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In other words, we're actually looking for a college which is LIKE Reed, but not filled with students who feel the need to "out-compete" one another.

Are you thinking that Reed is a place where people try to out-compete each other? To be fair, I've never actually been to Reed, but my alma mater is very similar, and there was very little competition, though people were interested in each others' interests. I was specifically looking for this after having gone a cuthroat, status-obsessed public high school where class rank was All.

From what you say of your DD, I think someplace Reed-ish sounds pretty promising. I'm obviously biased, though.