FAITH + IDEAS =: last updated 10/16/2012

How Your Vote Counts, Literally, and Why It Matters

October 9, 2012            Volume 5, Issue 13
Faith+ Ideas= an e-conversation with the faculty of Gordon College
By Karl-Dieter Crisman
In 2000, the world held its breath while Florida laboriously judged voter intent for thousands of presidential ballots with “hanging chads.” The U.S. Supreme Court eventually stepped in, and George W. Bush eked out the narrowest of victories over Al Gore—less than one percent of one percent.
Ignoring the controversy, the basic story was about who got more votes. True, there were—and continue to be—calls for removing the Electoral College, since Gore received more votes than Bush in the country as a whole, but the news then was mainly about Bush versus Gore.
There is another story, though. If even a few of the voters for Green party candidate Ralph Nader had chosen Gore, he would have easily won. In fact, voters for at least four third-party candidates in Florida could have turned the election by a vote for Gore or Bush. 
But in any election, like the upcoming one next month, the rules are clear: You only get one vote, and whoever gets the most wins, right? Not really. There is actually a wide range of ways your vote could be counted, a mathematics of voting, if you will, that goes far beyond the usual, “plurality vote.”
Consider, first, that to decide who even gets to run for president, there is an entire first round of primaries. (Vote #1.) Similarly, many countries would have a runoff in an election where neither candidate got a strict majority of votes, like in 2000.
Next, various professional societies allow members to vote for as many candidates for leadership as they deem qualified, all votes counting the same. (Vote #2.) This is called “approval voting.”
Another possible rule is the so-called “Instant Runoff” vote, where if your candidate is a clear loser, your second (or third or fourth . . .) choice is used to help decide the runoff. (Vote #3.) Burlington, Vermont, for example, used this method‚ and then repealed its use.
Given the variety of methods, I’m sure voters and candidates would love to know which one would provide the “best” results—a problem economists, political scientists, and mathematicians have been studying for decades. But the question is far older.
In 1433, Nicholas of Cusa, a church lawyer, discovered that the Emperor was coming to the church council where he was serving. It seemed an ideal opportunity to push for reform, and he went so far as to create what he called a most “righteous, just [and] honest” way to tally the votes for the Emperor.
Fast forward to today’s local sports page and you’ll find (essentially) his method in the college football polling, where teams are ranked from highest to lowest, and then points assigned based on ranking for each voter. The most points wins. (Vote #4.)
Unfortunately, though, just like the plurality vote, Cusa’s method (also called the Borda count) suffers from a basic problem. Namely, you should expect strange paradoxes if you compare head-to-head results (like the exit polling for Bush/Gore without third-party candidates), with the actual election results including all candidates.
In fact, Nobel laureate Kenneth Arrow's celebrated theorem (along with some extensions) shows that every election procedure is flawed in a similar way long before we even start counting the votes—much less ballot-box stuffing, voter fraud, or inept officials. 
Along similar lines, recall how some Romney supporters appealed in the primaries to his Republican rivals’ voters. They wanted those voters to misrepresent their true choices by voting for Romney on the basis of “electability” in the final election—presumably leading to a more desirable outcome vis-à-vis re-electing Obama. It turns out one can rigorously prove that every method can give voters the opportunity to get a better (in their perception) final outcome by casting an “insincere” ballot.
On November 6, we’ll enter the polls and hopefully we’ll be informed about what can go wrong. But do we know what can go right? On that count, the mathematics of voting today has become as opinionated and diverse as the approaches themselves. So a different part of Cusa’s proposal might be worth revisiting instead.
Considering the history of intrigue and corruption in elections of his day, Cusa called each voter to, “in the name of God . . . ponder, directed by his conscience, who among all candidates,” is best. Maybe we should urge everyone to make the best decisions they can, even while knowing that sometimes, without any foul play, the outcome will seem unfair.
That doesn't mean we shouldn’t change our system. It does mean that if we challenge it, we need to know all the consequences. Then, directed by our conscience, we can ponder who among all candidates deserves our votes. Because they really do count.

Karl-Dieter Crisman is an assistant professor of mathematics at Gordon College in Wenham, MA. He and his family live in Lynn, MA. 


Karl-Dieter Crisman