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Characterizing conflicts in fair division of indivisible goods using a scale of criteria

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Abstract

We investigate five different fairness criteria in a simple model of fair resource allocation of indivisible goods based on additive preferences. We show how these criteria are connected to each other, forming an ordered scale that can be used to characterize how conflicting the agents’ preferences are: for a given instance of a resource allocation problem, the less conflicting the agents’ preferences are, the more demanding criterion this instance is able to satisfy, and the more satisfactory the allocation can be. We analyze the computational properties of the five criteria, give some experimental results about them, and further investigate a slightly richer model with \(k\)-additive preferences.

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Notes

  1. Whereas most CUF—except Nash—only make sense if the utilities are expressed on a common scale or normalized.

  2. Actually, this result remains valid even with unrestricted combinatorial preferences.

  3. Moulin [37], although in a continuous context (divisible goods), introduces a seminal idea of this concept under a “though experiment” in which an agent considers the case of other agents having the same preference as her, deriving from that experiment a lower bound of her utility.

  4. Moreover, even if a common utility scale is used, an agent could manipulate the game to her advantage by decreasing her utilities. A way to overcome this problem is to normalize utilities such that the whole set of objects gives the same utility to each agent. This is known as the Kalai-Smorodinsky approach, see for example [35], p 67.

  5. Actually, Steinhaus [46] does not give any name for this criterion, later named Equal Split Guarantee by Moulin [36].

  6. This notion is actually called maximin share by Budish [16].

  7. We use here a very similar idea to the one used by Lipton et al. [32], page 4.

  8. See also [10] for a summary of the original Fisher’s equations and for a brilliant account of his work (actually, I. Fisher had also developed a hydraulic device (!) for calculating equilibrium prices). Vazirani [47] is a recent key reference on this subject, building on the linear case of the Fisher model (hence eliminating the notion of marginal utility present in the original one).

  9. In our settings, the Nash CUF is the function \(g_N : \overrightarrow{\pi }\mapsto \prod _{i\in {\fancyscript{A}}} u_i(\pi _i)\).

  10. For instance, consider an instance for which no envy-free allocation exists.

  11. Note that in this discrete model, money is not “real”, in the sense that it is just a modelling artifact used to define the “choice set” of buyers. This is an important difference with the combinatorial auction problem [19], in which a buyer is supposed to keep the money not spent if any. This important point is discussed in details by Othman et al. [39].

  12. The work by Othman et al. [39] is devoted to the computation of approximate competitive equilibria.

  13. Even if in our examples we use normalized weights. Actually, four of the criteria are even purely ordinal—proportional fair-share is not.

  14. If an allocation is not Pareto-efficient, then it is dominated by at least one Pareto-efficient allocation, which satisfies the criterion as well.

  15. Actually a similar result holds if weights are normalized such that \(u^{\mathrm {MFS}}_i\) is equal for all agents.

  16. This min-optimal allocation is also leximin-optimal. The leximin ordering [45] is a refinement of the min ordering for which a lexicographic comparison of sorted vectors of weigths is used, instead of comparing their min values.

  17. This property is sometimes known as full-correlation [8].

  18. For the meaning of the term “picking protocol”, see the beginning of Sect. 6.1.

  19. This also seems to be true for more demanding criteria as our experiments show, see Sect. 7.

  20. The best resort in this case would be a normalized leximin-optimal allocation.

  21. We have seen in Sect. 4.5 that EFP is a necessary condition for having a CEEI when preferences are strict. We believe that the CEEI and EFP criteria are not equivalent in the context of this discrete model.

  22. For example, when computing the most demanding criterion for a given instance, we first consider proportionality and envy-freeness, as these criteria can be tested in polynomial time.

  23. As a reminder, testing the existence of a complete and envy-free allocation (fourth column) is NP-complete [32], and testing the existence of a Pareto-efficient and envy-free allocation (last column) is \(\mathsf {\Sigma _2^P}\)-complete [30].

  24. In an ordinal setting, additivity should be replaced by its ordinal counterpart, separability.

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Acknowledgments

We thank Ariel Procaccia, Hervé Moulin, members of COST Action IC-1205, and the anonymous reviewers for helpful discussions, comments and suggestions.

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Authors and Affiliations

  1. LIG, Université Grenoble-Alpes, 681 rue de la Passerelle, 38402, Saint-Martin-d’Hères, France

    Sylvain Bouveret

  2. Onera Toulouse, Toulouse, France

    Michel Lemaître

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  1. Sylvain Bouveret
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  2. Michel Lemaître
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Correspondence to Sylvain Bouveret.

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A previous version of this article appeared in the proceedings of AAMAS’14.

Michel Lemaître is now retired from Onera Toulouse.

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Bouveret, S., Lemaître, M. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Auton Agent Multi-Agent Syst 30, 259–290 (2016). https://doi.org/10.1007/s10458-015-9287-3

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