November 2004

The Choice of Choices

A Mathematical View of the Electoral College

By Daniel Ullman

Much has been written about our political processes leading up to presidential elections and about the rules that govern how we attempt to change the minds of voters. Pundits complain about corrupt campaign financing and negative advertising, about spin doctors and staged political rallies and such.

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 be changed.

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.

Why do I say that our current method is “unfair”? Because not every voter is treated equally. John Banzhaf, GW professor of law, points out that voters in California have a more powerful vote for president than do voters in Montana. A popular misconception is that the opposite is true, that voters in small states are the beneficiaries of the Electoral College system. Either way, everyone can agree that not all voters are treated equally and that someone’s vote is therefore given less consideration in our system than it deserves. Our democratic values require that we hold elections and that each person be permitted to vote. But that is not enough; we must also tally those votes and then use a decision procedure that gives a result. We expect that decision procedure to adhere to certain criteria of acceptability, among them: (1) the anonymity criterion: if two voters swap their ballots, then the result of the election should not change; (2) the Condorcet criterion: if one candidate would win a run-off against every other, then that candidate should be the one chosen by the procedure; and, (3) the majority criterion: if one candidate is the top choice of more than half the electorate, then that candidate should be the one chosen.

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 for reasonableness.

A direct election by popular vote — by which is meant: whoever has the most votes wins — would be an improvement over the current Electoral College method. It is both simple and fair. It satisfies the anonymity criterion as well as the majority criterion. Like our current method, it fails the Condorcet criterion, but unfortunately this flaw cannot be corrected without introducing violations of other natural criteria. No voting method meets all criteria for reasonableness. This, roughly speaking, is the content of a theorem from 1951 of Kenneth Arrow, who won the 1972 Nobel Prize in Economics in part for this work.

If we in the US insist on preserving our tradition of forbidding voters from expressing any preferences at the ballot box other than their first choice, then it is hard to think of any simple and fair voting method other than direct election by popular vote. But if voters were given an opportunity to express their preferences more fully — there are many ways one might do this — then a number of competing voting methods could be considered. I can imagine a lively debate in which the mathematical advantages and disadvantages of several reasonable methods were discussed. One thing is for sure: our current Electoral College method would not even be on the table.

Daniel H. Ullman is a professor of mathematics in the Columbian College of Arts and Sciences.

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