  Games with continuous strategy sets arise in several machine learning
problems (e.g. adversarial learning). For such games, simple no-regret learning
algorithms exist in several cases and ensure convergence to coarse correlated
equilibria (CCE). The efficiency of such equilibria with respect to a social
function, however, is not well understood. In this paper, we define the class
of valid utility games with continuous strategies and provide efficiency bounds
for their CCEs. Our bounds rely on the social function being a monotone
DR-submodular function. We further refine our bounds based on the curvature of
the social function. Furthermore, we extend our efficiency bounds to a class of
non-submodular functions that satisfy approximate submodularity properties.
Finally, we show that valid utility games with continuous strategies can be
designed to maximize monotone DR-submodular functions subject to disjoint
constraints with approximation guarantees. The approximation guarantees we
derive are based on the efficiency of the equilibria of such games and can
improve the existing ones in the literature. We illustrate and validate our
results on a budget allocation game and a sensor coverage problem.
