@milosh The comments above are very good, but hearing it direct from Yale might be even better:
The Yale Daily News ran a story a few months ago, including quotes from the Yale Fin Aid Director that covered in detail how to negotiate a better award.
Some excerpts from the article:
"Every student interviewed who showed Yale a more generous offer from another Ivy received a subsequent readjustment in their Yale package after conversations with the financial aid office.
Since each Ivy League school can only award financial aid based on a student’s estimated need, as opposed to merit or athletic ability, Storlazzi (Yale’s director of financial aid) said that if another Ivy calculates a higher aid package for a student than Yale does, Yale will typically do its best to calculate a similar number.
Ideally, Storlazzi said, Yale is eventually able to offer students a package that will enable them to attend.
“We don’t say ‘match,’” Storlazzi said. “We don’t use the m-word. But we want to see what we can do within Yale’s philosophy of needs analysis, and, frankly, we do want to see if we can come close to what the other school offered. We want students to decide on fit instead of on finance.”
The full article can be found here: http://yaledailynews.com/blog/2016/11/11/how-generous-is-generous-enough/