  Arrow's theorem implies that a social choice function satisfying
Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant
Alternatives (IIA) must be dictatorial. When non-strict preferences are
allowed, a dictatorial social choice function is defined as a function for
which there exists a single voter whose strict preferences are followed. This
definition allows for many different dictatorial functions. In particular, we
construct examples of dictatorial functions which do not satisfy Transitivity
and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not
provide a complete characterization of all social choice functions satisfying
Transitivity, the Pareto Principle, and IIA.
  The main results of this article provide such a characterization for Arrow's
theorem, as well as for follow up results by Wilson. In particular, we
strengthen Arrow's and Wilson's result by giving an exact if and only if
condition for a function to satisfy Transitivity and IIA (and the Pareto
Principle). Additionally, we derive formulas for the number of functions
satisfying these conditions.
