Borda, Condorcet, and Pareto optimality in ordinal group activity selection

We consider the situation in which group activities need to be organized for a set of agents. Each agent can take part in at most one activity, and the agents’ preferences depend both on the activity and the number of participants in that activity. In particular, the preferences are given by means of strict orders over such pairs “(activity, group size)”, including the possibility “do nothing”. Our goal will be to assign agents to activities on basis of their preferences, the minimum requirement being that no agent prefers doing nothing, i.e., not taking part in any activity at all. We aim at establishing such an assignment by two approaches. On the one hand, taking a voting-theoretical perspective, we apply Borda scores and the Condorcet criterion to our setting. On the other hand, we target at a Pareto optimal assignment. We analyse the computational complexity involved in finding such desired assignments, with focus on two natural special cases of the agents’ preferences.