Equitable Allocations of Indivisible Goods

Abstract

We study fair allocation of indivisible chores (i.e., items with non-positive value) among agents with additive valuations. An allocation is deemed fair if it is (approximately) equitable, which means that the disutilities of the agents are (approximately) equal. Our main theoretical contribution is to show that there always exists an allocation that is simultaneously equitable up to one chore (EQ1) and Pareto optimal (PO), and to provide a pseudopolynomial-time algorithm for computing such an allocation. In addition, we observe that the Leximin solution-which is known to satisfy a strong form of approximate equitability in the goods setting-fails to satisfy even EQ1 for chores. It does, however, satisfy a novel fairness notion that we call equitability up to any duplicated chore. Our experiments on synthetic as well as real-world data obtained from the Spliddit website reveal that the algorithms considered in our work satisfy approximate fairness and e ciency properties signi cantly more often than the algorithm currently deployed on Spliddit.

Finding Fair and Efficient Allocations for Matroid Rank Valuations

“…(see, e.g. Caragiannis et al [2019b]; Freeman et al [2019] and references therein for further details) for matroid rank valuations. We present our results from a preliminary exploration of these questions in Appendix D. It is worthwhile to summarize here one of these results that extends a recent paper by Freeman et al [2019].…”

“…Caragiannis et al [2019b]; Freeman et al [2019] and references therein for further details) for matroid rank valuations. We present our results from a preliminary exploration of these questions in Appendix D. It is worthwhile to summarize here one of these results that extends a recent paper by Freeman et al [2019]. This paper shows that an allocation that is equitable up to one item or EQ1 (a relaxation of equitability in the same spirit as EF1) and PO may not exist even for binary additive valuations; however, for this valuation class, it can be verified in polynomial time whether an EQ1, EF1 and PO allocation exists and, whenever it does exist, it can also be computed in polynomial time (for the time complexity result, they show that such an allocation is MNW).…”

“…An allocation 𝐴 is said to be equitable or EQ if the realized valuations of all agents are equal under it, i.e. for every pair of agents 𝑖, 𝑗 ∈ 𝑁 , 𝑣 𝑖 (𝐴 𝑖 ) = 𝑣 𝑗 (𝐴 𝑗 ); an allocation 𝐴 is equitable up to one item or EQ1 if, for every pair of agents 𝑖, 𝑗 ∈ 𝑁 such that 𝐴 𝑗 ≠ ∅, there exists some item 𝑜 ∈ 𝐴 𝑗 such that 𝑣 𝑖 (𝐴 𝑖 ) ≥ 𝑣 𝑗 (𝐴 𝑗 \ {𝑜 }) [Freeman et al 2019]. 15 We can further relax the equitability criterion up to an arbitrary number of items: an allocation 𝐴 is said to be equitable up to 𝑐 items or EQ𝑐 if, for every pair of agents 𝑖, 𝑗 ∈ 𝑁 such that |𝐴 𝑗 | > 𝑐, there exists some subset 𝑆 ∈ 𝐴 𝑗 of size |𝑆 | = 𝑐 such that 𝑣 𝑖 (𝐴 𝑖 ) ≥ 𝑣 𝑗 (𝐴 𝑗 \ 𝑆).…”

“…17 Freeman et al [2019] use an example with 3 agents having binary additive valuations (Example 1). But it is easy to construct a fair allocation instance with only two agents having binary additive valuations that does not admit an EQ1 and PO allocation: 𝑁 = [2]; 𝑂 = {𝑜 1 , 𝑜 2 , 𝑜 3 , 𝑜 4 }; 𝑣 1 (𝑜) = 1 for every 𝑜 ∈ 𝑂; 𝑣 2 (𝑜 1 ) = 1 and 𝑣 2 (𝑜) = 𝑜 for every 𝑜 ∈ 𝑂 \ {𝑜 1 }.…”

On Fair and Efficient Allocations of Indivisible Public Goods

“…However, we show stronger results that PublicMNW and PublicLex remain NP-hard even when the valuations are binary. These results are in stark contrast to the PrivateGoods case, which admits polynomial-time algorithms for binary valuations [16,20]. Further, our reductions between PublicGoods and PublicDecisions also directly enable us to show NP-hardness of DecisionMNW and DecisionLex.…”