Unless your GMAT score is very unbalanced (i.e. 50th percentile Quant/99th percentile Verbal) I wouldn't retake the GMAT. A 720 alone isn't going to keep you out of programs even if you come from an over represented group. My guess is that you didn't get in last year due to applying in R3. Since Stanford is on your list you may want to retake just to bump your score to their average (730), but retaking the GMAT for 10 extra points is kind of a waste of time and I doubt it would measurably improve your chances at Stanford. They tend to care a lot about undergrad GPA and prestige of undergrad and employer/job. If those aren't there then even a 780 isn't likely to gain you an admit. I guess my point is that the ROI on retaking the GMAT is pretty low. You can do it if it makes you sleep better at night, but I wouldn't expect it to be the difference maker.

As for which round to apply in, I'd say that it doesn't matter if you apply R1 or R2. The only school that encourages applicants to apply in R1 (if they are ready) is Stanford. Other than that admissions rates are basically the same b/w R1 and R2. The majority of the class is admitted in R2, but that's simply because that's when the most applications come in. R1 has fewer applications and all of the spots in the class are open. However, the applicant pool tends to be very strong and schools like to WL people in order to see who else they will get in R2 and R3. I would say to apply when you are ready. If that's R1 then go for it. If it's R2 then that's cool too. I would just avoid R3 because it's a scary land where people who would have gotten in during the first two rounds wind up as reapplicants next year.
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Thanks for sharing !!