Northwestern sociology traditionally does consequential faculty deliberations like this: the person to the left of the chair is given the floor and says their piece, then the person to their left says theirs, and so on around their room, and only afterward is a more free-for-all discussion format used. I’ve only seen this system in action for a couple meetings, and I’m already smitten with it. It feels contemplative, orderly, and fair.
Still: I’ve become a teensy bit obsessed with the question of, given this system, what’s the best place to sit from a strategic standpoint–what’s the seat from which one can exercise the most influence on the ultimate decision that is made. There is a bunch of deliberating groups research that would suggest that, typically, you want to be the first to speak. However, consequential faculty deliberations are not the task for which these studies were done. Among other things, at least in collegial places, there is strong normative pressure to maximize the extent to which you sound like you deeply respect and see the reasoning of previously expressed opinions, even when actually you completely disagree with them.
I think, certainly, if you have an opinion that you think weakly conflicts with the consensus going into the meeting, you are best off going at the beginning and trying to get a domino effect going. If you are hoping to carry the day with an opinion that goes more strongly against consensus, I suspect maybe you should sit toward the end and try to rattle people with some kind of melodramatic or dire statement, although I’m skeptical of that working unless you’ve built up a lot of intradepartmental street cred.
For decisions where the consensus is unclear and could go any of various ways, my suspicion is that you want to sit somewhere between 2/5 and 3/5 of the way through the group: enough that you have a lot of input to work with, so you can fashion your opinion as consistent with general views but pointing toward your desired conclusion, but still not so late that people feel like the decision has already solidified by the time it is your turn.
Of course, a countermeasure to any such gaming of the system would be to wait until everyone is seated and then randomly choose a starting person (perhaps with a rousing round of “Spin the Treo”). My guess is that most faculty anywhere would be against doing that, because faculty members are professionals, and if there is anything being a professional does to your habitus, it is to convince you that you are insulated from the mundane social psychological dynamics that affect regular folks.