One Quarter Each (on Average) Ensures Proportionality

TL;DR

This paper demonstrates that allocating one quarter of the total subsidy per agent guarantees proportionality in fair division of chores or goods with additive costs, and establishes the tightness of this bound.

Contribution

It introduces tight bounds on the subsidy needed for proportional allocations in divisible and weighted cases with additive costs.

Findings

A total subsidy of n/4 dollars suffices for proportional chores allocation.

The n/4 bound is tight, as some instances require at least n/4 subsidy.

A total subsidy of (n-1)/2 suffices for weighted proportional allocation.

Abstract

We consider the problem of fair allocation of $m$ indivisible items to a group of $n$ agents with subsidy (money). Our work mainly focuses on the allocation of chores but most of our results extend to the allocation of goods as well. We consider the case when agents have (general) additive cost functions. Assuming that the maximum cost of an item to an agent can be compensated by one dollar, we show that a total of $n/4$ dollars of subsidy suffices to ensure a proportional allocation. Moreover, we show that $n/4$ is tight in the sense that there exists an instance with $n$ agents for which every proportional allocation requires a total subsidy of at least $n/4$. We also consider the weighted case and show that a total subsidy of $(n-1)/2$ suffices to ensure a weighted proportional allocation.