Llull Score (COMSOC)

Definition

Llull Score ( COMSOC)

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

$${\mathrm{Llull}}_R(x):=|\{y\in C\setminus \{x\}:N_R(x,y)\ge N_R(y,x)\}|,$$

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

$$0\le {\mathrm{Llull}}_R(x)\le m-1.$$

Equivalently,

$${\mathrm{Llull}}_R(x)=(m-1)-d^{-}(x),$$

where $d^{-}(x)$ 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 $B$. 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 $B,C$ on each ballot. Four voters prefer $B$ and one prefers $C$, so $B$ wins this contest: $4>n/2=5/2=2.5$.

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

Now compare $B$ with every other alternative. It loses to $A$, beats $C$, and beats $D$, giving $0+1+1=2$. The victories by $4:1$ and $3:2$ count equally. There are $m-1=3$ opponents and one loss, so the same score is $(m-1)-d^{-}(B)=3-1=2$.

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.