I'm following an equation from a published paper in order to calculate probabilities using a Markov Chain. The equation says:
Construct a set $U$ that consists of all items that appear in the top-$k$ in at least one list.
For each pair of items $i$ and $j$ in $U$, let the preference for $i$ over $j$, $m_i{_j}$, equal $1$ if the majority of the lists ($>=50$%) that rank both $i$ and $j$ rank $j$ above $i$ and $0$ otherwise. Let $m_i{_j} = m_j{_i} = 0.5$ if items $i$ and $j$ are never directly compared in any list.
My problem with the above is: $U$ is composed of the top-k items in at least one list. Wouldn't this mean that $m_i{_j} = 0.5$ would never be possible because if $U$ is composed of items that must all appear in one list, each item is directly compared in at least one list.
I just want someone to confirm my reading of this.