Wednesday, February 20, 2008

What Makes A Fair Election?

In a democracy we always say, majority rules. But, what exactly is a majority?

The simplest type of voting involves only two choices. With two choices, the most common way of deciding is majority rule, that is, the candidate with more than 50% of the vote wins. While this rule is taken for granted, it is important to check three properties:

1. Every vote has the same weight.

2. There is symmetry between the candidates. If all votes were reversed, the loser would become the winner.

3. If a vote for the loser were changed to a vote for the winner, the outcome of the election will not be changed.

Of course, in a democracy, there are many variations of this majority rule. A Constitutional Amendment can only be passed if it is approved by 2/3 of Congress and 3/4 of the states. A jury can only convict on criminal charges if there is 100% agreement.

Voting with more than two choices is much more complex. Sometimes, more often than not, no one gets a majority. Mo gets 35% of the votes, Larry gets 33% of the votes and Curly gets 32% of the votes. Mo cleary has the most votes, but not a majority. He has what is known as a plurality. These votes are very close. If Larry had decided to drop out and all his supporters decided to vote for Curly, Curly would win. Problems are definitely going to arise in this system. Studying this, makes it clear as to why some of our presidential hopefuls are dropping out early and giving their support to others.

In the mathematics of voting, we study different ways of holding an election, using preference schedules. We hold elections using run-offs of the top two candidates, sequential run-offs (similar to American Idol), Borda Counts (similar to figuring out GPAs) and pairwise comparisons. Holding the same election each way can give different results.

One of the things we always want is fairness. Judging fairness is subjective but mathematicians and political scientists have come up with four basic fairness criteria that must be met by a fair voting system.

Criterion 1: If a candidate recieves a majority of the first-place votes, that candidate should be the winner.
Criterion 2: If a candidate is favored over every other candidate in pairwise races, that candidate should be declared the winner.
Criterion 3: If a candidate is declared a winner and a second election is held which ranks the candidate higher, that candidate will still be the winner.
Criterion 4: If a candidate is declared a winner and a second election is held in which one of the losing candidates drops out, then that candidate should still be the winner.

In 1973 Kenneth Arrow received the Nobel Prize in Economics for his mathematical analysis of voting systems that led him to discover that no voting system satisfies all four fairness criteria.

I've only touched on this topic here, but it is one that I enjoy teaching. Delving further into it, students can appreciate how the electoral system we have now was developed. It brings real life applications of mathematics into my student's lives. It shows them that there is more to mathematics than the quadratic formula. There is a strong correlation to what they are learning in their government and in their math classes. For my weaker students, it is a way to practice arithmetic and calculator skills in an interesting manner. Kids who can't do the basics, are still drilling their basics (percents, fractions, decimals) and are not being treated like babies.

Source: Using and Understanding Mathematics by Jeffrey Bennett and William Briggs.


Anonymous said...

Not exactly true, some of the Democratic presidential candidates stand no chance in competing with the popular Hillary Clinton or Barack Obama. Do you remember Mike Gravel in the debate? It is definitely unfair because the media give all (well 99% of media time) attention to these two candidates.
Mike Gravel simply ran out of money and had to drop out. The U.S.A. government and ways in the election is definitely unfair, and the big guys are the only one who stand the chance. Good people never win in this country, and has it always been, people vote for the lesser evil.

Besides all that, our vote don't even count for much at all. Representative government is not a democratic government.

This is not a democracy. This country has become a capitalistic, bureaucratic, consumerist, republic

Son Of PissedOffTeeacher said...

WE actually don't have a democracy, we have a form of republic. They forget to explain that in schools today.

Pissed Off said...

This lesson just explains how different voting systems are set up and the mathematics that are involved in setting them up. No one says it is fair and Arrow proved that it is impossible to have a totally fair election.

Winston Churchill said:

No one pretends taht democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of government except for all those that have been tried from time to time.

David said...

In fact, (1) and (4) are inconsistent. Suppose that voters cast their ballots as follows:

6 votes for A > B > C

5 votes for B > C > A

4 votes for C > A > B

Then A cannot be the winner, because a majority prefer C to A, and C would win if B withdrew from the race. B cannot be the winner, because a majority prefer A to B. Finally, C cannot be the winner, because a majority prefer B to C.

Pissed Off said...

David, you are confusing a plurality and a majority. A majority winner must get more than 50% of the votes while a plurality winner is the one who receives the most votes.

If you study the different types of elections systems, all of the above are possible, and in fact happen. I know my post did not go into enough detail to fully explain the different systems.

David said...

I did not confuse plurality with majority. Perhaps my explanation was unclear.

I listed a scenario involving three candidates: A, B, and C. None of the candidates has a majority of first-place votes. However, a majority prefer A to B, a majority prefer B to C, and a majority prefer C to A.

Suppose that A is declared the winner. Then B is an irrelevant alternative. This means that the result should have been the same if the race were between A and C only. But nine of the 15 voters (a majority) prefer C to A, thus C should win the two-person contest. This is a contradiction, so A cannot be the winner under criteria (1) and (4).

Similarly, B cannot be declared the winner, and C cannot be declared the winner. It is not possible to select a winner in a manner that is consistent with both the majority criterion and IIA.

Pissed Off said...

That is the whole premise, that there is no system that satisfies all 4 criteria.
A plurality will sometimes violate 2 and 4
Top-two ruhn off will sometimes violate 2, 3, 4
Sequential run-off will sometimes violate 2, 3, 4
Borda count will sometimes violate 1, 2, 4

I'm sorry I misunderstood what you were saying.
pairwise will sometimes violate 4