There was a story in Analog a few years ago called "Dibs" if I recall the title correctly about a world that worked like that.

Whenever someone could be saved by a transplant they would find possible donors and send them a notification that one of their organs could save someone. Usually after a few weeks the potential donor would get notification that the person who needed the organ has died. During the time between those two notifications the dying person was said to have dibs on the organ.

Occasionally someone would get a second notification about someone having dibs on another one of the organs while someone already had dibs on one of their organs. Again what usually is that those people would die soon and the person would go back to nobody having dibs on any of their organs.

Sometimes though a person with people having dibs on two of their organs would get notified that a third person now had dibs on one of their organs. That was enough that the needs of the many thing kicked in and they were required to give up those organs, which would usually be fatal.