Disagreement Set (COMSOC)

Definition

Disagreement Set ( COMSOC)

The disagreement set of two complete, strict rankings over the same alternatives $C$ contains the unordered pairs they rank oppositely:

$$\mathrm{disag}({\succ }_i,{\succ }_j):=\{\{x,y\}:x,y\in C,x{\succ }_{i}y\wedge y{\succ }_{j}x\}.$$

Each element $\{x,y\}$ is one pair of alternatives, not a voter or a position. The pair is included once, regardless of its written order. The Kendall–tau distance is the cardinality of this set.

Example

Three rankings

For $a{\succ }_{1}b{\succ }_{1}c$, $c{\succ }_{2}a{\succ }_{2}b$, and $c{\succ }_{3}b{\succ }_{3}a$,

\begin{aligned} \operatorname{disag}(\succ_1,\succ_2)&=\{\{a,c\},\{b,c\}\},\\ \operatorname{disag}(\succ_1,\succ_3)&=\{\{a,b\},\{a,c\},\{b,c\}\},\\ \operatorname{disag}(\succ_2,\succ_3)&=\{\{a,b\}\}. \end{aligned} Their distances are respectively $2,3,1$.