rules that satisfy strategy-proofness are still voting by committees, with bal-
lots indicating the best feasible set of objects. Second: the committees for
different objects must be interrelated, in precise ways which depend on what
families of sets of objects are feasible. Third: unlike in Barberà, Sonnen-
schein, and Zhou (1991), the class of strategy-proof rules when preferences
are additively representable can be substantially larger that the set of rules
satisfying the same requirement when voters’ preferences are separable.
Our characterization result for separable preferences is quite negative:
infeasibilities quickly turn any non-dictatorial rule into a Inanipulable one,
except for very limited cases. In contrast, our characterization result for
additive preferences can be interpreted as either positive or negative, because
it has different consequences depending on the exact shape of the range of
feasible choices. The contrast between these two characterization results is
a striking conclusion of our research, because until now the results regarding
strategy-proof mechanisms for these two domains had gone hand to hand,
even if they are, of course, logically independent.
Notice that here, as in Barberà, Sonnenschein, and Zhou (1991), we could
identify sets of objects with their characteristic function, and our objects of
choice as (some of) the vertices of a ^-dimensional hypercube. Barberà,
Gul, and Stacchetti (1993) extended the analysis to cover situations where
the objects of choice are Cartesian products of integer intervals, allowing for
possibly more than two values on each dimension. In there and in other con-
texts of multidimensional choice where the range of the social choice rule is
a Cartesian product, strategy-proof rules are necessarily decomposable into
rules which independently choose a value for each dimension, and are them-
selves strategy-proof (see Le Breton and Sen (1997) and (1999) for general
expressions of this important result, which dates back to the pioneering work