Donald Saari is a professor of mathematics and economics at UC Irvine and director of the Institute for Mathematical Behavioral Sciences.

Los Angeles Times

October 21, 2004

Proposition 62, which will appear on the November ballot, would revamp the way the political primary system works in California. Under the new system, there would be only one primary per election season, open to all voters, who would be allowed to select any candidate on the ballot — regardless of the party affiliation of either the voter or the candidate. The top two winners would then compete in a November runoff.

It's another shot at electoral reform, motivated at least in part by the realization that our elections are not perfect — and that tinkering with the rules might provide a more democratic outcome. Indeed, there are many ways to choose among candidates (just as there are many ways to rank college football teams or judge figure-skating contests), and it would be a mistake to assume that our current system (or, for that matter, the one proposed by Proposition 62) is necessarily the best.

In fact, a close look suggests that we don't always select the preferred candidate when we go to the polls to pick a senator or a governor or a member of Congress. Let me give you an example of why a simple "plurality vote" system (whoever gets the most votes, wins) doesn't always work.

Imagine for a moment a group of 15 people. Of these, six people prefer milk, like wine second-best, and like beer least of all; five like beer the best, followed by wine and then milk; and four prefer wine, then beer, then milk. You might want to write this down.

Add it up under the plurality vote system, and you see that six find milk the best of all the drinks, five find beer the best and four say wine is their first choice. In a standard American election, milk would be selected as the best beverage.

But there are other ways to measure the outcome. For instance, you could just as easily conclude that nine out of the 15 voters — or 60% — really prefer wine to milk. And 10 out of 15 — 66% — prefer wine to beer. Looking at it that way, you could select wine as the preferred beverage.

Or, here's yet another way of determining the most popular drink: You could hold a runoff between the two highest plurality vote-gatherers — milk (with six votes) and beer (with five). That's the Proposition 62 solution.

In other words, if you take the same data but change the rules by which the decision is made, the winners can change too. When you really think about it, many elections more accurately reflect the rule rather than the voters' wishes or the data.

The first person to recognize the complexity of picking winners was Jean-Charles de Borda, a French mathematician with an eclectic life story (among other things, he helped introduce the metric system by determining the basic unit of length, the meter, and he led six ships supporting the American side in the Revolutionary War). In 1770, Borda worried whether the selection of French Academy members — done by plurality vote, like our elections today — was suspect.

Because he recognized that second- and third-place preferences also were important, Borda recommended still another system: tallying three-candidate ballots by assigning two points to a first-place candidate, one to a second and zero to a third. If we plug that system into the beverage example, wine would win and milk would come in last.

After Borda, "decision theory" fell into its own Dark Ages until, essentially, 1952, when Kenneth Arrow, an economist who is now professor emeritus at Stanford, used the muscle power of mathematics to search for a "fair election method." His conclusion, cited in his 1972 Nobel Prize, is stunning. Among other things, he looked at races with more than two candidates, concluding that when voters are choosing between, say, Ann and Barb, their view of Connie should be irrelevant.

But although that may sound obvious and reasonable, in fact, our elections generally violate this rule. An example is the 2000 Florida election: Head to head, Al Gore would have defeated George W. Bush, but with Ralph Nader in the race, he lost.

Since Arrow, academics have been working to identify optimal decision methods that reflect the "will of the people" or capture "what the data really mean." Using mathematical symmetries from higher dimensional spaces and chaotic dynamics, for instance, I recently proved that the Borda count method is the best expression of the will of the voters.

To illustrate, suppose your local school system decides to rank students strictly by the number of A's they receive. Sounds good, until you realize that a student with all Bs ranks below someone with one A and the rest Fs. The "all A's" approach, which ignores some very important information, is our plurality vote.

This analogy helps explain why Bob Dole lost to Pat Buchanan in the 1996 New Hampshire primary. In that crowded race, several candidates shared similar views, causing Dole to receive many Bs (meaning he was a voter's second choice — which gets him nothing under our system). But while Buchanan received low grades from most voters, he received enough A's (read: votes) to win.

This also helps explain the problem with Proposition 62. Because the runoff would be held between the two candidates with the most A's, it could lead to a runoff between two fringe candidates with lots of Fs but just enough A's.

Farfetched? Not really. This precise problem afflicted France in its 2002 presidential election. There were 16 candidates in the race, and the right-wing candidate, Jean-Marie Le Pen, (despite his many Fs) edged out the Socialist Prime Minister Lionel Jospin (who had many Bs).

In our schools, by contrast, our four-point grading system recognizes all grades: It corresponds to the Borda count method.

Should we adopt the Borda count method for elections as well? Well, only after modifying it to reflect pragmatic realities. Proposition 62, for that matter, would be improved if it allowed first- and second-place rankings. By understanding the mathematics of decision procedures, addressing this and other election failings becomes feasible.