Mark Franke: The mathematics of voting

Now that the primary election season has passed but the November campaigning cycle is not yet up and running, there is time for detached reflection on how our democracy works. Or doesn’t.

The past month was an unusual one for us residents of my hometown, Fort Wayne. In the Republican primary there were eight candidates for the 3rd District House of Representatives seat to replace incumbent Jim Banks, who was running in the Senate primary to replace Mike Braun, who was running in the gubernatorial primary to replace Eric Holcomb, who was retiring due to term limits. Are you keeping up?

What set my mind wondering and wandering was how does the electorate correctly indicate its favorite candidate when there are more than two choices?

Indiana’s voting rules require a simple plurality to win, what’s known as the “first past the post” system. The horse-race metaphor is appropriate in that the successful candidate need only finish ahead of each of the rest, regardless of how few votes he gets. A mathematical majority, 50 percent plus one, is not required. The advantage of this system is that there will be a winner on the first round.

Compare this with the Fort Wayne mayoral caucus which required a true majority among seven candidates, therefore needing a second round of voting after eliminating the first-round candidates with the fewest votes. The advantage of this system is that eventually you do get to someone who polls a majority of votes but it might take some time. Witness the 2022 senatorial election in Georgia which required a second round of voting a month later.

Which system is better? Or is there another methodology that surpasses both?

It may be serendipity but while I was musing over this question, I happened across a book directed toward this very issue. “Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present” by mathematician George G. Szpiro examining various alternatives.

Szpiro selects about a dozen philosophers and mathematicians through history to trace what one would hope to be a developmental path for discovering the perfect model to assure the most favored candidate wins. His earliest models, that proposed by Plato in “The Republic” and that of Pliny the Younger for Roman jury decision-making, set a foundation for future model-making. One of the medieval motivations to get this right was the papal elections and their frequent inability to come to a satisfactory decision.

One idea was to set each candidate against each other, one on one, and then count the win-loss record of each. Think of a typical athletic league’s method of determining its regular season champion. While we are using sports metaphors, another proposal was to use an NCAA March Madness approach, in which one-on-one winners advance to the next round.

Both of these systems will produce a winner but only after an extended period of time and tremendous expense when thinking of modern-day political spending. Each has its methodological flaws in that neither can assure that the most favored candidate is elected. Is the NCAA basketball tournament always won by the best overall team?

The problem is the lack of transitivity between successive votes. Let me try to simplify. If a voter prefers A to B and B to C, then it can be assumed that the voter prefers A to C. But now consider rock-paper-scissors. There is no transitivity in the above two methodologies, leaving the outcome suspect with either.

Eventually the idea of ranked-choice voting was proposed. This model has been resurrected by several contemporary election reform groups that see our current two-party system as systemically flawed. In this approach each voter ranks all the candidates from top to bottom. Votes are only cast once but counted multiple times as the last place candidate is thrown out and his first-place votes allocated to that voter’s second choice.

At first blush this system appears to address the issue of nonmajority candidates winning and the need for runoff elections. It is also supposed to reduce the influence of the extreme wings of our two national parties, making room for more moderate and third-party candidates.

But does it achieve Szpiro’s goal of assuring that the most favored candidate will always win? He thinks not as it opens the election to strategic voting, where voters intentionally rank their favorite candidate’s closest opponent as last. If this is properly, or should I say improperly, coordinated at a macro level, it can have a deleterious effect.

So if none of the alternatives work, is the answer to limit each ballot to only two candidates? Is our electoral system hardwired for a two-party system?

What none of these systems can guarantee is that the most favored candidate gets on the ballot. Voter dissatisfaction during the last several presidential elections is proof of that.

Mark Franke, an adjunct scholar of the Indiana Policy Review and its book reviewer, is formerly an associate vice-chancellor at Indiana University-Purdue University Fort Wayne. Send comments to [email protected].