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School X is offering me $12,000/year with 50% stipulation. Cost of attendance with cost of living: $120,000.

School Y is offering me $21,000/year with 3.00 stipulation. Cost of attendance: $60,000. Cost of living is negligible because I can live at home.

I have no desire to attend school Y for a number of reasons (less job prospects, do not want to live at home). Keep in mind that school Y is ranked somewhat lower (not a whole lot lower). Can I use the lower cost of attendance at school Y to get more money from school X? (I have specific reasons to offer them so they do not think it is just about prestige)

What's the median grade at school Y. If they curve to below 3 then the scholarship is worth much less than 50%. I would be more concerned about having them remove the stipulation than trying to get more money out of them.

You can certainly try to use school Y, but there is no guarantee they would even want to compete. It isn't just about ranking, but if they two schools are not really peers, they don't have a reason to compete.

It definitely can work. No guarantees, but in my own experiences I have been pleasantly surprised.

It also depends on how your LSAT is