Arrow's Impossibility Theorem is not so Impossible and Condorcet's Paradox is not so Paradoxical: the Adequate Definition of a Social Choice Problem

Abstract

In this article, we do two things: first, we present an alternative and simplified proof of the
known fact that cardinal individual utility functions are necessary, but not sufficient, and
that interpersonal comparability is sufficient, but not necessary, for the construction of a
social welfare function. This means that Arrow's impossibility theorem is simply a
consequence of forcing the individual utility functions to be ordinal. And second, based on
this proof, this article establishes two necessary conditions for the adequate definition of a
social choice problem. It is shown that, if these two conditions are satisfied, a number of
desirable properties for a social choice are satisfied, including transitivity. This means that
Condorcet's paradox is simply the result of a social choice problem that is not well defined.