In this section, we will explore the following questions.
What is a voting system?
What are some common means of choosing winners of elections?
What are some drawbacks for these common systems?
Our main task in this chapter is to explore the mathematics of voting systems. A voting system refers both to the ways votes are cast in elections and the way those votes are used to determine a winner.
We will see in this section that while two-candidate systems are fairly straightforward, they provide a fertile ground for exploring the characteristics we desire in all systems. We first want to determine how each system treats the voters and the candidates. A fair system should aim to treat each voter and each candidate equally.
Activity7.1.1.
It is time for the citizens of Scranton to elect a new mayor, and they have two choices: Angela and Stanley. How could the winner of this election be determined? How should the winner of this election be determined?
Investigation7.1.2.
If Activity 7.1.1 seemed too easy, consider the following (likely alternative) suggestion.
Angela’s husband, Dwight, has been a longtime player in Scranton politics. Thus, the system Scranton has adopted is: everyone votes, but whomever Dwight votes for will be declared the winner, regardless of how the other votes are cast.
Of the 76,328 citizens of Scranton, suppose 76,327 vote for Stanley, and Dwight votes for Angela. Who wins the election?
The method described in Investigation 7.1.2 is known as a dictatorship, with Dwight as the dictator. We note that, in the context of voting theory, the word dictatorship only refers to how the outcomes of elections are determined and has nothing to do with the system of government.
Exploration7.1.3.
Does a dictatorship treat all voters equally? That is, does every vote count “the same”? Does a dictatorship treat the candidates equally? Explain.
Investigation7.1.4.
If a dictatorship isn’t your style, try this one: before the election runs and the citizens of Scranton vote, it is decided that Angela will win. Does this method treat all of the voters equally? That is, do all votes count the same? Does it treat the candidates equally? That is, could each candidate win, depending on the how the votes come in? Explain.
Consider the following system. Each citizen of Scranton votes, and the votes are counted. The winner is the candidate with the smallest number of votes. This is known as minority rule.
Suppose that the citizens of Scranton vote, with 76,327 voters voting for Stanley and one (Dwight, no longer a dictator) votes for Angela. Who wins under minority rule?
Now suppose that Dwight convinces 38164 of the prospective Stanley voters to switch and vote for Angela. Who wins in this case under minority rule?
Does minority rule treat all voters equally? Does it treat all candidates equally? Explain.
Under minority rule, is it beneficial or detrimental for a candidate to receive additional votes? Explain.
In our quest for a mathematically just voting system, we will find it helpful to define certain desirable qualities for a fair and just system to satisfy. However: When we say that a voting system satisfies a certain criterion, we mean that it always satisfies it; that is, it can never violate that criterion. We will refer to these important criteria as fairness criteria. We have already observed a few important criteria that we now formally define.
Definition7.1.6.
A voting system for a two-candidate election is anonymous if it treats all voters equally. That is, if any two voters switched their votes, the outcome of the election should remain the same.
A voting system for a two-candidate election is neutral if it treats both candidates equally. That is, if every voter changes their vote, the outcome of the election should also change.
A voting system for a two-candidate election is monotone if is impossible for a winning candidate to become a losing candidate by gaining votes (and not losing any others), or for a losing candidate to become a winning candidate by losing votes (and not gaining any others).
Investigation7.1.7.
Now suppose three members of the board of the Dunder Mifflin Paper Company are voting to determine who should take over the business. The candidates are Michael and Andy. In order to vote, their “friend”, Toby, has devised a voting system. Three possible combinations of votes by Jim, Pam, and Kevin are presented in Table 7.1.8.
Table7.1.8.The results of Toby’s voting system.
Jim
Pam
Kevin
Winner
A
M
M
A
A
A
M
M
M
M
A
M
Which of the three properties described in Definition 7.1.6 are satisfied by Toby’s voting system? Explain.
Is Toby’s sytem equivalent to any of the other three systems we’ve investigated thus far? Why or why not?
Investigation7.1.9.
Let’s investigate the three criteria in Definition 7.1.6 in relation to the three voting systems we’ve examined.
Which of the three fairness criteria are satisfied by dictatorships? Explain clearly for each of your answers.
Which of the three fairness criteria are satisfied by imposed rule? Explain clearly for each of your answers.
Which of the three fairness criteria are satisfied by minority rule? Explain clearly for each of your answers.
The two-candidate voting system we haven’t yet explored is probably the one that you actually suggested in Activity 7.1.1. Perhaps it is something like:
Each voter should vote for the candidate they want to win the election. Whichever candidate gets the most votes wins the election.
This is known as majority rule.
Investigation7.1.11.
Which of the three fairness criteria does majority rule satisfy? Explain.
In fact, for elections with two candidates, majority rule is the only system to satisfy all three fairness criteria.
Theorem7.1.12(K. May (1952)).
In a two-candidate election with an odd number of voters, majority rule is the only voting system that is anonymous, neutral, and monotone and avoids the possibility of a tie.
Thus, for elections with two candidates, there is a clear fair and just choice for a voting system subject: majority rule. However, as we know, most elections don’t have just two candidates, so in the remainder of this chapter we will consider the question of fairness in the context of these larger elections. We’ll refine the existing fairness criteria to handle three or more candidates, and develop additional criteria due to surprising situations that can arise.
