Abstract
In multiagent resource allocation with indivisible goods, boolean fairness criteria and optimization of inequality-reducing collective utility functions (CUFs) are orthogonal approaches to fairness. We investigate the question of whether the proposed scale of criteria by Bouveret and Lemaître [5] applies to nonadditive utility functions and find that only the more demanding part of the scale remains intact for k-additive utility functions. In addition, we show that the min-max fair-share allocation existence problem is NP-hard and that under strict preferences competitive equilibrium from equal incomes does not coincide with envy-freeness and Pareto-optimality. Then we study the approximability of rank-weighted utilitarianism problems. In the special case of rank dictator functions the approximation problem is closely related to the MaxMin-Fairness problem: Approximation and/or hardness results would immediately transfer to the MaxMin-Fairness problem. For general inequality-reducing rank-weighted utilitarianism we show (strong) NP-completeness. Experimentally, we answer the question of how often maximizers of rank-weighted utilitarianism satisfy the max-min fair-share criterion, the weakest fairness criterion according to Bouveret and Lemaître’s scale. For inequality-reducing weight vectors there is high compatibility. But even for weight vectors that do not imply inequality-reducing CUFs, the Hurwicz weight vectors, we find a high compatibility that decreases as the Hurwicz parameter decreases.
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Notes
- 1.
Note that proportional fair-share can also be defined differently under k-additive utility functions.
- 2.
A utility function u is strict if \(u(G) = u(H)\) implies \(G = H\).
- 3.
Alternatively, one could define the median for an even number of agents as . However, we follow the definition due to Chevaleyre et al. [9].
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Acknowledgments
We thank the anonymous reviewers for their helpful comments. This work was supported in part by DFG grant RO 1202/14-1.
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Heinen, T., Nguyen, NT., Rothe, J. (2015). Fairness and Rank-Weighted Utilitarianism in Resource Allocation. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_31
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DOI: https://doi.org/10.1007/978-3-319-23114-3_31
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