Millions of votes have already been cast in next month’s presidential election, and a lot of attention is focused on making sure they are counted correctly. Once votes are counted, however, they have to be translated into results—and that is where math comes in. Mathematicians have developed logical rules that allow us to test the fairness and efficacy of different voting systems, rather than just yelling at each other about fairness and seeing who yells the loudest.
In presidential elections, almost all states use a first-past-the-post system, in which the candidate who receives the most votes wins the state, whether they get a majority (more than half the votes) or only a plurality (more votes than any other candidate). The more candidates there are, the further from a majority a plurality can be. Even when there are only two major-party candidates, small parties might receive enough votes that the winning candidate has less than 50% of the vote.
First-past-the-post goes hand in hand with a “winner take all” system for awarding electoral votes, where the candidate with the most votes receives all of the state’s electors. Maine and Nebraska have a slightly different system, with each congressional district choosing its own elector.
This year, Maine is also using Ranked Choice Voting (RCV) for the first time in a presidential election. In this system, voters can rank all the candidates rather than picking one favorite. Then multiple rounds of balloting are simulated: The first-choice votes are counted, the candidate who comes last is eliminated, and their votes are redistributed to the second choice listed on each ballot. The process is repeated until the last remaining candidate becomes the winner.
RCV is designed to avoid a central problem of first-past-the-post voting, which was identified by the Nobel Prize-winning economist Kenneth Arrow. For an election to be fair, he argued, the existence of a third candidate shouldn’t change the relative ranking of the first two—a condition he called “the independence of irrelevant alternatives.” In first-past-the-post systems, however, this condition fails to apply, since voters often vote tactically. If your main priority is to defeat candidate A, you might vote for candidate B because you think they have the best chance of winning, even if your actual favorite is candidate C.
RCV largely avoids this problem, dramatically reducing the need for tactical voting. But it doesn’t eliminate the possibility of the electoral college result not agreeing with the popular vote, as happened in 2016. One way to solve that problem would be to stop awarding electors on a winner-take-all basis and shift to a proportional representation system, in which each candidate receives a share of the state’s electors proportional to their share of the vote.
Still, it’s important to remember that math offers only one criterion for the fairness of voting systems, while politics involves many other considerations as well. In voting as in other areas of life, math doesn’t give us the answers but offers us a set of tools to use in conjunction with those of other disciplines.
Appeared in the October 17, 2020, print edition as "The Logic Of Counting Votes."