Definition
Surjective Voting Rule ( COMSOC)
A voting rule is surjective in the alternative-wise sense if every alternative wins for at least one profile. “Has a chance to win” means that a suitable profile exists, not that the rule uses randomness.
Single-winner output
For
, This is ordinary surjectivity onto
. A singleton-valued representation gives the equivalent condition . Set-valued output
For
, the analogous requirement is Every alternative can be a co-winner. This is not surjectivity onto the full set-valued codomain: not every non-empty subset of
must occur as an output.