the question, is non-cooperative spectrum sharing desirable or not, we consider a scenario utilizing protected and shared bands. In a static non-cooperative setting consisting of two communication system pairs, we study the existence, uniqueness and efficiency of a fixed point of the iterative water-filling algorithm which corresponds to the Nash equilibrium. There exist several sufficient conditions for the convergence of the algorithm in the literature mostly based on the contraction mapping theorem. We derive necessary and sufficient conditions for convergence by relating the game to supermodular games. There, the best response dynamics is globally convergent when a unique Nash equilibrium exists. In order to understand the loss in efficiency due to non-cooperation, we study the Price of Anarchy of the system. We show that the performance of the non-cooperative system cannot fall below two third of that of the cooperative system in the high signal to noise ratio regime. Theoretical results are illustrated by numerical simulations for a simplified system scenario.