According to the advanced analytics applied in the Harvard litigation, there was an advantage to apply early to Harvard, at least during the periods analyzed. I think if you search the threads on CC, @Data10, I believe, broke out the analysis. There has not been any similar study of Princeton. That being said, if one is a clear favorite, choose that one.
Echo the above comments on having some matches/safeties in the mix, especially EA options in publics/public honors programs. In terms of application strategy/tactics for someone with your accomplishments:
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Apply your top pick EA/REA
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Apply to your instate flagship(s)/honors EA or rolling. If you get in, you can drop any safeties or potential matches that you do not rate as highly
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Choose 3 to 5 more applications to complete by the EA/REA school’s decision date so you pretty much can just hit the send button (or not) after your EA/REA decision is received. If the decision was a yes, then only send the application of schools you might consider over your EA/REA school, including on the basis of financial aid. If the decision was a deferral (at least for H and P), submit the already completed apps. If a reject (esp H), rethink your reach list and perhaps some of your essays or LoR’s. An early reject from H probably means there is some fatal defect in your app as H historically rejects very few in the early round.
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Choose 3 to 5 more backup schools, primarily matches and safeties with essays in outline or draft form, on track to be completed prior to RD deadlines based on the results of the EA/REA and public EA/rolling schools.
I don’t think it is a numbers game for highly selectives. It is not like buying extra lottery tickets. Yes, you should apply to several that match your desires, but among the thousands of average excellent applicants like you, it is the subjective aspects of your app that will get you in or not. Your EC’s at this point are what they are and your LoR’s, other than picking the right teachers, are out of your control. The only thing you can control now are the essays and short answers. Don’t stretch yourself too thin. The tendency then is to repurpose essays or make them generic enough that they can be repurposed. Not saying repurposing doesn’t make sense for identical or similar essay topics of an intellectual nature, but there are opportunities in many essays to showcase your specific fit within that school either as a user of their resources or as a contributor to aspects of the school community that they signal as valuable.