(joint work with Huaxia Zeng). We study random mechanism design in an environment where the set of alternatives has a
Cartesian product structure. We first show that all generalized random dictatorships are sd-strategy-proof on a minimally rich domain if and only if all preferences are top-separable. We call a domain satisfying top-separability a multidimensional domain, and furthermore generalize the notion of connectedness (Monjardet, 2009) to a broad class of multidimensional
domains : connected+ domains. We show that in the class of minimally rich and connected+domains, the multidimensional single-peakedness restriction is necessary and sufficient for the design of a flexible random social choice function that is unanimous and sd-strategy-proof.
Such a flexible random social choice function allows for a systematic notion of compromise. We prove an analogous result for deterministic social choice functions satisfying anonymity. Our characterization remains valid for a problem of voting under constraints where not all
alternatives are feasible (Barbera, Masso, and Neme, 1997).