  Arrow's Impossibility Theorem states that any constitution which satisfies
Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a
Dictator has to be non-transitive. In this paper we study quantitative versions
of Arrow theorem. Consider $n$ voters who vote independently at random, each
following the uniform distribution over the 6 rankings of 3 alternatives.
Arrow's theorem implies that any constitution which satisfies IIA and Unanimity
and is not a dictator has a probability of at least $6^{-n}$ for a
non-transitive outcome. When $n$ is large, $6^{-n}$ is a very small
probability, and the question arises if for large number of voters it is
possible to avoid paradoxes with probability close to 1.
  Here we give a negative answer to this question by proving that for every
$\eps > 0$, there exists a $\delta = \delta(\eps) > 0$, which depends on $\eps$
only, such that for all $n$, and all constitutions on 3 alternatives, if the
constitution satisfies: The IIA condition. For every pair of alternatives
$a,b$, the probability that the constitution ranks $a$ above $b$ is at least
$\eps$. For every voter $i$, the probability that the social choice function
agrees with a dictatorship on $i$ at most $1-\eps$. Then the probability of a
non-transitive outcome is at least $\delta$.
