Seating Arrangement Problem

[pococurante]

Randomly draw the seats, then start the button in front of the person whose turn it is to be the button.

If you are trying to be perfectly fair, then keep track of every starting position each person gets, and don't allow them to get the same spot twice. For instance, if someone is UTG in game 1 and draws the same spot in game 4, make them switch with someone who hasn't been UTG yet. This is way more work than it's worth, no one really cares that much.

As far as a mathematical way to do this, I don't think there is one. If there is, it's more complicated than its worth.