31 Majority Rule

When making group decisions, majority rule is often taken for granted as the obvious choice. But why should we use majority rule rather than other possible voting methods?

We will start by restricting attention to situations where a group must choose between exactly two alternatives. While real-world decisions often involve multiple options, many important choices do come down to two alternatives: yes/no votes on proposals, accepting or rejecting a candidate, or approving or denying a request.

Let’s call the two options $a$ and $b$. In the simplest case, each voter has two possible choices:

Vote for $a$, meaning they strictly prefer $a$ over $b$. We write this preference as $a \mathrel{P} b$
Vote for $b$, meaning they strictly prefer $b$ over $a$. We write this preference as $b \mathrel{P} a$

While voters might also abstain or express indifference between the options, in this course we will focus on these two basic choices where voters express a clear preference for one option over the other.

When there are only two candidates, most common voting methods end up being equivalent to majority rule.

Definition 31.1 (Majority Rule) Suppose there are two candidates, $a$ and $b$. Majority Rule determines the winner as follows:

$a$ is the unique winner if more voters rank $a$ above $b than rank $b$ above $a$.
$b$ is the unique winner if more voters rank $b$ above $a than rank $a$ above $b$.
$a$ and $b$ are tied if the number of voters who rank $a$ above $b$ is equal to the number of voters who rank $b$ above $a$.

More formally, in a profile $\mathbf{P}$ for two candidates $a$ and $b$, we denote the set of majority winners as $Maj(\mathbf{P})$, which is defined as: \[Maj(\mathbf{P}) = \begin{cases} \{a\} & Margin_\mathbf{P}(a, b) > 0\\ \{b\} & Margin_\mathbf{P}(b, a) > 0\\ \{a, b\} & Margin_\mathbf{P}(a, b) = 0\\ \end{cases}\]

Majority Rule is simple and intuitive, but what makes it the best procedure to use for making a group decision when only two alternatives are available? Let’s consider two fundamental principles for group decision-making:

Democratic Responsiveness: The outcome should reflect and respond to voters’ preferences.
Equal Treatment: Every voter should have the same opportunity to influence the result.

Majority rule clearly satisfies both principles. It responds directly to voter preferences - if enough voters change their minds, the outcome changes accordingly. It also treats all voters equally - no vote counts more than any other.

However, Majority Rule is not the only procedure that satisfies these principles. As discussed in (Saunders 2010), Lottery Voting also satisfies these principles.

Definition 31.2 (Lottery Voting) Suppose there are two candidates, $a$ and $b$. Lottery Voting proceeds as follows: Each voter selects their preferred candidate (either $a$ or $b$). A single vote is then randomly selected, and the candidate chosen on that ballot becomes the winner.

Lottery Voting is democratic because the probability of an outcome being selected is proportional to the number of votes it receives. It is also egalitarian since each voter has an equal chance of being decisive in the election. However, Lottery Voting differs from Majority Rule, as the candidate with the most votes is not guaranteed to be selected as the winner.

This shows us that satisfying democratic responsiveness and equal treatment isn’t enough to uniquely justify majority rule. So why prefer majority rule over Lottery Voting? See (Risse 2004) for a critical discussion of the arguments for majority rule. In the next chapters, we present two key mathematical results that provide a stronger justification for majority rule.