A Mathematical View of the Electoral College
Rather less attention has been paid to the rules that govern how we select a president after the campaigns end, after the voters express their preferences, after the hanging chads are sorted out. All that remains at that point is a clearly defined social choice function (or voting method), a decision procedure for selecting a winner based on the voters opinions.
The voting method we use to elect presidents in the
United States is curiously complicated, arbitrary, and unfair. It should
Why do I say that our current method is curiously complicated? Because votes for president are actually cast by members of the Electoral College, which begs the question of how many members each state gets to send to the Electoral College. This in turn depends on the preceding census and the apportionment bill that Congress passed after that census doling out seats in the House of Representatives to the states. Census data is notoriously unreliable, since counting residents is naturally more difficult than the now widely recognized difficulty of counting votes. Even ignoring the anomaly of the census, we still must rely on a method to assign a whole number of representatives to each state based on the census results.
The Constitution gives little guidance regarding how
this rounding should be done. The apportionment method that the US seems
to use regularly nowadays is the relatively modern method of Huntington-Hill,
which is itself curiously complicated and hard to defend against the alternative
classical methods of Jefferson, (Quincy) Adams, Webster, Dean or Hamilton,
or against the new so-called quota method of Balinski and
Young, proposed in 1975. An entirely different anomaly of our current
system is the perhaps sensible but irregular way Electoral College representatives
are chosen in the states of Nebraska and Maine.
Why do I say that our current method of electing a president is arbitrary?
Because there are many other methods to choose from,
and the Electoral College system we use cannot be singled out among these
methods as being optimal in any way. Only part of the arbitrariness comes
from the dependence on the debatable apportionment method we currently
use. Another arbitrary aspect of the current system is the possibility
of a faithless elector, an anomaly that introduces an element
of randomness that some may find charming but most would regard as unequivocally
inappropriate. There are many ways to run direct elections,
at least when there are more than two candidates. In France, there is
a run-off among the two top-scoring presidential candidates. In other
systems, voters are permitted to be more expressive than they are in the
US, where each vote must be cast 100 percent for one candidate, regardless
of how the voter might feel about the entire slate. For example, one might
permit voters to indicate not only their first-choice candidate but an
entire preference list of candidates. Armed with this richer information
about voters preferences, one is afforded numerous other reasonable
decision procedures. Some mathematicians advocate strongly for the Borda
count method, in which each voter casts one point for their favorite
candidate, two points for their second favorite candidate, three points
for their third, and so on through the entire slate of candidates, with
the candidate receiving the fewest points declared the winner. Others
favor a method in which candidates with the fewest first-place votes are
eliminated in successive rounds.
Our current method satisfies none of these.
In the 2000 election, Al Gore could have won merely by trading votes from some states with votes from other states, without changing the mind of any voter about the candidates. This illustrates that the voting method we use to elect presidents in the US violates the anonymity criterion. In 2000, evidence suggests that Al Gore would have won a head-to-head contest against each of the other candidates, again without changing the mind of any voter about the candidates. This illustrates the violation of the Condorcet criterion. In 2000, had Al Gore picked up 100 percent of the California vote, he would have been the top choice of a vast majority of the national electorate and still lost. This illustrates the violation of the majority criterion.
The point is not that Gore could or should have won;
the point is that the decision procedure we use violates natural criteria