Fair and Efficient Allocations with Limited Demands

We study the fair division problem of allocating multiple resources among a set of agents with Leontief preferences that are each required to complete a finite amount of work, which we term "limited demands". We examine the behavior of the classic Dominant Resource Fairness (DRF) mechanism in this setting and show it is fair but only weakly Pareto optimal and inefficient in many natural examples. We propose as an alternative the Least Cost Product (LCP) mechanism, a natural adaptation of Maximum Nash Welfare to this setting. We characterize the structure of allocation of the LCP mechanism in this setting, show that it is Pareto efficient, and that it satisfies the relatively weak fairness property of sharing incentives. While we prove that it satisfies the stronger fairness property of (expected) envy freeness in some special cases, we provide a counterexample showing it does not do so in general, a striking contrast to the "unreasonable fairness" of Maximum Nash Welfare in other settings. Simulations suggest, however, that these violations of envy freeness are rare in randomly generated examples.