Ch. 17.1: Voting Methods
The mathematical/sociological study of voting methods is more properly referred to as Social Choice Theory. It is a very important and hotly-debated subject, for obvious reasons! Methods of voting have been developed and debated since democracy in ancient Athens (probably earlier). It was in 1785 that a French philosopher named Jean Antoine Nicolas Caritat (the Marquis of Condorcet) published two foundational mathematical results, now known as Condorcet’s Jury Theorem, and Condorcet’s Paradox. The term “Social Choice Theory”, however, was not coined until after 1951, with the publishing of “Social Choice and Individual Values” by American economist Kenneth Arrow. These two men are often credited as the founders of the modern theory. In this section, we will look at seven different methods of casting and evaluating votes: i. Majority Rule ii. Plurality Rule iii. Borda Count iv. Hare Method v. Pairwise Comparison Method vi. Approval Method vii. Tournament Method Here on, we consider the general problem in which a group of n people must choose exactly 1 option (e.g. “C”) from a list of distinct options (A,B,C, etc.). These options could be candidates running for office, bills, legal decisions, etc. The one that is chosen we shall call the “winner”.

1

Plurality Voting
In plurality voting (also called first-past-the-post voting in reference to horse races), each voter casts exactly 1 vote in favour of just 1 option. The winner is the option that earns a certain number of votes.

• Majority Rule: The winner is the option that earns more than 50% of all votes (a simple majority). If the number of votes, n, is even, then a simple majority is at least n + 1. 2 If n is odd, then a simple majority is at least n+1 . 2 • Plurality Rule: The winner is the option that earns more votes than any other candidate individually (a plurality).

Example 1: