# Elections - can we do better?

It is highly unlikely that the Electoral College will be abolished anytime soon. A constitutional amendment requires ratification by three-fourths of all states, over half of which have single-digit numbers of electoral votes. Given that most people believe the Electoral College favors small states, these small states will probably not agree to abolish it.

Still, it seems clear that the country is in need of election reform. Currently we use plurality (within the larger umbrella of the Electoral College) to elect our president. There are many arguments against plurality, including the fact that when there are more than two candidates, plurality does not take into account enough information about the preferences of the voters; it ignores all but a voter’s first choice. Perhaps there is a better voting method out there that better represents the will of the people.

There certainly are many other voting methods to choose from. To be a little more rigorous, let’s say that a voting method is a function whose input is a sequence of individual voters’ ranked preference lists (typically with no ties) and whose output is a ranked listing of the candidates (perhaps with ties). The candidate on the top of the output list is the winner of the election. Note that, given a method of selecting a winner from a sequence of preference lists, iterating that method yields a ranked listing of the alternatives, so in describing a voting method, it’s enough to specify a method for selecting a winner.

Instant runoff voting (i.e. the Hare system) is one popular voting method, with variations used in mayoral elections in San Francisco, presidential elections in Ireland, and student council decisions at Williams. Unfortunately, this method has the troubling property that if a voter decides to rank candidate A higher on his preference list, that alone could cause candidate A to lose the election! (See monotonicity definition below.)

The Borda count is another popular voting method, based on assigning points to candidates based on where they fall on a voter’s preference list. The Borda count is also problematic, however; it has the property that a candidate might lose the election, even if that same candidate would defeat every other in a one-on-one contest.

Several other voting methods exist, but, unfortunately, all are flawed. Kenneth Arrow proved in 1950 in the paper “A Difficulty in the Concept of Social Welfare” that there is only one voting method for three or more candidates that satisfies the following three reasonable criteria:

1) Pareto: If all voters rank candidate A higher than candidate B, then B should not be the winner.

2) Monotonicity: If candidate A is the winner, and one voter moves A up higher on his preference list, then A should still be the winner.

3) Independence of Irrelevant Alternatives: The presence of a third candidate C should not affect society’s preference between candidates A and B. (Imagine you’re at a restaurant and have just decided on the tiramisu over the chocolate hazelnut mousse for dessert. Then the waiter tells you that they are serving a special dark chocolate cake that evening as well. While you might change your mind and get the cake, it would be strange for the cake option to cause you to choose the mousse instead.)

A dictatorship!

Arrow received the Nobel Prize in economics in 1972, in large part due to this work.

Ok, so maybe Arrow’s theorem is a little depressing – but just because we can’t achieve perfection with our voting system doesn’t mean that we shouldn’t find the very best method out there. I am personally a big fan of the Borda count, though on Tuesday my students will be holding a debate on this topic during class. There are five different groups arguing in favor of: plurality, the Borda count, approval voting, dictatorship, and Condorcet’s method if applicable; otherwise plurality. I am eager to see what they have to say. In the meantime, what method do you think we should use for our presidential elections?