  The purpose of this note is to prove the existence of a randomized mechanism,
a social decision scheme (SDS), with desirable fairness, efficiency, and
strategyproofness properties unmatched by all known SDSs. In particular, we
disprove a conjecture by Aziz et al. (2013). Additionally, we obtain a strong
existence result for the domain of random assignment. Both, the notion of
efficiency and strategyproofness are based on stochastic dominance and have
been studied extensively for random assignment. The proof makes crucial use of
Brouwer's fixed-point theorem and is hence non-constructive. To the best of our
knowledge, this is the first application of a fixed-point theorem to show the
existence of a social choice function or mechanism.
