21  Voting Methods

A voting method is a function that determines the winning candidate(s) based on the ballots submitted by the voters. Formally, a voting method is defined as follows: \[F: L(X)^V \rightarrow \wp(X) \setminus \{\varnothing\},\] where

For any profile \(\mathbf{P}\), we write \(F(\mathbf{P})\) to denote the set of winning candidates in the profile \(\mathbf{P}\) according to the voting method \(F\).

When there are only two candidates, say \(a\) and \(b\), all reasonable voting methods reduce to the Majority Rule:

When there are more than two candidates, an important issue arises: it is possible that no single candidate is ranked first by more than half of the voters. A candidate \(a\) is called the absolute majority winner if more than half of the voters rank \(a\) above every other candidate (i.e., rank \(a\) in first place). However, a central challenge in the study of voting methods is that an absolute majority winner may not exist in every election. For instance, consider the following profile:

\[\begin{array}{ccc} 30 & 30 & 40\\\hline a & b & c\\ b & a & a\\ c & c & b\\ \end{array}\]

There are 100 voters in this election. To be an absolute majority winner, a candidate must be ranked first by at least 51 voters. In this case, both \(a\) and \(b\) are ranked first by 30 voters, while \(c\) is ranked first by 40 voters. Although \(c\) is ranked first by the most voters, \(c\) is not the absolute majority winner because \(c\) is ranked first by fewer than half of the voters. Moreover, while \(c\) is ranked first by the most voters, \(c\) is also ranked last by 60 voters! So, who should win this election? This example illustrates the challenge of determining a winner when there is no absolute majority winner.

Because there is often no single candidate that is an obvious winner in many elections, a variety of voting methods have been proposed. These methods differ significantly in how they aggregate voters’ ballots to determine a winner, with each offering unique strengths and weaknesses. In this course, we will examine several of these methods, focusing on how they handle different types of elections.

We conclude this section with a useful definition. A voting method is said to be resolute if it always selects exactly one winner. Formally, this means that for every profile \(\mathbf{P}\), we have \(|F(\mathbf{P})| = 1\) (where \(|X|\) denotes the number of elements in a set \(X\)).

For example, consider a profile with two voters and two candidates, \(a\) and \(b\), where one voter ranks \(a\) above \(b\), and the other ranks \(b\) above \(a\). In this election, it is clear that \(a\) and \(b\) should be tied for the win (for instance, under Majority Rule, the winning set would be \(\{a, b\}\)). However, a resolute voting method will select only a single winner, which means it must include a tiebreaking rule as part of its definition. In this course, we will not focus on tiebreaking rules. Instead, our attention will be on non-resolute voting methods, which allow tied outcomes.