Unfixed Pair (COMSOC)

Definition

Unfixed Pair (COMSOC)

A pair of distinct alternatives $\{x,y\}\subseteq C$ is unfixed in a non-empty profile $R$ if it is not fixed, i.e., both must hold

$$R(x,y)\ne R\text{and}R(y,x)\ne R,$$

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, preserving repeated ballots. Thus $R(x,y)$ keeps precisely the ballots that rank $x$ above $y$, and $R(y,x)$ keeps those ranking $y$ above $x$. Neither equals the whole profile: at least one voter prefers $x$ to $y$, and at least one prefers $y$ to $x$.

In terms of the pairwise support function, $N_R(x,y)=|R(x,y)|$, so the condition is equivalently $0<N_R(x,y)<n$. This describes disagreement among voters, not necessarily disagreement between optimal Kemeny rankings.