Fixed Pair (COMSOC)

Definition

Fixed Pair (COMSOC)

A pair of distinct alternatives $\{x,y\}\subseteq C$ is fixed in a non-empty profile $R$ if either

$$R(x,y)=R\text{or}R(y,x)=R,$$

Here $R$ is the multiset of all voters’ submitted strict rankings, with one occurrence per voter. For $R=({\succ }_1,\dots ,{\succ }_n)$ in tuple notation, let

$$R(x,y):=\left\{\right\{{\succ }_i\mid i\in \{1,\dots ,n\},x{\succ }_{i}y\left\}\right\}.$$

The double braces denote a multiset: identical ballots from different voters remain separate occurrences. Thus $R(x,y)$ keeps precisely the ballots that rank $x$ above $y$, and $R(y,x)$ keeps those ranking $y$ above $x$. Equality with $R$ means that every ballot is retained. A fixed pair therefore has unanimous agreement on its order.

In terms of the pairwise support function, $N_R(x,y)=|R(x,y)|$, so the condition is equivalently $N_R(x,y)=n$ or $N_R(y,x)=n$.