This is a tough one! All three schools have top alumni networks.
( I am an MIT grad. )
Between the three, you will get the best science and math education at MIT,
just because the students are more focused in those subjects. (a lot of learning happens student to student at any college today) The MIT school of Architecture is well known and very well ranked for design. Art flourishes at MIT but its not the Rhode Island School of Design by any means, MIT does not offer a bachelor of fine arts.
MIT has a separate department of political science and economics and both are very strong.
Ask yourself do you want a bachelor of fine arts or do you want a strong technical degree with a focus on math content of your courses? Public policy at MIT will be more focused on science and technology policy than policy degrees at your other two choices.
http://web.mit.edu/polisci/undergraduate/minor/pp.html
Brown and Columbia are more well rounded than MIT for student interests and majors , and the Columbia Core offers a lot of readings in philosophy, English and history. You will read and write better by the end of four years, if you pick Brown or Columbia. I think the Columbia Core is very light weight on math and science, I have looked it over for my son. However Columbia has fantastic departments in economics, physics and math. Not sure about biology. I feel Columbia is not as strong in engineering if you might say want to major in bioengineering.
At MIT you will do more problem sets and have tremendous options for research in pubic policy, political science, biology or economics starting in freshman year. MIT’s problem set approach is very good preparation for analytical careers in business, law, medicine or science or engineering, or econometrics.
Columbia has a School of Visual arts. Is that important to you?
Check about where MIT will allow you to take classes. In the 1980s, students could only take Harvard courses if MIT does not offer that subject. Wellesley College, about 20 minutes away, a girls liberal arts college, is completely open to MIT students, however. So check if Wellesley College would be a place to study the fine arts you seem interested in.
The bus between MIT and Wellesley runs frequently so some MIT students are able to take lot of classes over there.
But if you did not visit Wellesley College, you have to think about whether you fit over there and would be happy taking art or humanities subjects at a women’s liberal arts college.
Boston and NYC are better for a college student, than Providence in my opinion, but that is a personal choice.
Providence is 20 minutes from Cape Cod, if you like the beach. Boston is 50 miles north of Providence, but ask yourself do you want to experience city life?