Who Matters in an Ad Hoc Coalition?

Who sets the agenda for multilateral military operations?  I am curious because the coalition engaged in the anti-ISIS effort had a meeting and I pondered thusly:

Who is at kids' table? mt @HayesBrown: The list of participants at today's anti-ISIS coalition meeting pic.twitter.com/vmBR7AlKNm
— Steve Saideman (@smsaideman) December 3, 2014

What do I mean by this?  When NATO was heavily involved in the Balkans, not every member mattered equally.  The five biggest force contributors met to discuss "the way ahead" and their agreements then framed the agenda for the NATO meetings.  The QUINT, as they were called, were the US, UK, France, Germany and Italy.  During my year on the Joint Staff, I organized the dinner for one of these meetings.  The fun part?  The program was supposed to read "QUINT DINNER" but I was not clear enough, so they printed "QUINT SPACE DINNER."  Good thing I never actually served in the military or my lack of attention to detail might get someone killed.

In Afghanistan, one could guess the group was similar--those who had responsibility for one of the Regional Commands: Germany, Italy, the UK/Canada/Dutch rotation, and the US (pretty sure running Regional Command-Capital didn't get Turkey to the table).  However, given that Germany and Italy had rather significant restrictions on what their contingents could do (those caveats that were much of the focus of our book), I am pretty sure that Germany and Italy might have been in the room but at the kids' table.  In other words, I am pretty sure that the decision-making for much of the effort was driven by the US (as always) with the UK and Canada having some voice since they were fighting in some of the toughest and most important spots in the country.  That is, it was not just about size but about what one was doing.

In the anti-ISIS fight?  The list is so long that it makes clear that many of these folks do not matter when it comes to the decision-making.  So, who is at the adults' table?  I have no certainty, but I have some guesses: US, UK, France, maybe Saudi Arabia (due to its regional heft), and UAE (for willing to bomb Syria--a rare commodity).  The kids' table would probably include Canada, Denmark, Belgium, the Dutch, and others that have engaged in significant airstrikes plus Kuwait and Qatar for providing key bases.  The rest are left outside at the dinner, but allowed to show up when the pictures are taken to show that the coalition is as broad as possible. 

The point is: the big meetings are for show and for ratifying the decisions made by smaller groups.  Who exactly is in the smaller group this time?  I made some guesses, but someone else will have to do the fieldwork to figure this one out.  My plate is full and not just with turkey, stuffing, apple pie and more stuffing.