I had a lot of success doing this for my doctoral program. The graduate director from School A made it very clear that they wanted me there, and so I laid out the offers I received from schools B and C, and asked if they could match them. I ended up getting an additional $15k a year in grants and fee waivers.
A good script to use:
"Your program is my first choice/I am very excited about your program, BUT in comparison with the other offers I've received, the funding package just isn't feasible for me. To make this work I would need [a guaranteed RAship/a tuition waiver/full health coverage/a stipend of $XXX]. Is there a way that we can make this work?"
Don't think of it as playing schools off each other, think of it as using the other offers as a jumping off point to articulate what you want. Talk to whoever is your point person at the program. If it's a PI, talk to them, if it's the graduate director, talk to them. You may have better luck calling them rather than emailing them.
