Fair Allocation of Indivisible Goods to Asymmetric Agents

Abstract

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang [20] wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with n agents, no allocation can guarantee better than 1/n approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a 1/n approximation guarantee.Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem w.l.o.g, we show it enables us to find a 1/2 approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

Weighted Maxmin Fair Share Allocation of Indivisible Chores

“…Different to the case of symmetric agents, we first prove that even with only two agents, no algorithm can simultaneously guarantee each agent's value to be higher than 4 3 of her weighted maxmin share. Moreover, we show that many greedy algorithms widely used in the literature, including [Farhadi et al, 2017] and [Aziz et al, 2017], may have arbitrarily bad performance. Then we design a polynomial-time algorithm which provides a 4-approximation to the minimal relaxation of WMMS value (OWMMS) under which a WMMS allocation exists.…”

“…When the instance I is clear from the context, we may use WMMS i for short. The definition above for WMMS fairness is exactly the same as that of WMMS as formalized by [Farhadi et al, 2017] for the case of goods except that the entitlement e i of an agent i is replaced by her share s i . As mentioned in the introduction, whereas a higher entitlement for goods is desirable for an agent, a higher share for chores is undesirable for the agent.…”

“…All these works make a typical assumption that agents are symmetric and should be treated in a similar manner. [Farhadi et al, 2017] were the first to consider MMS fairness for the case where indivisible goods are allocated and the agents are not symmetric because they may have different entitlement share of the goods. Ideally, an agent would expect to get a share of the total value that is proportional to her entitlement.…”

The Complexity of Computing Maximin Share Allocations on Graphs

“…The idea is that every agent i ∈ N must get a bundle B i ⊆ V of goods for which she gets an utility that is not worse than the utility she could guarantee to herself if asked to divide the set of goods in |N | bundles and then to keep the worst one. In fact, there are experimental and theoretical evidences showing that maximin share allocations can be singled out in many relevant settings (Kurokawa, Procaccia, and Wang 2016;Bouveret and Lemaître 2016); moreover, while their existence cannot be guaranteed in general (Procaccia and Wang 2014), allocations that "nearly satisfy" the maximin share criterion always exist and can be computed efficiently (Procaccia and Wang 2014;Amanatidis et al 2017; Barman and Krishna Murthy 2017;Ghodsi et al 2018;Nguyen, Nguyen, and Rothe 2017;Farhadi et al 2019).…”

Cake-cutting with different entitlements: How many cuts are needed?

“…The situation of agents with different entitlements has been studied also for allocation of indivisible items[1,9,17].…”