Unanimous Preference Profile (COMSOC)

comsoc

Definition

Unanimous Preference Profile (COMSOC)

In COMSOC, a preference profile is unanimous on the ordered pair $x\succ y$ if every voter ranks the alternative $x$ above the alternative $y$.

Formally, let

$$R=({\succ }_1,\dots ,{\succ }_n)$$

be a preference profile over alternatives $C$. For $x,y\in C$, define

$$R(x,y):=\{i\in \{1,\dots ,n\}:x{\succ }_{i}y\}.$$

The profile is unanimous on $x\succ y$ precisely when

$$R(x,y)=\{1,\dots ,n\}.$$

If a profile is called unanimous without specifying a pair, this usually means full unanimity: all voters have the same complete preference ranking. Pairwise unanimity is weaker; it only requires agreement about one ordered pair.

Notation

Some lecture notes write the condition as

$$R(x,y)=R.$$

Here $R(x,y)$ denotes the voters in the profile who rank $x$ above $y$, while $R$ denotes the whole profile or voter set. Thus the equality means that the supporting set for $x\succ y$ is the whole profile.

Example

Pairwise unanimity without full unanimity

Let

$$R=(abc,acb,cab).$$

Every voter ranks $a$ above $b$, so the profile is unanimous on $a\succ b$:

$$R(a,b)=\{1,2,3\}.$$

The profile is not fully unanimous, because the voters do not have the same complete ranking.