On the Fairness of Normalized p-Means for Allocating Goods and Chores
Owen Eckart, Alexandros Psomas, Paritosh Verma
TL;DR
This work introduces normalized p-mean objectives to fair-division problems with additive valuations across goods and chores, in both divisible and indivisible settings. By normalizing utilities and disutilities, the authors restore scale invariance and systematically map which p-values yield fairness (EF, PROP, EF1, PROP1) plus efficiency (Pareto or fractional PO) guarantees, often via market-equilibrium interpretations. Key results include: for divisible goods, normalized p-means with $p\le 0$ yield PROP and fPO; for divisible chores, minimization with $p\ge1$ yields $n^{1/p}$-PROP and fPO with tight bounds; for two-agent indivisible goods, $p\le0$ gives EF1 and PO; and for two-agent indivisible chores, $p\ge2$ yields EF1 and PO. Moreover, the paper develops rounding techniques that convert divisible, fair, efficient allocations into integral ones with PROP1 or EF1 guarantees, and it establishes strong impossibility results for welfarist rules in the chores domain. Overall, the normalized p-mean framework broadens the toolkit for fair and efficient allocation, highlighting notable separations between goods and chores and offering practical algorithms via market-based interpretations and rounding methods.
Abstract
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. The celebrated notion of Nash welfare is known to produce fair and efficient allocations for both divisible and indivisible goods; there is no known analogue for dividing chores. The Nash welfare objective belongs to a large, parameterized family of objectives called the p-mean welfare functions, which includes other notable members, like social welfare and egalitarian welfare. However, among the members of this family, only the Nash welfare produces fair allocations for goods. Incidentally, Nash welfare is also the only member that satisfies the axiom of scale invariance, which is crucially associated with its fairness properties. We define the class of "normalized p-mean" objectives, which imparts the missing key axiom of scale invariance to the p-mean family. 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) EF1 and Pareto efficient allocations for indivisible goods for two agents, and (iv) EF1 and Pareto efficient allocations for indivisible chores for two agents.