  Social choice functions (SCFs) map the preferences of a group of agents over
some set of alternatives to a non-empty subset of alternatives. The
Gibbard-Satterthwaite theorem has shown that only extremely unattractive
single-valued SCFs are strategyproof when there are more than two alternatives.
For set-valued SCFs, or so-called social choice correspondences, the situation
is less clear. There are miscellaneous - mostly negative - results using a
variety of strategyproofness notions and additional requirements. The simple
and intuitive notion of Kelly-strategyproofness has turned out to be
particularly compelling because it is weak enough to still allow for positive
results. For example, the Pareto rule is strategyproof even when preferences
are weak, and a number of attractive SCFs (such as the top cycle, the uncovered
set, and the essential set) are strategyproof for strict preferences. In this
paper, we show that, for weak preferences, only indecisive SCFs can satisfy
strategyproofness. In particular, (i) every strategyproof rank-based SCF
violates Pareto-optimality, (ii) every strategyproof support-based SCF (which
generalize Fishburn's C2 SCFs) that satisfies Pareto-optimality returns at
least one most preferred alternative of every voter, and (iii) every
strategyproof non-imposing SCF returns a Condorcet loser in at least one
profile.
