Fair Division via Quantile Shares
Pages 1235 - 1246
Abstract
We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely the proportional share and the maximin share, are not universally feasible, nor are any constant approximations of them.
We propose a novel share notion, where an agent assesses the fairness of a bundle by comparing it to her valuation in a random allocation. In this framework, a bundle is considered q-quantile fair, for q∈[0,1], if it is at least as good as a bundle obtained in a uniformly random allocation with probability at least q. Our main question is whether there exists a constant value of q for which the q-quantile share is universally feasible.
Our main result establishes a strong connection between the feasibility of quantile shares and the classical Erdős Matching Conjecture. Specifically, we show that if a version of this conjecture is true, then the 1/2e-quantile share is universally feasible. Furthermore, we provide unconditional feasibility results for additive, unit-demand and matroid-rank valuations for constant values of q. Finally, we discuss the implications of our results for other share notions.
References
[1]
Ron Aharoni and David Howard. 2017. A rainbow r-partite version of the Erdős-Ko-Rado theorem. Combin. Probab. Comput., 26, 3 (2017), 321–337. https://doi.org/10.1017/s0963548316000353
[2]
Hannaneh Akrami and Jugal Garg. 2024. Breaking the 3/4 Barrier for Approximate Maximin Share. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 74–91. https://doi.org/10.1137/1.9781611977912.4
[3]
Itai Arieli, Yakov Babichenko, Ron Peretz, and H Peyton Young. 2020. The speed of innovation diffusion in social networks. Econometrica, 88, 2 (2020), 569–594. https://doi.org/10.3982/ecta17007
[4]
Haris Aziz, Ioannis Caragiannis, Ayumi Igarashi, and Toby Walsh. 2022. Fair allocation of indivisible goods and chores. Autonomous Agents and Multi-Agent Systems, 36 (2022), 1–21. https://doi.org/10.1007/s10458-021-09532-8
[5]
Moshe Babaioff, Tomer Ezra, and Uriel Feige. 2021. Fair and Truthful Mechanisms for Dichotomous Valuations. Proceedings of the AAAI Conference on Artificial Intelligence, 35, 6, 5119–5126. https://doi.org/10.1609/aaai.v35i6.16647
[6]
Moshe Babaioff and Uriel Feige. 2022. Fair Shares: Feasibility, Domination and Incentives. In Proceedings of the 23rd ACM Conference on Economics and Computation. 435–435. https://doi.org/10.1145/3490486.3538286
[7]
Yakov Babichenko, Michal Feldman, Ron Holzman, and Vishnu V. Narayan. 2023. Fair Division via Quantile Shares. https://doi.org/10.48550/arxiv.2312.01874 arxiv:2312.01874.
[8]
Nikhil Bansal and Maxim Sviridenko. 2006. The santa claus problem. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing. 31–40. https://doi.org/10.1145/1132516.1132522
[9]
Siddharth Barman and Paritosh Verma. 2021. Existence and Computation of Maximin Fair Allocations Under Matroid-Rank Valuations. In Proceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems. 169–177. https://doi.org/10.5555/3463952.3463978
[10]
Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi, and Yair Zick. 2020. Finding fair and efficient allocations when valuations don’t add up. In Algorithmic game theory (Lecture Notes in Comput. Sci., Vol. 12283). Springer, Cham, 32–46. https://doi.org/10.1007/978-3-030-57980-7_3
[11]
Anna Bogomolnaia, Hervé Moulin, Fedor Sandomirskiy, and Elena Yanovskaya. 2017. Competitive division of a mixed manna. Econometrica, 85, 6 (2017), 1847–1871. https://doi.org/10.3982/ecta14564
[12]
Eric Budish. 2011. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119, 6 (2011), 1061–1103. https://doi.org/10.1086/664613
[13]
Ioannis Caragiannis, David Kurokawa, Hervé Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. 2019. The Unreasonable Fairness of Maximum Nash Welfare. ACM Trans. Econ. Comput., 7, 3 (2019), Article 12, sep, 32 pages. issn:2167-8375 https://doi.org/10.1145/3355902
[14]
Christopher P Chambers and Federico Echenique. 2009. Supermodularity and preferences. Journal of Economic Theory, 144, 3 (2009), 1004–1014. https://doi.org/10.1016/j.jet.2008.06.004
[15]
Geoffroy De Clippel, Kfir Eliaz, and Brian Knight. 2014. On the selection of arbitrators. American Economic Review, 104, 11 (2014), 3434–3458. https://doi.org/10.1257/aer.104.11.3434
[16]
Lester E Dubins and Edwin H Spanier. 1961. How to cut a cake fairly. The American Mathematical Monthly, 68, 1P1 (1961), 1–17. https://doi.org/10.2307/2311357
[17]
Jack Edmonds. 1970. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969). Gordon and Breach, New York-London-Paris, 69–87.
[18]
Jeff Edmonds and Kirk Pruhs. 2006. Cake cutting really is not a piece of cake. In SODA. 6, 271–278. https://doi.org/10.1145/1109557.1109588
[19]
Paul Erdős. 1965. A problem on independent r-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math, 8 (1965), 93–95.
[20]
P. Erdős, Chao Ko, and R. Rado. 1961. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2), 12 (1961), 313–320. https://doi.org/10.1093/qmath/12.1.313
[21]
Paul Erdős and Tibor Gallai. 1959. On maximal paths and circuits of graphs. Acta Mathematica Hungarica, 10, 3-4 (1959), 337–356. https://doi.org/10.1007/bf02024498
[22]
Shimon Even and Azaria Paz. 1984. A note on cake cutting. Discrete Applied Mathematics, 7, 3 (1984), 285–296. https://doi.org/10.1016/0166-218x(84)90005-2
[23]
Uriel Feige. 2006. On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM J. Comput., 35, 4 (2006), 964–984. https://doi.org/10.1137/s0097539704447304
[24]
Uriel Feige, Ariel Sapir, and Laliv Tauber. 2021. A tight negative example for MMS fair allocations. In International Conference on Web and Internet Economics. 355–372. https://doi.org/10.1007/978-3-030-94676-0_20
[25]
Duncan Karl Foley. 1966. Resource allocation and the public sector. Yale University.
[26]
Peter Frankl. 2013. Improved bounds for Erdős’ matching conjecture. J. Combin. Theory Ser. A, 120, 5 (2013), 1068–1072. https://doi.org/10.1016/j.jcta.2013.01.008
[27]
Peter Frankl. 2017. On the maximum number of edges in a hypergraph with given matching number. Discrete Applied Mathematics, 216 (2017), 562–581. https://doi.org/10.1016/j.dam.2016.08.003
[28]
Peter Frankl. 2017. Proof of the Erdős matching conjecture in a new range. Israel Journal of Mathematics, 222 (2017), 421–430. https://doi.org/10.1007/s11856-017-1595-7
[29]
Peter Frankl and Andrey Kupavskii. 2022. The Erdős Matching Conjecture and concentration inequalities. Journal of Combinatorial Theory, Series B, 157 (2022), 366–400. https://doi.org/10.1016/j.jctb.2022.08.002
[30]
Peter Frankl, Vojtech Rödl, and Andrzej Ruciński. 2012. On the maximum number of edges in a triple system not containing a disjoint family of a given size. Combinatorics, Probability and Computing, 21, 1-2 (2012), 141–148. https://doi.org/10.1017/s0963548311000496
[31]
Jun Gao, Hongliang Lu, Jie Ma, and Xingxing Yu. 2022. On the rainbow matching conjecture for 3-uniform hypergraphs. Sci. China Math., 65, 11 (2022), 2423–2440. https://doi.org/10.1007/s11425-020-1890-4
[32]
Brian Garnett. 2020. Small deviations of sums of independent random variables. Journal of Combinatorial Theory, Series A, 169 (2020), 105119. https://doi.org/10.1016/j.jcta.2019.105119
[33]
Mohammad Ghodsi, MohammadTaghi HajiAghayi, Masoud Seddighin, Saeed Seddighin, and Hadi Yami. 2018. Fair allocation of indivisible goods: Improvements and generalizations. In Proceedings of the 2018 ACM Conference on Economics and Computation. 539–556. https://doi.org/10.1145/3219166.3219238
[34]
Jiayi Guo, Simai He, Zi Ling, and Yicheng Liu. 2020. Bounding probability of small deviation on sum of independent random variables: Combination of moment approach and Berry-Esseen theorem. https://doi.org/10.48550/arxiv.2003.03197 arxiv:2003.03197.
[35]
Hadi Hosseini, Andrew Searns, and Erel Segal-Halevi. 2022. Ordinal Maximin Share Approximation for Goods. Journal of Artificial Intelligence Research, 74 (2022), 353–391. https://doi.org/10.1613/jair.1.13317
[36]
Hao Huang, Po-Shen Loh, and Benny Sudakov. 2012. The size of a hypergraph and its matching number. Combinatorics, Probability and Computing, 21, 3 (2012), 442–450. https://doi.org/10.1017/s096354831100068x
[37]
Xin Huang and Pinyan Lu. 2021. An algorithmic framework for approximating maximin share allocation of chores. In Proceedings of the 22nd ACM Conference on Economics and Computation. 630–631. https://doi.org/10.1145/3465456.3467555
[38]
Mamoru Kaneko and Kenjiro Nakamura. 1979. The Nash social welfare function. Econometrica: Journal of the Econometric Society, 47, 2 (1979), 423–435. https://doi.org/10.2307/1914191
[39]
Andrey Kupavskii. 2023. Rainbow version of the Erdős matching conjecture via concentration. Comb. Theory, 3, 1 (2023), Paper No. 1, 19. https://doi.org/10.5070/c63160414
[40]
László Lovász. 2007. Combinatorial problems and exercises (second ed.). AMS Chelsea Publishing, Providence, RI. https://doi.org/10.1090/chel/361
[41]
Hongliang Lu, Yan Wang, and Xingxing Yu. 2023. A better bound on the size of rainbow matchings. J. Combin. Theory Ser. A, 195 (2023), Paper No. 105700, 19. https://doi.org/10.1016/j.jcta.2022.105700
[42]
Tomasz Ł uczak and Katarzyna Mieczkowska. 2014. On Erdős’ extremal problem on matchings in hypergraphs. Journal of Combinatorial Theory, Series A, 124 (2014), 178–194. https://doi.org/10.1016/j.jcta.2014.01.003
[43]
Tomasz Ł uczak, Katarzyna Mieczkowska, and Matas Šileikis. 2017. On maximal tail probability of sums of nonnegative, independent and identically distributed random variables. Statist. Probab. Lett., 129 (2017), 12–16. https://doi.org/10.1016/j.spl.2017.04.024
[44]
Reshef Meir, Gil Kalai, and Moshe Tennenholtz. 2018. Bidding games and efficient allocations. Games and Economic Behavior, 112 (2018), 166–193. https://doi.org/10.1016/j.geb.2018.08.005
[45]
John F Nash Jr. 1950. The bargaining problem. Econometrica: Journal of the econometric society, 18, 2 (1950), 155–162. https://doi.org/10.2307/1907266
[46]
Ariel D Procaccia and Junxing Wang. 2014. Fair enough: Guaranteeing approximate maximin shares. In Proceedings of the fifteenth ACM conference on Economics and computation. 675–692. https://doi.org/10.1145/2600057.2602835
[47]
Malgorzata Romanowska. 1977. A note on the upper bound for the distance in total variation between the binomal and the Poisson distribution. Statistica Neerlandica, 31, 3 (1977), 127–130. https://doi.org/10.1111/j.1467-9574.1977.tb00759.x
[48]
S. M. Samuels. 1966. On a Chebyshev-type inequality for sums of independent random variables. Ann. Math. Statist., 37 (1966), 248–259. https://doi.org/10.1214/aoms/1177699614
[49]
Hugo Steinhaus. 1948. The problem of fair division. Econometrica, 16 (1948), 101–104.
[50]
Henry Teicher. 1955. An inequality on Poisson probabilities. The Annals of Mathematical Statistics, 26, 1 (1955), 147–149. https://doi.org/10.1214/aoms/1177728608
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June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
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