Preliminaries • A social decision scheme (SDS) is a function that maps a preference profile to a lottery over the alternatives. Formally, an SDS is a function f : RFN → ∆(A). • An SDS is majoritarian if it only depends on the (unweighted) majority comparisons between alternatives. • We compare lotteries using stochastic dominance (SD). A lottery p is preferred to another lottery q if the expected utility for p is at least as large as that for q for every von Neumann-Morgenstern utility function. Formally, p ≳ q iff ∑x≿y p(x) ≥ ∑x≿y q(x) for all y∈A • Random serial dictatorship (RSD) is the generalization of random dictatorship to weak preferences. RSD is defined by picking a sequence of voters uniformly at random and then invoking serial dictatorship. • BOR returns the uniform lottery over all Borda winners, i.e., alternatives that receive the highest Borda score.