33  Social Welfare Functions

So far, we have focused on voting methods that select winners from a set of candidates. However, in many situations, we want not just to identify winners, but to create a complete ranking of all alternatives. A Social Welfare Function provides such a ranking.

Definition 33.1 (Social Welfare Function) A Social Welfare Function \(f\) maps each profile \(\mathbf{P}\) from some domain \(\mathcal{D}\) to an ordering of the candidates. For each pair of candidates \(x\) and \(y\), this ordering specifies whether \(x\) is ranked above \(y\), \(y\) is ranked above \(x\), or \(x\) and \(y\) are tied.

Formally, for any profile \(\mathbf{P}\), the social ordering \(f(\mathbf{P})\) is a relation on the set of candidates where:

• \(x\) is ranked above \(y\) according to \(f\) in \(\mathbf{P}\) if \((x,y)\in f(\mathbf{P})\) and \((y,x)\notin f(\mathbf{P})\)
• \(x\) is tied with \(y\) according to \(f\) in \(\mathbf{P}\) if \((x,y)\in f(\mathbf{P})\) and \((y,x)\in f(\mathbf{P})\)

Key properties of Social Welfare Functions:

• Decisiveness: Every profile maps to exactly one ordering of the candidates.
• Completeness: The ordering ranks all pairs of candidates (either with one above the other or tied).
• Domain: The function’s domain \(\mathcal{D}\) specifies which profiles are allowed as inputs.

The following are some examples of Social Welfare Functions:

• Borda Ranking: The Borda ranking \(B(\mathbf{P})\) orders candidates based on their Borda scores:
  • Candidate \(x\) is ranked above \(y\) if \(x\) has a higher Borda score than \(y\).
  • Candidates with equal Borda scores are tied.
• Majority Ranking: The Majority ranking \(M(\mathbf{P})\) is based on pairwise comparisons:
  • \(x\) is ranked above \(y\) if \(Margin_\mathbf{P}(x, y) > 0\)
  • \(x\) and \(y\) are tied if \(Margin_\mathbf{P}(x, y) = 0\)
• Unanimity Ranking: The Unanimity ranking \(U(\mathbf{P})\) reflects universal agreement:
  • \(x\) is ranked above \(y\) if every voter ranks \(x\) above \(y\).
  • Otherwise, \(x\) and \(y\) are tied.
• Plurality Ranking: The Plurality ranking \(Pl(\mathbf{P})\) orders candidates by their first-place votes:
  • \(x\) is ranked above \(y\) if \(x\) has more first-place votes than \(y\).
  • Candidates with equal first-place votes are tied.
• Instant Runoff Ranking: The Instant Runoff ranking \(I(\mathbf{P})\) is based on the elimination order in the Instant Runoff algorithm (Chapter 23):
  • \(x\) is ranked above \(y\) if \(y\) is eliminated before \(x\).
  • Candidates eliminated in the same round are tied.
• Maximum Loss Ranking: The Maximum Loss ranking \(ML(\mathbf{P})\) is based on minimizing the maximum loss:
  • \(x\) is ranked above \(y\) if \[\max\{Margin_\mathbf{P}(z, x)\mid Margin_\mathbf{P}(z, x) > 0\} < \max\{Margin_\mathbf{P}(z, y)\mid Margin_\mathbf{P}(z, y) > 0\}.\]
  • \(x\) is tied with \(y\) if \[\max\{Margin_\mathbf{P}(z, x)\mid Margin_\mathbf{P}(z, x) > 0\} = \max\{Margin_\mathbf{P}(z, y)\mid Margin_\mathbf{P}(z, y) > 0\}.\]

These methods can produce different rankings of the same profile. Consider the following example profile with 100 voters and three candidates (\(t\), \(r\), and \(k\)):

\[\begin{array}{ccc} 40 & 35 & 25 \\\hline t & r & k\\ k & k & r\\ r & t & t \end{array}\]

The different methods introduced above produce the following rankings:

• Borda and Majority: \(k\) is ranked above \(r\) is ranked above \(t\);
• Unanimity: all candidates are tied;
• Plurality: \(t\) is ranked above \(r\) is ranked above \(k\);
• Instant Runoff: \(r\) is ranked above \(t\) is ranked above \(k\); and
• Maximum Loss: \(k\) is ranked above \(t\) is ranked above \(r\)

This example illustrates how different Social Welfare Functions can disagree about the proper ranking of a set of candidates, even with the same underlying voter preferences.

The next chapter explores properties of Social Welfare Functions and discusses one of the most important results in social choice theory: Arrow’s impossibility theorem. This theorem shows that no Social Welfare Function can satisfy certain basic and seemingly reasonable properties simultaneously.