[1] https://market.subwiki.org/wiki/Tragedy_of_the_commons [2] https://market.subwiki.org/wiki/Rival_good
The problem in this case is how to get people to pay for the creation of the good in the first place, because people know that even if they do not pay for the creation of the good, they can still enjoy the benefits.
https://market.subwiki.org/wiki/Free-rider_problemSure some cities will not contribute back but in open source it is the organizations most in need and most able to contribute back who do. Why is that a problem?
Also, in your example, I don't know how other cities' use would impede the funding city's use.
If the lab has to give the software away, all development ends, unless you have $300k/year of grants sitting around.
If they can't get the interested labs to contribute towards the development without the coercion of a commercial license then it sounds like it's either done, or the development roadmap is misaligned with the users' goals. In either case, stopping the abusive current practices would be an immediate improvement for everyone involved.
As many have pointed out I misused the term "Tragedy of commons" , when in fact it is "The Free Rider Problem"[1]
However the main point stands. Others will not invest in FOSS in hopes that someone else will first and they they can just consume for free.