The Mathematics of Elections

I originally wrote this entry on September 1, 2004, and published it on blogs.sun.com.

Should software and servers matter in breaking election ambiguities?

Well, it all depends on what sort of ambiguity we're talking about.
Here, I'm certainly not talking about hanging chads. I'm talking about
ambiguities that may arise because of the election method used.

Some election ambiguities require computational power, others may take
you to the Supreme Court. Let's focus on the first type of ambiguity,
for the moment.

The mathematics of elections is quite simple when everyone has
a chance to vote only for one of the candidates, i.e. when no allowance
is made for the voter to fully specify and assign his or her
preferences to each and as many of the candidates as he or she wishes.
In such an election, all we need to do is add the votes for each
candidate. Whoever has more votes wins. No puzzling ambiguities are
left.

Kenneth Arrow, the distinguished Stanford economist, who has also
written the forward to the anniversary edition of Chester Barnard's
classic (about which I've written earlier) has a famous theorem
(Arrow's Impossibility Theorem) which effectively says there are no
ideal methods for elections.

Arrow gives several criteria for "ideal" elections, the most
controversial of which is the "Independence from Irrelevant
Alternatives Criterion" (IIAC).

IIAC says, effectively, that removal or addition of a candidate should
make no difference unless that candidate was or will be the winner
(against all other candidates).

Arrow shows that it is impossible for an election method to satisfy all
of his criteria. So, according to Arrow's theorem, even if voters had a
chance to fully specify their preferences, it would make no difference.

The trouble is that the IIAC is not necessarily a valid criterion.

As noted in ElectionMethods.org, IIAC is too strong a criterion and near-ideal election methods do exist. The Condorcet election method is one such near-ideal election method:

The proper method of counting ranked
votes is called the Condorcet election method, named after the French
mathematician who conceived it a couple of centuries ago. The main idea
is that each race is conceptually broken down into separate pairwise
races between each possible pairing of the candidates. Each ranked
ballot is then interpreted as a vote in each of those one-on-one races.
If candidate A is ranked above candidate B by a particular voter, that
is interpreted as a vote for A over B. If one candidates beats each of
the other candidates in their one-on-one races, that candidate wins.
Otherwise, the result is ambiguous and a simple procedure is used to
resolve the ambiguity.

Well, it is the resolution of this ambiguity at a national (or any
other large-scale) level that may require some use of computing power.

ElectionMethods.org discusses Basic Condercet (BC) and Schwartz
Sequential Dropping (SSD) for resolving the ambiguity. It includes a
software implementation for the SSD method for solving cyclic
ambiguities.

BC method drops the weakest defeat (from the cyclic series of
defeats) until there's a candidate that is unbeaten. This may cause
strategy issues. Parties may have clone candidates, i.e. multiple
candidates running for the same party.

The SSD method has been described in ElectionMethods.org, where links to software and other useful information can also be found.

Personally, the Beatpath Winner (BW) method appears to me to be
adequate in resolving cyclicity in a Condercet voting result. It's
equivalent to the SSD method of resolving ambiguities. In the BW method
of resolving cyclic ambiguities, if A defeats B through a "path" (chain
of defeats) and B defeats A through another path (in the cycle), the
two paths (chains) are compared to see which one has the weakest defeat
in its sequence of defeats. The candidate which has the strongest
defeat paths (chains) against all other candidates is the winner.

Condercet method of elections seems like a very reasonable
method. The reason it has not been popular in the U.S. is probably
because people are very conservative and don't want surprises.
Furthermore, the strategy outcomes are hard to predict. It really
unleashes a marketplace for votes and makes results much more difficult
to predict. Another reason could be that Condercet will be truly bad
for the two-party system. Finally, with more parties having a chance,
there is also the question of political stability. In the absence of
political stability, economic stability may also be a rarity. So, I
would expect there may issue some arguments from institutional
economists against Condercet.