MAT 105 Fall 2008

Harder than you might think

There are many examples in history where the results were disputed

We care about this because we want the outcome of the election to be “fair”

One purpose of primary elections is to narrow such a wide field down to a single nominee for each party

A common criticism of the primary process is that some states are more important than others

Pennsylvania’s primary wasn’t until April 22nd, but the first was the Iowa caucus on January 3rd

Since the Democratic primary election was so close, many states with late primaries that ordinarily wouldn’t have been important were suddenly vital to each candidate’s success

Whenever an election is close, there is usually controversy

The most recent exampleof this was the Democratic primary involving Barack Obama and Hillary Clinton

Some Clinton supporters still don’t support Obama, even though they agree on many policies

This is far from the first time there was a controversial election that got national attention

We will look at several examples from recent (and not so recent) years

Al Gore vs. George W. Bush

There were two other candidates: Ralph Nader (Green Party) and Pat Buchanan (Reform Party)

The result of the popular vote was: Gore 48.4% Bush 47.9% Nader 2.7% Buchanan 0.4%

Even though Al Gore won the popular vote, George W. Bush won the electoral college

The result of Florida was in dispute for several weeks, but eventually Florida’s electoral votes were given to Bush after the Supreme Court ordered a stop to recounts

The official margin of victory for Bush in Florida was 537 votes, out of 5.8 million votes cast

The election in 2000 wasn’t the first time there had been a third-party “spoiler” in a national election

In 1912, after Theodore Roosevelt failed to get the Republican party nomination, he ran as a third-party candidate

This split the Republican vote, and Woodrow Wilson, the Democrat, won the election with only 42% of the popular vote

Woodrow Wilson

TheodoreRoosevelt

William Howard Taft

The example from 1912 teaches us that two candidates can “split” a large portion of the voters, leading to an unanticipated outcome

For example, if Mitt Romney had decided, after losing the Republican nomination to John McCain, to run as a third-party candidate, this would almost certainly guarantee that the Democrats would win the 2008 Presidential race

Three main candidates: Norm Coleman (R) Hubert Humphrey (D) Jesse Ventura (Reform Party)

The results of the vote were: Ventura (37%) Coleman (34%) Humphrey (28%)

Norm Coleman

Hubert Humphrey

Jesse Ventura

Few people expected the former professional wrestler to win the election

Most of the people who voted for Coleman or Humphrey probably had Ventura as their last choice

That means that Ventura was elected governor even though 63% of the voters would have ranked him last!

There are many different systems, as we will learn The most common system used in US elections is the

plurality system: the candidate who gets more votes than any other candidate is said to receive a “plurality”

A candidate receives a “majority” if they earn more than half of the total number of votes Al Gore won a plurality of the popular vote in 2000

Woodrow Wilson won a plurality of the popular vote in 1912

Jesse Ventura won a plurality of the vote in 1998 None of these candidates won a majority

In most US elections, voters can only cast a single ballot for the candidate he or she likes the best

However, most voters will have “preference lists”: a ranking of the candidates in order of most preferred to least preferred

For example, many (but not all) of the people who voted for Ralph Nader in 2000 would have had Al Gore as their second choice

Suppose a class of children is trying to decide what drink to have with their lunch

The choices are milk, soda, and juice

Each child votes for their top choice

The results are: Milk 6 Soda 5 Juice 4

Milk wins a plurality of the votes, but not a majority

What if we ask the children to rank the drinks in order of preference?

6 have the preference Milk > Soda > Juice 5 have the preference Soda > Juice > Milk 4 have the preference Juice > Soda > Milk

Is the outcome fair? If we choose Milk as the winner of this election, 9 of the 15 students are “stuck” with their last choice

We will not allow ties on individual preference lists, though some methods will result in an overall tie

All candidates must be listed in a specific order

We will sometimes assume that the number of voters is odd to avoid ties (remember we will think about applying these methods to situations where we have thousands or millions of voters)

We’ll start off simple and only consider the case where we have two candidates

There are only two preferences: A > B and B > A

Voters with preference A > B vote for A

Voters with preference B > A vote for B

The candidate with the most votes wins

This method is called majority rule

Notice that one of the two candidates will definitely get a majority (they can’t both get less than half of the votes)

Majority rule has three desirable properties anonymous neutral monotone

If any two voters exchange (marked) ballots before submitting them, the outcome of the election does not change

In this way, who is casting the vote doesn’t impact the result of the vote; all the voters are treated equally

If a new election were held and every voter reversed their vote (people who voted for A now vote for B, and vice versa), then the outcome of the election is also reversed

In this way, one candidate isn’t being given preference over another; the candidates are treated equally

If a new election were held and a single voter were to change his or her ballot from being a vote for the loser of the previous election to being a vote for the winner of the previous election, and everyone else voted exactly as before, then the outcome of the new election would be the same as the outcome of the previous election

Changing your vote from the loser to the winner shouldn’t help the loser

Majority rule is not the only way to determine the winner of an election with two candidates

May’s Theorem states that majority rule is the only method for determining the winner of an election with two candidates that satisfies all three conditions: anonymous, neutral, and monotone

Matriarchy: only the votes of women count Dictatorship: there is a certain voter called the dictator, and only the dictator’s vote counts (all other ballots are ignored)

Oligarchy: there is a small council of voters, and only their votes count (think of the oligarchs as “co-dictators”)

Minority rule: the candidate who gets the fewest votes wins

Imposed rule: a certain candidate wins no matter what the votes are

All of these methods are “unfair,” but fairness can be a very subjective concept

May’s Theorem gives us a way to think of fairness objectively

An election method that satisfies all three conditions is “fair,” and a method that does not isn’t