Almost (Weighted) Proportional Allocations for Indivisible Chores

Abstract

In this paper, we consider how to fairly allocate m indivisible chores to a set of n (asymmetric) agents. As exact fairness cannot be guaranteed, motivated by the extensive study of EF1, EFX and PROP1 allocations, we propose and study proportionality up to any item (PROPX), and show that a PROPX allocation always exists. We argue that PROPX might be a more reliable relaxation for proportionality in practice than the commonly studied maximin share fairness (MMS) by the facts that (1) MMS allocations may not exist even with three agents, but PROPX allocations always exist even for the weighted case when agents have unequal obligation shares; (2) any PROPX allocation ensures 2-approximation for MMS, but an MMS allocation can be as bad as Θ(n)-approximation to PROPX.We propose two algorithms to compute PROPX allocations and each of them has its own merits. Our first algorithm is based on a recent refinement for the well-known procedure -envy-cycle elimination, where the returned allocation is simultaneously PROPX and 4/3-approximate MMS. A by-product result is that an exact EFX allocation for indivisible chores exists if all agents have the same ordinal preference over the chores, which might be of independent interest. The second algorithm is called bidand-take, which applies to the weighted case. Furthermore, we study the price of fairness for (weighted) PROPX allocations, and show that the algorithm computes allocations with the optimal guarantee on the approximation ratio to the optimal social welfare without fairness constraints.* The authors thank Haris Aziz and Warut Suksompong for their valuable comments and suggested related works.

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

“…It is easy to see that in the case of an instance with bads only, EFX (resp.EFX 0 ) implies PropX( [24], Lemma 3.2). We extend this observation and show that this implication continues to hold in the mixed manna setting.…”

“…Given that computing EFX (and EFX+PO) allocations is a challenging problem even for general instances of goods, a significant amount of past works have focused on sub-classes by restricting the values that agents have for the items, see [3,27,10,19,1,2] and references therein for practical scenarios involving different sub-classes. Among these, the valuation classes of identical [28,10,1,2], identical order preferences (IDO) [28,24], restricted additive [7], and binary [10,21,11] are well-studied for goods (bads) manna. We study all of these for 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].…”

“…Notice that in an IDO instance, the cardinal preferences can be significantly different across different agents. Afterwards, in Section 4.2, we present a reduction from [24], from an instance which is IDO for bads to a general instance for bads, and show that it holds even in the mixed manna setting. This allows us to obtain a PropMX 0 allocation for any separable instance.…”

“…Similarly, for bads, an allocation is said to be PropX if every agent can receive her proportional share after removing any one bad assigned to her. 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.…”