  Social decision schemes (SDSs) map the ordinal preferences of individual
voters over multiple alternatives to a probability distribution over the
alternatives. In order to study the axiomatic properties of SDSs, we lift
preferences over alternatives to preferences over lotteries using the natural
-- but little understood -- pairwise comparison (PC) preference extension. This
extension postulates that one lottery is preferred to another if the former is
more likely to return a preferred outcome. We settle three open questions
raised by Brandt (2017): (i) there is no Condorcet-consistent SDS that
satisfies PC-strategyproofness; (ii) there is no anonymous and neutral SDS that
satisfies PC-efficiency and PC-strategyproofness; and (iii) there is no
anonymous and neutral SDS that satisfies PC-efficiency and strict
PC-participation. All three impossibilities require $m\geq 4$ alternatives and
turn into possibilities when $m\leq 3$. We furthermore settle an open problem
raised by Aziz et al. (2015) by showing that no path of PC-improvements
originating from an in
