We study social decision schemes (SDSs), i.e., functions that map a collection of individual preferences over alternatives to a lottery over the alternatives. Depending on how preferences over alternatives are extended to preferences over lotteries, there are varying degrees of efficiency and strategyproofness. In this paper, we consider four such preference extensions: stochastic dominance (SD), a strengthening of SD based on pairwise comparisons (PC), a weakening of SD called bilinear dominance (BD), and an even weaker extension based on Savage’s sure-thing principle (ST). While random serial dictatorships are PC-strategyproof, they only satisfy ex post efficiency. On the other hand, we show that strict maximal lotteries are PC-efficient and ST-strategyproof. We also prove the incompatibility of (i) PC -efficiency and PC -strategyproofness for anonymous and neutral SDSs, (ii) ex post efficiency and BD-strategyproofness for pairwise SDSs, and (iii) ex post efficiency and BD-group-strategyproofness for anonymous and neutral SDSs.