Columnist: It's How Votes Get Counted That Counts

The Academy Awards are all over except for the incessant movie ads that will trumpet the results for months. But it's how the academy voted that has surprising connections to other topical issues.

The voting system used by the members of the Academy of Motion Picture Arts and Sciences often has a pronounced effect on its selections. This may appear to be a rather myopic commentary on the Oscars that only a mathematician would make, but how one keeps score helps determine who wins not only in Hollywood, but in Palm Beach, in Salt Lake, and almost everywhere there are competitions.

No one has forgotten the many types of dimpled chad and the infamous butterfly ballots that clouded the presidential election of 2000. The Electoral College always generates controversy, and strategic positioning for the 2004 Democratic presidential primaries is already being discussed by potential candidates.

Everyone is aware, too, of the vagaries of Olympic figure skating scoring, where the judging was sometimes quite subjective, to say the least. Even in those racing events such as skiing and sledding, where the scoring was objective, the times were measured in imperceptible thousandths of a second, differences that were much less than those caused by the changing weather and course conditions that the athletes confronted.

In short, much often depends on such fine discriminations and arcane measurements.

Hollywood No Different

Hollywood is no different. The winners there were chosen by a plurality system whereby the nominated movies and artists receiving the most first-place votes received the Oscars. This sounds unobjectionable until one realizes that with five nominees for each award, the winner is unlikely to be a favorite of a majority of the voting member of the academy.

The nominees, however, were chosen by a more complicated system designed to ensure variety and, some argue, quality.

Roughly it goes as follows: Each of the members of the academy ranks up to five of his or her preferences, and those nominees who receive enough first-place votes are selected. Those with the fewest first-place votes are eliminated. The votes are then redistributed among the remaining candidates. Once again those with enough first-place votes are selected and the process continues until the nominees are selected.

An unappreciated mathematical point is that who the winners are depends as much on the system used as the votes cast. There's certainly nothing wrong with this. In fact, it couldn't be otherwise.

Different Methods, Different Winners

For illustration, consider the following imaginary scenario.

Let's reduce the numbers involved and say there were only three candidates for Best Actor and only 100 voting members. Assume that 37 members rated Denzel Washington (the real winner) their favorite, Russell Crowe their second choice, and Sean Penn their third. Further assume that 18 members rated Crowe first, Washington second, and Penn third, and that the remaining 45 rated Penn first, Crowe second, and Washington third. After the presenters exchange banter, the envelope is opened and …

If the academy uses the plurality system, then Penn, who has 45 first-place votes, wins handily.

If it first eliminates the nominee with the fewest first-place votes (Crowe) and has a runoff between the two remaining nominess, then the winner is Washington, whom 55 of the 100 voters prefer to Penn.

If it stipulates that each first-place vote an actor receives is worth 3 points, each second-place vote 2 points, and each third-place vote one point, then who wins? (Answer below)

There are many alternative systems, including the increasingly popular approval voting system in which voters vote for or approve of as many nominees as they care to. A famous theorem by economist Kenneth Arrow states, however, that all systems satisfying some very simple conditions have their flaws.

It's appropriate that this year's best picture award went to A Beautiful Mind. The movie's protagonist, mathematician John Nash, made seminal contributions to game theory, a discipline not unrelated to voting systems and the ways voters and candidates respond to them. The movie was my favorite, despite the fact that there was less math in it than there is in this short column.

Answer: Given the rankings (37 - W, C, P; 18 - C, W, P; 45 - P, C, W), Crowe wins because his total number of points is (37 x 2) + (18 x 3) + (45 x 2), which equals 218. Penn's point total is (37 x 1) + (18 x 1) + (45 x 3), which equals 190, while Washington's is (37 x 3) + (18 x 2) + (45 x 1), which equals 192.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears every month.