here is a cynical precept of politics that says that those who cast the votes decide nothing, but those who count the votes decide everything.

A third group, though, decides quite a bit, too: those who set the election rules.

New York City's election rules could be rewritten soon, if the voters approve Mayor Michael R. Bloomberg's plan to abolish party primaries in 2009.

The mayor says his push for nonpartisan elections will lead to fairer elections. But fair can be in the eye of the beholder. While the pros and cons of the mayor's proposal will be debated in the coming months, it is worth noting that recent history is riddled with well-intentioned election reforms that ended up having unintended consequences.

Think of the decision to hold the old community school board elections in the spring, so they would not be overshadowed in November. Voters forgot they were being held, or did not care, and turnout had to be measured with electron microscopes.

Then there is New York's system of nominating candidates for State Supreme Court at judicial conventions, which was supposed to insulate the bench from politics. The thought now seems risible.

Or take a look at what happened with redistricting, which was supposed to preserve the one-person-one-vote standard by periodically redrawing district lines to take population shifts into account. In practice it regularly becomes an exercise in partisan gerrymandering, with the political parties redrawing electoral maps to guarantee themselves safe, uncontested seats. Redistricting, it has often been observed, turns the age-old principle of having the electorate choose its leaders on its head.

Some scholars would go so far as to question whether any system of choosing a winner by way of a vote can truly be fair, at least when more than two candidates are concerned.

Kenneth J. Arrow, a Stanford University economist, won the Nobel Prize in economics in 1972, in part for his work in the late 1940's and early 1950's for developing what came to be known as Arrow's Impossibility Theorem. It provided proof that there is no way to arrange an electoral system to perfectly reflect the will of the people when it comes to choosing a winner from more than two candidates.

"It was a little disquieting," Professor Arrow recalled last week, explaining that he had originally set out to find a good system.

What's the problem? Depending on how an election is structured, it is almost bound to have some imperfections from a mathematical point of view.

In the American system, the candidate who wins the most votes wins the election, even if he (and it still usually is a he) falls short of a majority. That sounds fair. But does it accurately reflect the will of the people, to the extent it can be known?

Consider the last presidential election results from Florida. If you accept the final, much-disputed tally, it showed that George W. Bush got slightly more votes than Al Gore, and both men trounced Ralph Nader. So Mr. Bush won the state, and the presidency. But most people in Florida voted for someone other than Mr. Bush. And since most Nader voters would presumably have preferred Mr. Gore to Mr. Bush, it follows that Mr. Gore would have been a more acceptable choice to most Floridians than Mr. Bush. But second choices do not count in the American system.

A number of alternative election systems exist. A runoff election between the top vote-getters can be held if no candidate gets a majority of the vote. Such a system, though, can keep members of minority groups — racial, ethnic or ideological — from winning, even when they finish first in the initial count.

In the Democratic mayoral primary in 2001, Fernando Ferrer, who forged a black-Latino coalition, beat Mark Green. But since neither man got more than 40 percent of the vote, a runoff election was held. Mr. Green won the runoff (only to lose the general election), and some of Mr. Ferrer's supporters complained that the runoff system was unfair.

Other countries use other voting systems. Many parliamentary systems use proportional representation — an alternative to the American winner-take-all system — in which parties are allocated seats based on their share of the popular vote. Such an arrangement gives a voice to smaller parties, giving their representatives a small number of seats. But critics argue that they create unstable governments held together by fragile coalitions.

Then there are systems in which voters rank their choices in some way. One such system of ranking candidates, and choosing the one at the end with the largest number of points, was championed by Jean-Charles de Borda, an 18th-century French scientist. And why was he interested in voting systems?

"He thought the wrong people were being elected to the French Academy of Sciences," said Donald G. Saari, a professor of economics and mathematics at the University of California at Irvine who has written several books about election theory. He said that the Borda system appears to be the fairest way of voting.

Many groups in the United States now advocate a ranking system called Instant Runoff Voting. Known by its avuncular-sounding acronym, IRV, this is a system in which voters rank their choices. If a candidate wins a majority of the vote on the first count, that candidate wins. Otherwise, the losing candidate is eliminated and those votes are accorded to those voters' second-choice candidate, and so on, until a winner gets a majority.

But changing voting systems can be even tougher than figuring out a butterfly ballot. For one thing, governing parties like to stick with the systems that brought them to power. For another, many new systems sound as if they came out of that New York State Math Regents exam, the one whose results were discounted because it was too difficult.

But it is not an academic question. Dan King, a professor of mathematics at Sarah Lawrence College, offered a hypothetical example to demonstrate how the choice of election system can actually determine the outcome of the election.

Imagine nine people trying to figure out which actor was the best James Bond: Sean Connery, Roger Moore or Pierce Brosnan. (O.K., actually Professor King's example said that the nine people were trying to choose between candidates A, B and C, but this is not math class.) Four of them like Connery best, followed by Moore, followed by Brosnan. Three voters like Brosnan, followed by Moore, followed by Connery. And two voters like Moore the best, followed by Brosnan, followed by Connery. So who is the best Bond?

In a plurality system, Sean Connery would be the winner: he got four votes, while Pierce Brosnan got three and Roger Moore got two. But that's no majority.

But what if a runoff election were held? Roger Moore would be eliminated, because he got the least number of votes. But the Moore fans liked Brosnan second best. That would leave four voters preferring Connery to Brosnan, and five voters preferring Brosnan to Connery. Pierce Brosnan would be the winner of the 007 runoff.

And in a Borda count — giving 1 point to the least favorite, 2 points to the second and 3 points to the favorite — Roger Moore would be the Bond of choice.

"Three procedures, three different winners," Professor King noted.

In the coming weeks Mayor Bloomberg's Charter Revision Commission will come out with proposals for how a nonpartisan election system might work: how people will qualify to get on the ballot, whether runoffs will be held, what the thresholds will be. Some of those details could be as significant as their primary goal of abolishing primaries.

"You can skew the outcome of any election," said Burt Neuborne, the legal director of the Brennan Center for Social Justice at New York University School of Law, "depending on how you structure the democratic system."