If I have to venture a guess, my guess is perhaps two rounds of picking a numbered ping-pong ball from a bag and...

The first ping pong ball round determining to box number the player is assigned and the second ping pong ball round determining which position the player will be standing at (or vice versa).

With respect to this, my guess here is for place purposes, 1-11 are for West Wing and 12-22 for East Wing.

Did I have this approximately correct?

Sounds reasonable enough to me. Then again, maybe they just gave each player a box and a place to stand and said 'Have fun'

Quite a few actually do stand in their original positions - Russell, Sally, Helen, Jeff, Lucy, 1p Nick and John G all do I believe.

I also think they gave Morris 15 as it's good to make a poem from.