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Consider the following setting:

  1. A committee has $N$ members.
  2. There are $m$ types of positions in the committee.
  3. For each type of positions $1 \leq i \leq m$, there are $n_i$ identitical positions for the $i$-th type.
  4. Each member must be assigned exactly one position, thus we have the equality $$ \sum_{i=1}^m n_i = N$$
  5. Each member has a strict linear preference over the types of positions that he/she wants to be assigned. For example, someone (call her Jessica) may have the following preference $$ \text{President} \succ \text{Vice President} \succ \text{Treasurer} \succ \cdots $$ which means Jessica wants to be president the most, followed by vice president, followed by treasurer, etc. In total, there are $N$ such preference relations.
  6. For each type of positions $1 \leq i \leq m$, each member $1 \leq j \leq N$ has a strict linear preference over the members that member $j$ thinks are suitable to do the type $i$ positions. For example, Jessica may have the following preference for the positions of treasurers $$ \text{Peter} \succ \text{Jessica} \succ \text{Alex} \succ \cdots $$ which means Jessica thinks that Peter is the most suitable member for treasurers, followed by herself, followed by Alex, etc. Remember that there can be more then one positions of treasurers. In total, there are $mN$ such preference relations.

Questions:

  1. What does social choice theory say about how to assign positions in this situation?
  2. Is it always possible to have a pareto-optimal assignment?
  3. I don't know what kind of conditions on the preferences would lead to what kind of properties of the assignments. What kind of conditions can we impose on the preferences to guarantee stronger properties of the assignments?
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    $\begingroup$ The goal of social choice theory is to say what we should do in such circumstances, if we want certain desirable properties of the final assignment. It is up to you to decide what properties you would like, there is no "correct" answer we nor our theories could provide. $\endgroup$ Commented Nov 16, 2021 at 16:49
  • $\begingroup$ @WalrasianAuctioneer Well, I don't know what kind of conditions on the preferences would lead to what kind of properties of the assignments. That's why I'm asking this question. Anyway, I've edited my question to make it more specific. $\endgroup$
    – user141240
    Commented Nov 17, 2021 at 5:49

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