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Games with continuous strategy sets arise in several machine learningproblems (e.g. adversarial learning). For such games, simple no-regret learningalgorithms exist in several cases and ensure convergence to coarse correlatedequilibria (CCE). The efficiency of such equilibria with respect to a socialfunction, however, is not well understood. In this paper, we define the classof valid utility games with continuous strategies and provide efficiency boundsfor their CCEs. Our bounds rely on the social function being a monotoneDR-submodular function. We further refine our bounds based on the curvature ofthe social function. Furthermore, we extend our efficiency bounds to a class ofnon-submodular functions that satisfy approximate submodularity properties.Finally, we show that valid utility games with continuous strategies can bedesigned to maximize monotone DR-submodular functions subject to disjointconstraints with approximation guarantees. The approximation guarantees wederive are based on the efficiency of the equilibria of such games and canimprove the existing ones in the literature. We illustrate and validate ourresults on a budget allocation game and a sensor coverage problem.

Bounding Inefficiency of Equilibria in Continuous Actions Games using Submodularity and Curvature