  In the year 1876 the mathematician Charles Dodgson, who wrote fiction under
the now more famous name of Lewis Carroll, devised a beautiful voting system
that has long fascinated political scientists. However, determining the winner
of a Dodgson election is known to be complete for the \Theta_2^p level of the
polynomial hierarchy. This implies that unless P=NP no polynomial-time solution
to this problem exists, and unless the polynomial hierarchy collapses to NP the
problem is not even in NP. Nonetheless, we prove that when the number of voters
is much greater than the number of candidates--although the number of voters
may still be polynomial in the number of candidates--a simple greedy algorithm
very frequently finds the Dodgson winners in such a way that it ``knows'' that
it has found them, and furthermore the algorithm never incorrectly declares a
nonwinner to be a winner.
