praguepride:
This depends heavily by state. I think Michigan is a no-fault state where everyone just pays their own damages. Some states have clear cut fault, others assign values like 90/10 or 80/20 etc.
It's more confusing than that. "No Fault" compels YOUR insurance company to pay YOU regardless of fault.
It does not necessarily prevent OTHER insurance company from paying you, or paying your insurance company, when THEIR insured has caused the damage.
In such circumstances you can recover through subrogation IF its clear that their insured was at fault- hence, if they were issued a citation and you weren't, it works.
It might depend on local laws, I have done this twice in my life, both times in "no fault" states, and it has worked. (Once, it got me to switch insurers, the guy who rear ended me had State Farm, which was easier to deal with for me and far more helpful than my OWN insurance company at the time, so maybe there's a sales angle to it!)