# Derive logistic choice probabilities symbolically

More generally, I am interested in learning what the current limitations of Mathematica are when using it for doing pure mathematics. A recent blog post by Stephen Wolfram (see: http://blog.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/) outlines his vision, but I am interested in what is currently feasible, and I have a specific question to narrow things down (but broader comments/rules of thumb are also welcome).

Problem setup: Suppose I want to derive the logistic choice probabilities. To start, take as given that we know the probability with which person $n$ chooses alternative $i$. This is (in a discrete choice setting where the person chooses the option that maximizes his/her utility):

$\quad \quad P_{ni} = Pr(U_{ni}>U_{nj} \forall j \neq i) = \int^\infty _{-\infty} \prod_{j\neq i} e^{-e^{(s+V_{ni} - V_{nj})}} e^{-s} e^{-e^{-s}} ds \quad \quad (1)$

This expression has a known closed form:

$\quad \quad P_{ni} = e^{V_{ni}} / \sum _{j} e^{V_{nj}} \quad \quad (2)$

Question: Can Mathematica handle $\prod_{i\neq j}$ and $\sum$ here? That is, is it possible to write general code that would allow me to go from (1) to (2) when considering a product over an arbitrary number of alternative choices $j$?