Pairwise Support Function (COMSOC)

Definition

Pairwise Support Function ( COMSOC)

The pairwise support function of a profile $R=({\succ }_1,\dots ,{\succ }_n)$ of $n$ complete, strict rankings over alternatives $C$ counts the voters who prefer one alternative to another:

$$\begin{array}{rl}{N}_R& :C\times C⟶\{0,\dots ,n\},\\ N_R(x,y)& :=\underset{\text{voters preferring }x\text{ to }y}{\underbrace{|\{i\in \{1,\dots ,n\}:x{\succ }_{i}y\}|}}\\ & =|R(x,y)|.\end{array}$$

For distinct $x,y$, every voter supports exactly one side, so

$$N_R(x,y)+N_R(y,x)=n.$$

Also $N_R(x,x)=0$. The count $N_R(x,y)$ is support for $x$ against $y$, not the majority margin $N_R(x,y)-N_R(y,x)$.

The lecture notation ${\#}_R(x,y)$ denotes this same count: ${\#}_R(x,y)=N_R(x,y)$.