On the Fairness of Normalized p-Means for Allocating Goods and Chores

Allocating items in a fair and economically efficient manner is a central problem in fair division. We study this problem for agents with additive preferences, when items are all goods or all chores, divisible or indivisible. We define the class of “normalized p-mean” objectives, which imparts the missing key axiom of scale invariance to the family of p-mean welfare functions. Our results show that optimizing the normalized p-mean objectives produces fair and efficient allocations when the items are goods or chores, divisible or indivisible. For instance, the normalized p-means gives us an infinite class of objectives that produce (i) proportional and Pareto efficient allocations for divisible goods, (ii) approximately proportional and Pareto efficient allocations for divisible chores, (iii) $\text{EF}1$ and Pareto efficient allocations for indivisible goods for two agents, and (iv) $\text{EF}1$ and Pareto efficient allocations for indivisible chores for two agents.

Funding: This research was supported by an NSF CAREER award [CCF-2144208], a Google AI for Social Good award, and research awards from Google and Supra.