Voting Rule (COMSOC)

Definition

Voting Rule ( COMSOC)

A voting rule maps the reported preferences of voters to winning alternatives. For $n$ voters and a non-empty alternative set $C$, its input is a profile $R\in \mathcal{L}(C{)}^n$ of rankings.

Set-valued rule

A social choice correspondence returns a non-empty set of winners:

$$\lambda :\mathcal{L}(C{)}^n\to 2^C\setminus \{\varnothing \}.$$

This is the convention used in the course; several alternatives may tie as winners.

Single-valued rule

A social choice function returns exactly one winner:

$$f:\mathcal{L}(C{)}^n\to C.$$

The rule specifies the outcome for every admissible profile; an algorithm specifies how to compute that outcome.