## Abstract

We investigate the efficiency of fair allocations of indivisible goods using the well-studied *price of fairness* concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of *strong price of fairness*, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation as in the price of fairness. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions, for both the price of fairness and the strong price of fairness.

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## Notes

Indeed, the instance that Caragiannis et al. used to show that the price of proportionality is at least

*n*âˆ’â€‰1 + 1/*n*admits no envy-free allocation. Thus, it is still possible that the price of envy-freeness is lower than the price of proportionality for indivisible goods.See SectionÂ 2 for the formal definitions of these notions.

See the example in Theorem 3.5.

Moreover, a round-robin allocation is likely to be envy-free and proportional as long as the number of goods is sufficiently larger than the number of agents [28].

In addition to these exceptions, MNW, MEW, and leximin allocations are hard to compute regardless of price of fairness considerations (see, e.g., [33], footnote 7).

Interestingly, this stands in contrast to our result that the price of MNW for indivisible goods is Î˜(

*n*).See [17] for the definitions of MMS and PMMS.

Recently, Chaudhury et al. [18] showed that the existence is also guaranteed for three agents.

In case there are ties between goods, we may assume worst-case tie breaking, since it is possible to obtain an instance with infinitesimal difference in welfare and any desired tie-breaking between goods by slightly perturbing the utilities.

In the case where the maximum Nash welfare is 0, an allocation is an MNW allocation if it gives positive utility to a set of agents of maximal size and moreover maximizes the product of utilities of the agents in that set.

To see the first and third inequalities, one may prove by induction that in general, if we have \(\frac {a_{1}}{b_{1}}\geq \dots \geq \frac {a_{k}}{b_{k}}\), then \(\frac {a_{1}}{b_{1}}\geq \frac {a_{1}+\dots +a_{k}}{b_{1}+\dots +b_{k}}\geq \frac {a_{k}}{b_{k}}\). The case

*k*=â€‰2 holds because \(\frac {a_{1}}{b_{1}}\geq \frac {a_{1}+a_{2}}{b_{1}+b_{2}}\) is equivalent to \(\frac {a_{1}}{b_{1}}\geq \frac {a_{2}}{b_{2}}\), and similarly for \(\frac {a_{1}+a_{2}}{b_{1}+b_{2}}\geq \frac {a_{2}}{b_{2}}\).

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## Acknowledgements

This work is partially supported by NSF Award CCF-1815434, by the European Research Council (ERC) under grant number 639945 (ACCORD), and by an NUS Start-up Grant. We are grateful to the reviewers of IJCAI 2019 and Theory of Computing Systems for many helpful comments, and to Ioannis Caragiannis for pointing us to the price of envy-freeness result by Bertsimas et al. [10].

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A preliminary version of the paper appeared in Proceedings of the 28th International Joint Conference on Artificial Intelligence [7]. This version includes several proofs that were omitted or partially omitted in the preliminary version (Theorems 3.2, 3.3, 3.4, 3.8, 3.9, 4.1, 4.2, 5.2, 5.3, 5.4, 5.5, 6.1, 6.2, and 6.3). We also included additional discussion of our results and updated the state of related work. The third author is now at Google Research. Most of the work was done while the fourth author was at the University of Oxford.

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Bei, X., Lu, X., Manurangsi, P. *et al.* The Price of Fairness for Indivisible Goods.
*Theory Comput Syst* **65**, 1069â€“1093 (2021). https://doi.org/10.1007/s00224-021-10039-8

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DOI: https://doi.org/10.1007/s00224-021-10039-8