alternatives share nothing relevant in common. They can still be represented
as vectors of zeros and ones, but now it is better to embed them in JR4, and
identify them as a = (1,0,0,0), b = (0,1,0,0), c = (0,0,1,0), and d =
(0, 0, 0,1). There may still be intermediate cases where three characteristics
are necessary and sufficient to distinguish between these four alternatives.
Two examples may be given by the cases
a = (1, 0, 0) ,b = (1,1, 0) ,c = (1, 0,1), andd = (0, 0, 0)
or
a = (1, 0, 0) ,b = (0,1, 0) ,c = (0, 0,1), andd = (0,1,1).
In the four-dimensional and three-dimensional cases, these four alternatives
are only some of the conceivable vertices of the corresponding cubes. Other
combinations of zeros and ones represent conceivable but unfeasible choices.
These examples suggest that the objects in our model (interpreted as
characteristics) may be taken as partial aspects of the overall alternatives
(whose role is played in our model by the feasible sets). This interpretation
is not restrictive: any alternative (out of a Unite set) can be described by
a (finite) set of characteristics. What is restrictive is that once we identify
each alternative with a set of characteristics (thus embedding it into some
/-dimensional cube), we also determine the shape of the set of feasible alter-
natives, and this has consequences on the class of preferences which pass the
test of additivity (or separability).
In fact, thanks to the above observations, we can conclude by arguing
that the Gibbard-Satterthwaite theorem arises as a particular corollary of our
Theorem 1. Indeed, take any finite set A of к alternatives (⅛ > 2). Identify
them with the к unit vectors. Notice that all preferences over A are restric-
tions of some additive preference on the ^-dimensional cube. Hence, we are
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