You may discover that a grant has been turned down by another candidate and is now available to you. You may also catch a certain school that is falling a little short of its target enrollment, which could spark some willingness to negotiate. An insider at one large Delaware Valley university told me recently that the school frequently obliges these requests, as long as a student can produce written confirmation of the competing offer. But at the time we were going through this process, I was also advised by an experienced high school guidance counselor (and former college admissions officer), as well as several contacts at universities, that it usually doesn't work that way.
In most cases, a college's financial aid officers have already scoured the numbers and carefully decided what a kid with your son or daughter's record is worth to them. College admissions officers have a big advantage in this sort of negotiation, in that they've done it a lot more than you. And no matter how great your kid is, there are very likely a number of other great kids who are going to take the available seat if yours doesn't. Yes, they'd like to have your son or daughter---they wouldn't have offered them admission if this wasn't the case. But there is a fairly large "you win some, you lose some" attitude out there which most admissions officers can afford to adopt.
Some colleges also request a list of other schools to which you've applied, because they like to know their chief competitors. Although I was never told that the information was used in any other way, it's easy to imagine the data coming in handy in assessing the aid these other schools generally offer, and how much they need to toss out there in order to get their desired share of top candidates.
In short, I would encourage you to ask whether a given school might be willing to match another school's offer. But I wouldn't go into the college search assuming that this strategy is going to work. A better approach to finding an affordable option is to apply to a variety of schools, including smaller private colleges and larger public colleges whose base tuition prices are less than others on the list. After all, with or without aid, you can be assured of some level of cost containment if you have options that have a lower bottom line.
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