Two Desirable Fairness Concepts for Allocation of Indivisible Objects under Ordinal Preferences

Fair allocation of indivisible objects under ordinal preferences is an important problem. Unfortunately, a fairness notion like envy-freeness is both incompatible with Pareto optimality and is also NP-complete to achieve. To tackle this predicament, we consider a different notion of fairness, namely proportionality. We frame allocation of indivisible objects as randomized assignment but with integrality requirements. We then use the stochastic dominance relation to define two natural notions of proportionality. Since an assignment may not exist even for the weaker notion of proportionality, we propose relaxations of the concepts optimal weak proportionality and optimal proportionality. For both concepts, we propose algorithms to compute fair assignments under ordinal preferences. Both new fairness concepts appear to be desirable in view of the following: they are compatible with Pareto optimality, admit efficient algorithms to compute them, are based on proportionality, and are guaranteed to exist.