Plackett-Luce models paper

Question

I am reading the following paper.

Embarrassingly I am stuck on section 2. Plackett-Luce models.

I believe I understand how the set of $N$ rankings will look like. Something of the following form, if we have $3$ items and $2$ judges:

$$\{(3,1,2),(2,1,3)\}$$

Where the first $3$-tuple would tell us that judge $1$ (provided the first $3$-tuple corresponds to the first judge and the second to the second one) gave ranking $3$ to item $1$, $1$ to item $2$ and $2$ to item $3$.

If the reasoning above is correct, then I totally do not understand what the author is saying when talks about rankings having associated ordering $\omega^{(n)} \equiv (\omega_1^{(n)},...,\omega_K^{(n)})$. If someone could clarify what is meant by this, I would be very grateful.

Answer 1

He is saying that each judge (judge $(n)$)provides a permutaion $\omega^{(n)}$ which for the full range of $k$ contestants places contestant $k$ into the $i$-th slot in the permutation. For your example, judge $1$ provides $$ \omega^{(1)} = \pmatrix{3&1&2\\1&2&3} $$ and judge $2$ provides $$ \omega^{(2)} = \pmatrix{2&1&3\\1&2&3} $$

Then the right hand side describes the particular $\omega^{(n)}$ provided by judge $(n)$ in terms of the components of the permutation. In your example, $$ \omega^{(1)}_1 = 3\\ \omega^{(1)}_2 = 1\\ \omega^{(1)}_3 = 2\\ \omega^{(2)}_1 = 2\\ \omega^{(2)}_2 = 1\\ \omega^{(2)}_3 = 3 $$