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By definition of positive ROI, the university has more money than before, which in particular means it can spend more money on research by building football stadia.

On the wider scale, extracurriculars only affect[2] share-of-students rather than increase the total number of students[1], so such programs have a globally negative ROI. But each individual university is making a rational decision.

[1] I'm assuming there's a negligible percentage of students that would avoid college entirely if no or very few colleges had football programs. It's safe to ignore football scholarships, because you still have the option of giving the students free money, which is cheaper than giving them free money and also running a football program.

[2] I'm also assuming the football program itself doesn't generate enough revenue to offset its costs, and only affects enrollment. I honestly don't know if they make enough money in tickets and trinkets to offset the debt service for a stadium, salaries for coaches, free tuition for students, etc. If the ROI is positive(or even negative, but with a positive cap rate), then it might be rational economically to continue them.

You're assuming that the ROI generated by football isn't a transfer from other universities who lose students to the spending uni. More likely, funding football is a zero sum game which generates no overall benefit for the research community.

You think that football game attendees (and T-shirt buyers, and...) would write checks to math researchers if football got cancelled?

That's what I said in my second sentence.

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