Cadet branch matching satisfies traditional assumptions: Ravi Jagadeesan in AEJ: Micro

Cadet-Branch Matching in a Kelso-Crawford Economy
By Ravi Jagadeesan
American Economic Journal: Microeconomics 2019, 11(3): 191–224https://doi.org/10.1257/mic.20170192191


Abstract: "Sönmez (2013)  and  Sönmez  and  Switzer  (2013)  used  matching  theory  with  unilaterally  substitutable  priorities  to  propose  mechanisms  to  match  cadets  to  military  branches.  This  paper  shows  that,  alternatively,  the  Sönmez  and  Sönmez–Switzer  mechanisms  can  be  constructed  as descending  salary  adjustment  processes  in Kelso-Crawford (1982)  economies  in  which  cadets  are  (grossly) substitutable.  The  lengths  of  service  contracts  serve  as  (inverse) salaries. The underlying substitutability explains the unilateral substitutability of the priorities utilized by Sönmez and Sönmez-Switzer."

...
"This paper shows that cadet-branch matching does not formally require matching theory with weakened substitutability conditions or many-to-many matching. I restore substitutability(in the sense of Kelso and Crawford 1982 and Hatfield and Milgrom 2005) by changing priorities to systematically favor long contracts. This change  of  priorities  does  not  affect  the  deferred  acceptance  mechanism.  Defining  the “salary” corresponding to a contract to be any decreasing function of the service time, the substitutable priorities are generated by maximizing a quasi-linear utility function.4  If  cadets  prefer  short  contracts,  then  the  cadet-branch  economy  can  be  regarded  as  a  job  market  in  the  Kelso-Crawford  (1982)  model,  and  the  Sönmez  (2013) and Sönmez-Switzer (2013) mechanisms correspond to the descending salary adjustment process. Thus, the Sönmez and Sönmez-Switzer mechanisms feature cadets bidding against each other in an ascending auction in service length."
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This paper admirably tidies up this corner of the stable matching literature.

There is of course also a market design question lurking in the background, concerning how the armed forces should match service members to jobs. (The present paper doesn't take a position on that.)

It isn't obvious that stability, in the sense of avoiding blocking pairs between service members and military assignments, is an appropriate market design objective for military job assignments.  This is because the military has an immense amount of unified control over military assignments, and so there isn't a lot of room for such blocking pairs to form, i.e. there isn't much opportunity for one military branch to recontract with a service member to change his or her assignment once it has been made.  One of the problems that military assignment mechanisms actually do have to deal with is that service members periodically come to the end of their enlistment period, and have to be given incentives to re-enlist. So the relevant blocking pairs might be those between service members and civilian jobs. How best to address these in the contexts of military assignments is an open question.

The National Academies of Sciences, Engineering, and Medicine have very recently convened a committee to study aspects of the management of human capital in the military, whose work may  begin to shed some light on this. Here's a link:
Strengthening Air Force Human Capital Management