Harvard has the 380 year old prestige as the top school that is hard to beat. Stanford is objectively just as strong a university and in some fields significantly stronger which is why it is able to attract a good portion of the cross admits and has comparable yield figures. Its lay prestige has been growing exponentially in the past few decades and is right behind Harvard’s now however it is not 100% on par with Harvard yet. There is no denying though that Harvard, Stanford are the undisputed top dogs in higher ed and each other’s main competition.