Share-Based Fairness for Arbitrary Entitlements
Moshe Babaioff, Uriel Feige
TL;DR
The paper develops a comprehensive framework for share-based fairness with arbitrary entitlements in allocating indivisible goods and chores. By introducing ρ-domination and tight dominating shares, it proves that for additive valuations over goods there exists a poly-time 1/2-ex-post dominating allocation (and near 1/2 via ˆTPS), with a sharp ex-ante bound 1/γ (γ ≈ 1.69103), and shows BoBW limitations for goods but a strong BoBW result for chores. The authors deploy a bidding-game methodology to realize the ex-post guarantees, deriving structural and complexity insights for worst-case-optimal strategies and offering poly-time ε-approximation strategies. For chores, they obtain a 2-dominating Rounded-Responsibilities Round Robin share and a BoBW that jointly bound ex-ante and ex-post fairness, along with tight impossibility results. Overall, the work delivers tight, implementable guarantees across ex-post and ex-ante settings for both goods and chores, clarifying the trade-offs and guiding practical fair allocation under asymmetric entitlements.
Abstract
We consider the problem of fair allocation of indivisible items to agents that have arbitrary entitlements to the items. Every agent $i$ has a valuation function $v_i$ and an entitlement $b_i$, where entitlements sum up to~1. Which allocation should one choose in situations in which agents fail to agree on one acceptable fairness notion? We study this problem in the case in which each agent focuses on the value she gets, and fairness notions are restricted to be {\em share based}. A {\em share} $s$ is an function that maps every $(v_i,b_i)$ to a value $s(v_i,b_i)$, representing the minimal value $i$ should get, and $s$ is {\em feasible} if it is always possible to give every agent $i$ value of at least $s(v_i,b_i)$. Our main result is that for additive valuations over goods there is an allocation that gives every agent at least half her share value, regardless of which feasible share-based fairness notion the agent wishes to use. Moreover, the ratio of half is best possible. More generally, we provide tight characterizations of what can be achieved, both ex-post (as single allocations) and ex-ante (as expected values of distributions of allocations), both for goods and for chores. We also show that for chores one can achieve the ex-ante and ex-post guarantees simultaneously (a ``best of both world" result), whereas for goods one cannot.