Yeah, this is an interesting distributed problem, but totally doable.
One way you could approach this is to have a Blink broadcast the lowest score it sees from direct neighbors. This will result in everyone sharing the lowest score. If the individual Blink is that score, it knows it is among the lowest (does not insure uniqueness).
This is a fun problem to work through, I believe that @Khaos created a fun dice game that did something similar.
Are you thinking something like this?