Voting and Elections

I. May's Theorem

Suppose that you have two candidates Peter and Liz running for senior student body president. How should the winner of the election be decided?
Jason, the current student body president, suggest that all 101 will be seniors vote. We have several options for how to determine the winner of this election.
All of these are legitimate voting systems, i.e. the way in which the winner of an election is determined from the individual ballots. However, most would not call these systems "fair". In order for us to objectively determine what a "fair" voting system is let's list some properties that a fair voting system should have.

Problem Set 1 : Look at the three voting systems we have considered and decide if they do, or do not, have each of these properties.
Imposed Rule
Minority Rule

End of Problem Set 1

Notice something about your solution to activity 1? None of those voting systems have all three properties. What about using majority rule for the election? Does majority rule have those three properties? Well indeed it does. These observations lead us to the title of the section.

May's Theorem (Kenneth 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 that avoids the possibilites of ties.

Now, to be able to prove this theorem we will need to develop some terminalogy and make a few observations.

A voting system is called a quota system if there is some number q, called the quota, such that a candidate will be declared the winner if and only if they receive at least q votes. Note that in a quota system there could be two winners in an election or two losers, also that the quota could depend on the number of voters in the election (i.e. the quota could mean that a candidate must get 75% of the votes cast to become the winner).

Lemma: If a voting system (call it V) for an election with two candidates is anonymous, neutral, and monotone, then it is a quota system.

First, we will prove this lemma. Then, we will see how it implies May's Theorem.

Problem Set 2 : In an election with two candidates A and B, and n voters (labeled v1, v2, v3, ...,vn-1, vn). Using the voting system V answer the following serious of questions

Now, explain how you can use the answers to these questions to find a value that might work as a quota for V. Also, why can we label the voters in any order we want to? (Hint: what properties does V have?)

Now, with this potential quota q for V, using the three properties of V, cleary explain why the following statements are true.
Now, using the answers you have so far from Activity 2, explain why the Lemma is true.
End of Problem Set 2

So, now that we have shown the validity of the Lemma, how do we show that the Lemma gives us May's Theorem? Back to the statement of May's theorem, since our voting system is anonymous, neutral, and monotone, by the Lemma it is a quota system. Also, we have an odd number of voters n. All that remains to show is that the quota q is not (n+1)/2 (this is the quota that gives us majority rules), then there is a possibility for ties.

Note: Using the same line of reasoning, you could show that there exist no voting system for two candidates, when there are an even number of voters, that is anonymous, neutral, and monotone, and that avoids the possibility of ties!

References and Further Reading:
[1]Jonathan K. Hodge and Richard E. Klima.The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, R.I., 2005.