Neutral Voting Rule (COMSOC)

Definition

Neutral Voting Rule ( COMSOC)

A voting rule is neutral if renaming the alternatives only renames its outcome. Let be a profile over , and let be a permutation. Apply the same renaming to every ballot:

For every profile and every permutation , the following condition must hold for the rule’s output type. One renaming that changes the outcome beyond merely renaming it disproves neutrality.

Single-winner output

If wins before renaming, wins afterwards.

Set-valued output

The whole winner set is renamed; alternative names do not favour particular winners.

Ranking-valued output

For a rule returning a set of collective rankings,

This is the lecture’s condition for the set of Kemeny consensuses: rename every alternative in every optimal ranking, without changing its positions.

Intuition

Start with a renaming of alternatives: , , . It exchanges the names , not their positions on a ballot:

The same symbol is then used for renaming larger objects, entry by entry:

These are different operations induced by the same renaming. Strictly, the ranking operation has type ; writing it as is a shorthand, not a claim that rankings belong to .

Now suppose . Neutrality compares two computations:

The first equality follows from the renaming. The second is what neutrality requires of the voting rule. Both computations must agree: changing names must not change which positions and preferences lead to victory.