One advantage of HYP are grad school admissions. Two of the best law schools in the country are Yale’s and Havard’s. If you look at the class profile for Yale, the most matriculates come from Yale, then Harvard with Princeton a distant third. Harvard has the same ranking, flipping Yale with Harvard. After that come the various Ivies and NE LAC’s in the relative order of their prestige.
With the exception of Stanford and Berkeley, schools outside the Ivy/NE LAC world struggle to get more than a handful of graduates into YLS or HLS.
You can see the figures on page 160 from the Yale Law Bulletin.
http://bulletin.printer.yale.edu/pdffiles/law.pdf
The top schools were:
Yale: 69
Harvard: 65
Princeton: 37
Columbia: 30
Brown: 22