Abstract
We study fair allocation of indivisible goods among agents. Prior research focuses on additive agent preferences, which leads to an impossibility when seeking truthfulness, fairness, and efficiency. We show that when agents have binary additive preferences, a compelling rule—maximum Nash welfare (MNW)—provides all three guarantees. Specifically, we show that deterministic MNW with lexicographic tie-breaking is group strategyproof in addition to being envy-free up to one good and Pareto optimal. We also prove that fractional MNW—known to be group strategyproof, envy-free, and Pareto optimal—can be implemented as a distribution over deterministic MNW allocations, which are envy-free up to one good. Our work establishes maximum Nash welfare as the ultimate allocation rule in the realm of binary additive preferences.
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Notes
- 1.
There are two popular definitions of EFX [3]; this result holds for the stronger one: an allocation is EFX if the envy that one agent has toward another can be eliminated by removing any good from the envied agent’s bundle.
- 2.
There are two popular definitions of EFX [3]. The original definition by Caragiannis et al. [16] asks that agent i not envy agent j after removal of any good from agent j’s bundle that has positive value for agent i, whereas a latter definition omits the requirement of “positive value”. Under binary additive valuations, the former definition is equivalent to EF1 whereas the latter definition is stronger than EF1.
- 3.
We note that tie-breaking by agent index is without loss of generality. One can break ties according to any given ordering of the agents, and the corresponding rule will still satisfy all the desiderata.
- 4.
In case of Pareto optimality of a fractional allocation, we require that no other fractional allocation Pareto-dominate it.
References
Aleksandrov, M., Aziz, H., Gaspers, S., Walsh, T.: Online fair division: analysing a food bank problem. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI), pp. 2540–2546 (2015)
Amanatidis, G., Birmpas, G., Christodoulou, G., Markakis, E.: Truthful allocation mechanisms without payments: characterization and implications on fairness. In: Proceedings of the 18th ACM Conference on Economics and Computation (EC), pp. 545–562 (2017)
Amanatidis, G., Birmpas, G., Filos-Ratsikas, A., Hollender, A., Voudouris, A.A.: Maximum Nash welfare and other stories about EFX. In: Proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI) (2020, forthcoming)
Aziz, H., Rey, S.: Almost group envy-free allocation of indivisible goods and chores. In: Proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI) (2020)
Babaioff, M., Ezra, T., Feige, U.: Fair and truthful mechanisms for dichotomous valuations. arXiv preprint arXiv:2002.10704 (2020)
Barman, S., Krishnamurthy, S.K., Vaish, R.: Greedy algorithms for maximizing Nash social welfare. In: Proceedings of the 17th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 7–13 (2018)
Benabbou, N., Chakraborty, M., Igarashi, A., Zick, Y.: Finding fair and efficient allocations when valuations don’t add up. arXiv preprint arXiv:2003.07060 (2020)
Birkhoff, G.: Three observations on linear algebra. Universidad Nacional de Tucumán, Revista A 5, 147–151 (1946)
Bogomolnaia, A., Moulin, H.: Random matching under dichotomous preferences. Econometrica 72, 257–279 (2004)
Bogomolnaia, A., Moulin, H., Stong, R.: Collective choice under dichotomous preferences. J. Econ. Theory 122(2), 165–184 (2005)
Bouveret, S., Lemaître, M.: Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Auton. Agents Multi-Agent Syst. 30(2), 259–290 (2016)
Brams, S.J., Fishburn, P.C.: Approval Voting, 2nd edn. Springer, New York (2007). https://doi.org/10.1007/978-0-387-49896-6
Brams, S.J., Taylor, A.D.: Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press, Cambridge (1996)
Budish, E.: The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J. Polit. Econ. 119(6), 1061–1103 (2011)
Budish, E., Che, Y.K., Kojima, F., Milgrom, P.: Designing random allocation mechanisms: theory and applications. Am. Econ. Rev. 103(2), 585–623 (2013)
Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A.D., Shah, N., Wang, J.: The unreasonable fairness of maximum Nash welfare. ACM Trans. Econ. Comput. 7(3), article 12 (2019)
Cheng, Y., Jiang, Z., Munagala, K., Wang, K.: Group fairness in committee selection. In: Proceedings of the 20th ACM Conference on Economics and Computation (EC), pp. 263–279 (2019)
Cole, R., Gkatzelis, V.: Approximating the Nash social welfare with indivisible items. SIAM J. Comput. 47(3), 1211–1236 (2018)
Cole, R., Gkatzelis, V., Goel, G.: Mechanism design for fair division: allocating divisible items without payments. In: Proceedings of the 14th ACM Conference on Economics and Computation (EC), pp. 251–268 (2013)
Darmann, A., Schauer, J.: Maximizing Nash product social welfare in allocating indivisible goods. Eur. J. Oper. Res. 247(2), 548–559 (2015)
Foley, D.K.: Resource allocation and the public sector. Yale Economics Essays, vol. 7, pp. 45–98 (1967)
Freeman, R., Shah, N., Vaish, R.: Best of both worlds: ex-ante and ex-post fairness in resource allocation. In: Proceedings of the 21st ACM Conference on Economics and Computation (EC) (2020, forthcoming)
Freeman, R., Sikdar, S., Vaish, R., Xia, L.: Equitable allocations of indivisible goods. In: Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pp. 280–286 (2019)
Klaus, B., Miyagawa, E.: Strategy-proofness, solidarity, and consistency for multiple assignment problems. Int. J. Game Theory 30, 421–435 (2001)
Kurokawa, D., Procaccia, A.D., Shah, N.: Leximin allocations in the real world. ACM Trans. Econ. Comput. 6(3–4), article 11 (2018)
Lackner, M., Skowron, P.: A quantitative analysis of multi-winner rules. In: Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pp. 407–413 (2019)
Lipton, R.J., Markakis, E., Mossel, E., Saberi, A.: On approximately fair allocations of indivisible goods. In: Proceedings of the 6th ACM Conference on Economics and Computation (EC), pp. 125–131 (2004)
Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2003)
von Neumann, J.: A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Kuhn, W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. 2, pp. 5–12. Princeton University Press (1953)
Orlin, J.B.: Improved algorithms for computing fisher’s market clearing prices: computing fisher’s market clearing prices. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), pp. 291–300 (2010)
Ortega, J.: Multi-unit assignment under dichotomous preferences. Math. Soc. Sci. 103, 15–24 (2020)
Plaut, B., Roughgarden, T.: Almost envy-freeness with general valuations. In: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2584–2603 (2018)
Varian, H.: Equity, envy and efficiency. J. Econ. Theory 9, 63–91 (1974)
Végh, L.A.: Concave generalized flows with applications to market equilibria. Math. Oper. Res. 39(2), 573–596 (2013)
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Halpern, D., Procaccia, A.D., Psomas, A., Shah, N. (2020). Fair Division with Binary Valuations: One Rule to Rule Them All. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham. https://doi.org/10.1007/978-3-030-64946-3_26
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