Applying Combinatorics to College Applications From a list of 4,000 colleges, I have a list of 50 colleges. From that list of 50 colleges, I now plan to choose 15 colleges to apply to.
Each of the 50 colleges has a scale value (how well the university's offerings allign with my goals) and an admissions percentage I tweaked to reflect the likelihood of my acceptance into the college.
I want to be able to apply combinatorics (game theory, whatever the academic field may be!) to frame and answer these questions:
1) if I want to receive acceptance letters from n universities, how should I craft my application list to maximize the scale value of the n universities
2) is standard application strategy -- apply (in the case of 15 universities) to 3 reaches, 9 matches and 9 safeties -- sound advice 
Nobody need explicitly answer this question! I just need someone to point me in the right direction, to offer a reference I can use to learn the principles behind a decision like this one
 A: Warning: this answer is a purely formal treatment of this problem. Getting accurate enough numbers to actually apply this approach would be basically impossible.
Suppose that there are $n$ schools to which you can apply. Give each school an index $i \in [n]$, with your top choice numbered $1$, your second choice $2$, and so on. For each $i$, let $v_i$ be the value of attending school $i$, by whatever measure you like, $p_i$ be the probability of admission, and $c_i$ be the cost of applying, factoring in both fees and time. Because of how we have numbered the schools, we have $v_1 > v_2 > \cdots > v_n$.
Ultimately, you will only attend one school, so you need to maximize the expected value of the highest ranked school to which you are admitted, minus all costs of application.
Let $S \in [n]$ be the set of schools to which you apply. Since not applying to a school effectively sets your chance of admission to $0$, let $p_i^S$ be equal to $p_i$ if $p \in S$, and zero otherwise.
Now we can finally write out your utility function. Given the set $S$ as above, the chance of getting in to your top choice school is $p_1^S$, and if you get in, you will go there, so this contributes the term $p_1^S v_1$. You will only go to your second choice if you get in there and not to your first, so the next term in your expected utility is $(1-p_1^S)p_2^Sv_2$. This pattern continues for the later schools. Taking costs of applying into account, the final utility function is 
$$ U(S) = \sum_{i=1}^n -c_i + v_i p_i^S \prod_{j<i} (1-p_j^S)$$
Your goal is to find the subset $S$ maximizing $U(S)$. One way to do this is just to check all $2^n$ possible subsets, which isn't too unreasonable for small $n$. Since this is essentially a nonlinear program, I don't think that there are likely to be any subexponential time algorithms.
