Llull Winner (COMSOC)

Definition

Llull Winner ( COMSOC)

A Llull winner is an alternative with maximal Llull score in a profile over :

The Llull rule returns all such alternatives. Several may be co-winners; a unique winner has strictly more points than every other alternative. A tie in total score does not imply that the co-winners tie their head-to-head contest.

Majority Graph

In the majority graph , let and let be the number of incoming arcs at . Each incoming arc records one lost contest. Therefore

A Llull winner has the fewest incoming arcs. The lecture expression is one larger than the Llull score, but adding the same constant to every score leaves the winners unchanged.

Ties have no arc in either direction and earn a point under Llull. Thus maximising outgoing arcs alone is equivalent only when there are no tied contests.