Approximation Algorithms for Computing Maximin Share Allocations

Abstract

We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a 2/3-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in [14], exploiting the construction of certain, appropriately selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances there exists a maximin share allocation with high probability. This can be seen as a justification of the experimental evidence reported in [5,14], that maximin share allocations exist almost always.Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of 3 agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of 2 agents). We provide a 7/8-approximation algorithm, improving the previously known result of 3/4 [14]. The second case is when all item values belong to {0, 1, 2}, extending the {0, 1} setting studied in [5]. We obtain an exact algorithm for any number of agents in this case.

Fair Task Allocation When Cost of Task Is Multidimensional

“…This fact initiated research on approximate MMS allocations of goods in which each agents gets some fraction of her MMS guarantee [18]. There have been several works on algorithms that find an approximate MMS allocation [19][20][21][22][23]. [7] showed that although an MMS allocation of goods may not always exist, but there exists a polynomial-time algorithm that returns a 2 3 -approximate MMS allocation.…”

“…Algorithms for computing approximate MMS allocations of goods are being used in practice for fair division in real-world problems [24]. The authors of [19][20][21] proposed an algorithm that can guarantee an approximate MMS allocation. The work mentioned above only focused on the case of goods.…”

How to Be Helpful to Multiple People at Once

“…Harsanyi (1975) argued in favor of expected utilitarianism, loosely the maxsum principle, except in cases where the maximin choice was similar to the maxsum choice, as in the case in our experiments. There has been a strong focus on maximin mathematically and in economics (e.g., Amanatidis, Markakis, Nikzad, & Saberi, 2017; Barman & Krishna Murthy, 2017; Dubois, Fargier, & Prade, 1996; Escoffier, Gourvès, & Monnot, 2013; Kurokawa, Procaccia, & Wang, 2016; Procaccia & Wang, 2014), including applications to networking (e.g., bandwidth‐sharing) (Salles & Barria, 2008). Choices aligned with the maximin , maxsum metric, and IA metrics are often compared in studies, and results are often mixed, with participants trading off between different metrics depending on the numbers and contexts involved (Ahlert, Funke, & Schwettmann, 2013; Charness & Rabin, 2002; Engelmann & Strobel, 2004; Faravelli, 2007; Fehr et al, 2006; Gaertner et al, 2001; Gaertner & Schwettmann, 2007; Konow, 2001, 2003; Konow & Schwettmann, 2016; Mitchell, Tetlock, Mellers, & Ordonez, 1993; Ordoñez & Mellers, 1993; Pelligra & Stanca, 2013; Schwettmann, 2009, 2012).…”

Matroid Constrained Fair Allocation Problem

“…The problem of fair allocation of indivisible goods has recently garnered a lot of interest (Amanatidis, Birmpas, and Markakis 2018;Barman et al 2018;Endriss 2017;Brandt et al 2016;Amanatidis et al 2015;Bouveret and Lemaître 2014;Procaccia and Wang 2014;Kurokawa, Procaccia, and Wang 2016). Several important practical problems-for eg., matching courses, resolving inheritance issues, allocating cloud computing resources-naturally lend themselves to the problem of assigning goods to agents, when the goods cannot be fractionally allocated.…”