Preference Profile (COMSOC)

Definition

Preference Profile ( COMSOC)

In COMSOC, a preference profile is the tuple of preferences of all voters over a common set of alternatives . For voters reporting strict preference rankings,

where means that voter prefers to . The profile keeps one entry per voter, including repeated rankings; it is not a single collective ranking. Other models replace rankings by approval sets or scores.

Further, for alternatives , define as the multiset of indices preferring over , i.e.,

Rankings as Relations

A ranking can be represented by a binary relation , where means “ is at least as preferred as ”. It is complete and transitive:

These conditions allow ties. To exclude them, additionally require antisymmetry:

which gives a total order, whose strict preference relation is

The notation above uses this no-ties convention.

Number of Rankings

For alternatives, a ranking without ties is a permutation: choose the first alternative in ways, the second in ways, and so on:

Thus there are exactly rankings, and . This count assumes no ties; completeness and transitivity alone do not imply it. For labelled voters, independently choosing their rankings gives possible profiles.