Lets say my friends and I ($n$ people in total) are moving together to an $n$-room house and each one uf us has a precise ordering of the rooms we'd like to live in. That is, if $\{R_j\}_{j=1}^n$ is the set of rooms and $\{A_i\}_{i=1}^n$ the set of people then, for every $1\le i\le n$ there exists a total ordering $\le_i$ of $\{R_j\}_{j=1}^n$. For example, $R_3\le_4 R_5\le_4 R_{n-1}\le_4\dots\le_4 R_2$ means that person number four thinks room 2 is the best and room 3 the worst.
How should the rooms be distributed so as to make everyone the happiest? There are a lot of discussions here and there about what constitutes a "fair" or "envy-free" allocation and some algorithms. I for once am interested in the practical side of the matter. Is there a pen and paper procedure or a matlab function?
Why is this different to fair cake-cutting? And why is it harder than a simple optimization problem?