We join the goals of two giant and related fields of research in group decision-making that have historically had little contact: fair division, and efficient mechanism design with monetary payments. To do this we adopt the standard mechanism design paradigm where utility is assumed to be quasilinear and thus transferable across agents. We generalize the traditional binary criteria of envy-freeness, proportionality, and efficiency (welfare) to measures of degree that range between 0 and 1. We demonstrate that in the canonical fair division settings under any allocatively-efficient mechanism the worst-case welfare rate is 0 and disproportionality rate is 1; in other words, the worst-case results are as bad as possible. This strongly motivates an average-case analysis. We then set as the goal identification of a mechanism that achieves high welfare, low envy, and low disproportionality in expectation across a spectrum of fair division settings. We establish that the VCG mechanism is not a satisfactory candidate, but the redistribution mechanism of [Bailey, 1997; Cavallo, 2006] is.