ANY of the schools at this level will have a preponderance of students who are accustomed to being among the smartest in any room they enter before arriving at college. Yale, IME, is among the schools whose students tend not to “wear it on their sleeves.” I can only speak to the CS/math types, but DS reports that they are collaborative, frequently work together on psets (within ethical guidelines), don’t consider education to be a zero sum game (i.e., one student’s success does not detract from another’s), and genuinely encourage each other to go for it.
There are exceptions, but IME they are discovered early and their selfishness is noted and discounted.
Competitiveness exists: DS would rather win than lose at poker, would rather win a hockey game than lose, would rather get an A than a B, etc. I don’t think, though, that someone describing him or his circle of friends would choose “competitive” among the first 10 words describing them.