  Social decision schemes (SDSs) map the preferences of a group of voters over
some set of $m$ alternatives to a probability distribution over the
alternatives. A seminal characterization of strategyproof SDSs by Gibbard
implies that there are no strategyproof Condorcet extensions and that only
random dictatorships satisfy ex post efficiency and strategyproofness. The
latter is known as the random dictatorship theorem. We relax
Condorcet-consistency and ex post efficiency by introducing a lower bound on
the probability of Condorcet winners and an upper bound on the probability of
Pareto-dominated alternatives, respectively. We then show that the SDS that
assigns probabilities proportional to Copeland scores is the only anonymous,
neutral, and strategyproof SDS that can guarantee the Condorcet winner a
probability of at least 2/m. Moreover, no strategyproof SDS can exceed this
bound, even when dropping anonymity and neutrality. Secondly, we prove a
continuous strengthening of Gibbard's random dictatorship theorem: the less
probability we put on Pareto-dominated alternatives, the closer to a random
dictatorship is the resulting SDS. Finally, we show that the only anonymous,
neutral, and strategyproof SDSs that maximize the probability of Condorcet
winners while minimizing the probability of Pareto-dominated alternatives are
mixtures of the uniform random dictatorship and the randomized Copeland rule.
