Surveys and experiments have uncovered a strong connection between perceptions of self-determination and preferences for redistribution. This paper presents results from a vignette-based survey that examines whether redistributive preferences are further affected by the dimensionality of luck and self-determination. The vignettes specify whether economic advantages are the result of one self-determined dimension (effort or independence), two self-determined dimensions (effort and independence), one lucky dimension (natural ability or dependence) or two lucky dimensions (natural ability and dependence). Results provide evidence that the presence of two dimensions of luck or self-determination can shift redistributive preferences relative to the presence of one dimension, but only when the decision task is cognitively simple and the vignette’s context is easily applicable to everyday life. More generally, whenever the decision task is cognitively complex or the context is ethically simplistic, the dimensionality of selfdetermination and luck is found to have insignificant effects on redistributive preferences. The results also demonstrate that the presence of a single, salient dimension of luck can increase support for redistribution more than the joint presence of multiple dimensions of luck.