We propose a cascade model of wave turbulence designed to simplify the study of this phenomenon in the way that shell models simplify the study of Navier-Stokes turbulence. The model consists of resonant quartets, in which some modes are driven and damped and others are shared by pairs of quartets and transferring energy between them, mimicking the natural energy transfer mechanism in weakly turbulent waves. A set of detailed-balance conditions singles out the case of the cascade model in equilibrium, for which we can explicitly derive a Gaussian equilibrium measure and a maximum-entropy principle using a Kolmogorov forward equation. Away from equilibrium, we can approximate the second-moment dynamics of the mode amplitudes using kinetic equations. In a nonequilibrium steady state, we can also approximate the higher moments of the driven-damped mode amplitudes and characterize the distribution of the shared-mode amplitudes as Gaussian. For this latter distribution, we find an information-theoretic argument, akin to entropy maximization, which lets us conclude that arbitrary initial shared-mode amplitude distributions approach Gaussian form in forward time. The cascade model may provide insight into mechanisms governing weakly turbulent wave systems and perhaps afford computational savings as compared to direct numerical simulations of the corresponding wavelike equations.