<p>In my humble opinion, the difference between the quality of CS education/opportunities at CMU vs. Columbia is greater than the difference between the quality of business/entrepreneurship education/opportunities at Columbia vs. CMU. Deciding where you should go is a complex problem.</p>
<p>Let’s say that before you go to college, you have Xa ability/knowledge in CS and Xb ability/knowledge in business. Let’s further say that CMU has Ya quality/strength in CS and Yb quality/strength in business, whereas Columbia has Za quality/strength in CS and Zb quality/strength in business. Attending CMU would change your ability/knowledge into (Xa’, Xb’) = f(Xa, Xb, Ya, Yb) and attending Columbia would change your ability/knowledge into (Xa’’, Xb’’) = g(Xa, Xb, Za, Zb). Let us finally assume that your probability of success is a function P(a, b), where (a, b) is either (Xa’, Xb’) or (Xa’’, Xb’’).</p>
<p>Notice all of the assumptions that have been made; this is a huge simplification of the real world. Even given that, and further assuming that we can say that Ya > Za, Yb < Zb, and Ya - Za > Zb - Yb (my initial sentiment expressed in this framework), we still have no idea what might be the form of the functions f and g (which depend on myriad factors related to how well the school is able to influence your ability and/or knowledge), nor do we have a clue as to how to calculate the probability P (corresponding to the myriad economic factors which decide how likely someone with given ability/knowledge is to succeed).</p>
<p>If we assume that f and g simply impart a fraction of the school’s strength to the student, then perhaps we can let f(a, b) = (a + c<em>Ya, b + c</em>Yb) and g(a, b) = (a + c<em>Za, b + c</em>Zb). We might also assume that the higher the ability/talent/knowledge, the higher the probability of success in the market; and why not assume that business and CS knowledge are equally important? Then we want to determine whether a + c<em>Ya + b + c</em>Yb < a + c<em>Za + b + c</em>Zb, which boils down to Ya + Yb < Za + Zb. Rearranging, we get Ya - Za < Zb - Yb; by a previous (opinion-based) assumption, this inequality is untrue; indeed, the opposite is true.</p>
<p>Therefore, f produces an optimal outcome, under all of these random assumptions. In other words, if you find this analysis convincing, go with CMU. Otherwise, you may find the model useful as a way to formalize assumptions you might make about the nature of yourself and the world, to try to reach a more informed conclusion. You might even look for data to support your models.</p>