ECONOMY

donald g. saari 399
where the plurality ranking reverses if any candidate is dropped, but the pairwise
outcomes are cyclic? As there are many profiles with precisely this property, this listing
defines another property of the plurality vote.
By modifying notions from “chaotic dynamics” (the intrepid reader can check
Saari 1995), I was able to find everything that could ever happen for any number
of candidates, any number of voters, and all combinations of positional voting
methods (Saari 1989, 1990). To explain the discouraging results with, say, candidates
{A, B, C, D, E }, rank them in any desired manner; e.g. A ≻ B ∼ C ≻ D ∼ E .For
each way to drop a candidate, rerank the remaining four in any desired manner; e.g.
dropping E , select D ≻ C ≻ B ≻ A,droppingD choose E ≻ A ≻ B ∼ C ...Next,
dropping two candidates creates ten three-candidate subsets: rank each in any desired
manner. Finally, rank each pair in any desired manner. For any listing designed in
this almost random fashion, there is a profile so that for each subset of candidates,
the sincere plurality election outcome is the selected ranking. This conclusion, which
holds for any number of candidates, is a discouraging commentary on our standard
election method: it means that with the plurality vote “anything can happen.”
Beyond the plurality vote, select a positional method for each subset of candidates;
e.g. maybe the “vote for three” scheme for five-candidate sets, “vote for two” for
four-candidate subsets, and a (7, 6, 0) method for all triplets. The same assertion
holds: for almost all choices of positional methods, anything can happen. This “almost
all” modifier provides hope by suggesting that by carefully selecting voting methods,
we might provide consistency among election outcomes. This is the case. It turns
out that there are certain special combinations of positional methods that prohibit
some ranking lists from ever occurring, so they impose some consistency among
the election rankings as candidates are added or dropped. But the choices can be
complicated. As an illustration, if four candidate elections are tallied by assigning
3, 1, 0, 0 points, respectively, to a top-, second-, third-, and bottom-ranked candidate,
then the four-candidate outcome never bottom-ranks a candidate who wins all threecandidate
plurality contests. Rather than describing these complicated results, let me
cut to the chase by identifying the unique method with the ultimate consistency.
With n-candidates, the Borda Count assigns a candidate the same number of points
as there are lower-ranked candidates on the ballot. So for five candidates, the Borda
Count assigns 4, 3, 2, 1, 0 points, respectively, to a top-, second-, third-, fourth-, and
bottom-ranked candidate. A main result is that the maximum consistency in election
rankings is attained only by using the Borda Count with all subsets of candidates. More
precisely, any ranking list coming from the Borda Count also arises with any other
combination of voting rules! But if a non-Borda method is assigned to any subset of
candidates, the system generates ranking outcomes that never occur with the Borda
method. Moreover, the differences in kinds and numbers of unexpected election
outcomes (paradoxes) are mind boggling. To illustrate with seven candidates, it would
be impressive if, for instance, the number of plurality ranking lists is three times that
of the Borda Count: this multiple would measure the increased inconsistency of the
plurality vote. This assertion, however, is far too modest: the plurality vote generates
more than 10 50 more lists than the Borda Count! This number, a 1 followed by fifty