PROPm Allocations of Indivisible Goods to Multiple Agents

Abstract

We study the classic problem of fairly allocating a set of indivisible goods among a group of agents, and focus on the notion of approximate proportionality known as PROPm. Prior work showed that there exists an allocation that satisfies this notion of fairness for instances involving up to five agents, but fell short of proving that this is true in general. We extend this result to show that a PROPm allocation is guaranteed to exist for all instances, independent of the number of agents or goods. Our proof is constructive, providing an algorithm that computes such an allocation and, unlike prior work, the running time of this algorithm is polynomial in both the number of agents and the number of goods.

(Almost) Envy-Free, Proportional and Efficient Allocations of an Indivisible Mixed Manna

“…We prove this theorem by presenting an algorithm which returns a PropMX 0 allocation for this instance, combining ideas from previously known algorithms for instances with goods only [6] and pure bads instances [24].…”

“…Our algorithm first uses the algorithm of Baklanov et al [6] for goods to obtain a PropM 0 allocation with respect to M ≥0 . Note that this allocation will also trivially be PropMX 0 , since we have only assigned items from M ≥0 .…”

“…For the remainder of this section, we use Goods to refer to Algorithm 1 of [6], and we assume that the bads in M − are ordered such that v i1 ≤ v i2 ≤ • • • ≤ v im − (recall that v ij < 0 for all i ∈ N, j ∈ M − , i.e. the bads are ordered from most painful to least painful).…”

“…While PropX need not exist for goods [26,5], Li et al [24] showed that a PropX allocation always exists for bads and can be computed in polynomial time. Since PropX need not exist for goods, Baklanov et al [6] introduce the notion of proportionality up to the maximin good (PropM), a weaker version of PropX. They show that PropM allocations always exist for the setting of goods and agents with additive valuations, and that they can be computed in polynomial time.…”

“…x i ← x i + j 11: return x Since Goods runs in polynomial time, Algorithm 4 runs in polynomial time as well. The following lemma about allocation x ≥0 (Line 3) follows directly from [6].…”