The quirky math of voting

John Allen Paulos

is a professor of mathematics

at Temple University

April is Mathematics Awareness Month. This year's theme is the math of voting (

» READ MORE: http://www.mathaware.org/mam/08

) and math will play its inevitable role in this month's primary in Pennsylvania, which possibly may decide the Democratic presidential nominee. So, some math notes are in order.

The outcome of an election depends on what the rules are, who's running, and who gets to vote. These simple facts have generated volumes on the mathematics of voting.

Oddly, only one of various possible voting methods has been deployed in the primaries. The "plurality winner" in a state - the candidate receiving the most votes, although not necessarily a majority - is always declared the winner. Aside from caucuses and superdelegates, which are slight deviations from plurality, the parties have totally ignored other methods such as runoffs, range voting, assigning of numbers to candidates, ranking of candidates, and approval voting.

The parties used plurality to decide the winner in the individual state primaries, but the Republicans opted for a winner-take-all version that led to a relatively quick decision. (Even more determinative for them was the slate of candidates deciding to compete. Had either Mike Huckabee or Fred Thompson not entered the race, or had they dropped out early, the conservative votes that they split would have benefited Mitt Romney, and John McCain would very likely not have been the Republican nominee.) The Democrats opted for dividing the delegates roughly in proportion to the candidates' vote totals in the various states, so there is a seemingly close contest between Sen. Barack Obama and Sen. Hillary Clinton. I say

seemingly

because mathematically it is quite doubtful Clinton can catch up to Obama no matter how the Keystone State votes.

Even with the plurality standard, candidates find a way to highlight their strengths. Clinton stresses her big-state victories (even the unofficial ones in Florida and Michigan) and Obama stresses his lead in pledged delegates.

This natural tendency to spin matters recalls the advice the old lawyer gave to his protegé. "When the law's on your side, pound the law. When the facts are on your side, pound the facts. And when neither is on your side, pound the table." One of the more humorous instances of pounding the table occurred during the brief 1992 presidential campaign of Iowa Sen. Tom Harkin. Some of his staffers noted that, though he was far behind in the primaries, their man was actually the front-runner. They argued that Harkin, who had won caucuses or primaries in Minnesota, Iowa, Idaho and Montana, had captured the largest

land mass

of any of the contenders.

In addition to the rules used and the candidates entered in the fray, there is a thorny imponderable: which side is more effective at getting the vote out. Organization, of course, is critical, but to be cold-blooded about it, parties want to enfranchise as many supporters and disenfranchise - or at least discourage - as many opponents as reasonably possible. The time-honored custom of stuffing the ballot box (or altering its software) illustrates the former; opposition to women's suffrage and the practice of apartheid are historical examples of the latter. Variants of these tendencies extend to virtual voters. Antiabortion activists sometimes enlist the "votes" of the unborn, while environmentalists occasionally go further and appeal to the "electoral" support of unconceived future generations.

No matter what system is used, oddities and anomalies are always a possibility. In fact, the Nobel Prize-winning economist Kenneth J. Arrow demonstrated that there was never a foolproof way to derive group preferences from individual preferences that can be absolutely guaranteed to satisfy a few minimal conditions: not allowing dictators, requiring that the electorate's preferring X to Y and Y to Z should result in its preferring X to Z, and the like. The injunction to be democratic is a formal one. The real question is:

How

should we be democratic? Politicians and their supporters who are the beneficiaries of a particular electoral system naturally wrap themselves in the mantle of democracy and need to be reminded that this mantle can come in different styles, all of them with patches.

Finally, I should mention that every once in a while,

issues

are important, and this election might very well be one of those once-in-a-whiles. The issues are stark: the relentless and disastrous war in Iraq, the recession and growing economic disparities, health care, the disturbing conflation of church and state, an educational system not up to the requirements of the 21st century, intrusive and unauthorized spying on Americans, the restriction of habeas corpus, and other violations of our Constitution.

Because of his position on these and most other issues, I, like many other voters in Pennsylvania, have decided to support Barack Obama for president. I would feel this way even had Bill Clinton come over to my house to schmooze with me during March Madness.