Llull Score (COMSOC)

Definition

Llull Score ( COMSOC)

The Llull score of an alternative is the number of opponents it defeats or ties under the Llull rule. For a profile over alternatives ,

where is the pairwise support function. A win or tie contributes one point; a loss contributes zero, regardless of the vote margin. Thus

Equivalently,

where counts incoming arrows (losses) in the majority graph. The alternatives with maximal score are Llull winners.

Intuition

Use the five-voter profile from the Llull example and follow . Unlike a Borda score, a Llull score is not computed by awarding points on each ballot separately. First combine the ballots into head-to-head contests.

Keep only on each ballot. Four voters prefer and one prefers , so wins this contest: .

That victory contributes one point to , not four. The four supporting voters determine the contest’s outcome; the outcome determines the point. receives zero from this contest because it loses.

Now compare with every other alternative. It loses to , beats , and beats , giving . The victories by and count equally. There are opponents and one loss, so the same score is .

The five-voter profile has no tied contests. With an even number of voters, however, a tie can occur: both sides then receive one point. Llull counts opponents beaten or tied, not only outgoing majority arrows. Repeat this count for each alternative, then compare their totals.