Plurality Score and Plurality Rule
The plurality score is the positional scoring vector that puts all weight on the top rank: an alternative earns $1$ from each voter who ranks it first and $0$ otherwise. As a rank-indexed vector (rank $0$ is the top), $$ \mathrm{pluralityScore}(m, r) = \begin{cases} 1 & r = 0 \\ 0 & r \ne 0,\end{cases} $$ independent of the number of alternatives $m$.
The plurality rule is the positional scoring rule built from this vector. At a profile $P$ its winner set is the alternatives maximising total plurality score — those ranked first by the most voters, with ties allowed: $$ \mathrm{plurality}(P) = \arg\max_{a \in A}\ \#\{i \in N \mid a \text{ is } i\text{'s top choice}\}. $$
In Lean: SocialChoice.Voting.pluralityScore (the score vector) and
SocialChoice.Voting.plurality (the rule, scoringRule pluralityScore, a
set-valued VotingRule). As with Borda ([[social_choice.voting.borda_score]]),
these are noncomputable because aggregating a bare-Prop strict preference
uses classical decidability, and ties are kept in the winner set.
Properties (informal)
- Plurality is the simplest positional rule (all weight on the top rank).
- Plurality is not a Condorcet method: examples exist where the Condorcet winner is not even a plurality winner.
- Plurality satisfies unanimity ([[social_choice.voting.unanimity]]) but is strategy-manipulable for $|A| \ge 3$ ([[social_choice.voting.gibbard_satterthwaite]]).
References
- [AGT Chapter 10] Nisan, Roughgarden, Tardos, and Vazirani, Algorithmic Game Theory. Plurality and positional scoring rules.