## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## May 18, 2007

### Tricky Choices

Voting sounds so simple. Whether you're picking a class president or deciding which snack food you and your pals should get, you just make a choice, someone counts the votes, and the majority wins. Right?

That's true if there were only two candidates or two choices. But as soon as you have more than two choices, things can get pretty complicated and crazy. The winner may depend on the voting procedure you use. And the most popular choice might not even win.

How could that be? Here's an example. Suppose your class of 15 kids must decide which one of three snacks (pizza, popcorn, or cookies) to get for a party. Six kids like pizza best, cookies next, then popcorn. Five kids like popcorn best, cookies next, then pizza. Four kids like cookies best, popcorn next, then pizza.

If each person voted only for his or her favorite snack, pizza would win, popcorn would come second, and cookies would come third.

Would everyone be happy? If you take a close look at everyone's preferences, you'll see that nine kids actually like popcorn more than they like pizza. Similarly, nine kids like cookies more than pizza, and ten kids like cookies more than popcorn. So if you get a pizza, you voted, but the majority did not rule.

How could you get a truer vote? Suppose you dropped the choice with the fewest votes in each round and voted again until the top choice had more than half the votes. In this case, cookies would be dropped first. In the runoff, however, popcorn would beat pizza!

The winner has changed, and, hey, what about cookies? More people like them than like popcorn.

This is very confusing. What if you try a voting system in which you give two points to your first choice, one point to your second choice, and no points to your third choice. The winner is the one with the most points. In this case, cookies would win. The answer changed again, and you ended up with cookies even though most kids wanted pizza or popcorn rather than cookies!

So just by changing the voting procedure, you can get a different result—and still not satisfy the majority of the voters.

Mathematicians who have studied voting procedures involving three or more choices have come to a startling conclusion. It turns out that no matter which system you use, it's always possible to get results that don't look fair.

It can happen in all sorts of voting when you have a lot of choices—whether you're picking pizza or a president.

More choices (Oct. 27, 2004)

Lots of things can go wrong in elections. In national, state, and local contests, people worry about voting machines that don't work properly, poorly designed ballots that make it hard to vote for the candidates you want, and votes that are miscounted.

Officials are trying to fix these kinds of problems to make sure elections are fair and run smoothly. In some places, they're even trying out different voting rules.

For example, instead of picking just one candidate from a list, voters in San Francisco will now rank their choices, from first to third. These votes are counted in a special way to determine the winner.

Here's what happens in elections for San Francisco's Board of Supervisors.

• Voters pick three candidates from a list. They rank their choices, from first to third.
• All the top choices are counted first. If any candidate gets more than 50 percent of the vote, that candidate wins.
• If no candidate has a majority, the candidate with the fewest first-place votes is eliminated. Voters who marked the losing candidate as their first choice will have their votes counted for their second-choice candidate.
• The process continues until one candidate receives a majority of the vote.

Does that sound confusing? Will the system discourage people from voting? Does it look like it might take a long time to get the final results? Will the final result be any fairer?

The first test of the San Francisco system will take place in November. It'll be interesting to see what happens.

Muse, October 2002, p. 24.