Abstract
We focus on a fundamental problem in discrete fair division, where the goal is to divide indivisible goods among a set of agents “fairly”. Ideally, one would aim to divide the goods such that no agent envies another agent. However, since the goods are indivisible, such allocations may not always exist (a simple scenario involving two agents and a single good). Therefore, relaxations of envy-freeness have been proposed and extensively studied. We focus on one of the most fundamental and sought out relaxations – envy-freeness up to any good (EFX), where no agent envies another, following the removal of any single good from the other’s bundle. Despite substantial effort from the community, the existence of EFX allocations has not been settled. In this paper, we sketch the proof of existence of “almost” EFX allocations and the existence of EFX allocations when there are only three agents. In the end, we reduce the problem of finding improved guarantees on EFX allocations to a problem in zero sum extremal combinatorics.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
- 2.
Bundle that minimizes disutility in the case of bads or the bundle that maximizes utility in the case of goods.
- 3.
A valuation v is additive if \(v(S) = \sum _{s \in S} v(\{s\})\) for all S.
- 4.
Note that n is the number of agents which is typically much smaller than the number of goods.
- 5.
An allocation where not all the goods are allocated.
- 6.
An allocation with maximum Nash welfare is also Pareto-optimal (an alternate measure of efficiency).
- 7.
One would need exponentially many value queries to get any sublinear approximation of Nash welfare when agents have subadditive valuation functions.
- 8.
An allocation \(X =\langle X_1,X_2, \dots , X_n \rangle \) is a \((1-\varepsilon )\)-EFX allocation if and only if for all pairs of agents i and \(i'\), we have \(v_i(X_i) \ge (1-\varepsilon ) \cdot v_i (X_{i'} \setminus \{g\})\) for all \(g \in X_{i'}\).
References
Anari, N., Mai, T., Gharan, S.O., Vazirani, V.V.: Nash social welfare for indivisible items under separable, piecewise-linear concave utilities. In: Proc. 29th Symp. Discrete Algorithms (SODA). pp. 2274–2290 (2018)
B. Budish, E., Cantillon, E.: The multi-unit assignment problem: Theory and evidence from course allocation at harvard. American Economic Review 102 (2010)
Barman, S., Krishnamurthy, S.K., Vaish, R.: Finding fair and efficient allocations. In: Proceedings of the 19th ACM Conference on Economics and Computation (EC). pp. 557–574 (2018)
Brams, S.J., Taylor, A.D.: Fair division - from cake-cutting to dispute resolution. Cambridge University Press (1996)
Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A.D., Shah, N., Wang, J.: The unreasonable fairness of maximum Nash welfare. In: Proceedings of the 17th ACM Conference on Economics and Computation (EC). pp. 305–322 (2016)
Chaudhury, B.R., Garg, J., Mehlhorn, K.: EFX exists for three agents. In: Proceedings of the 21st ACM Conference on Economics and Computation (EC). pp. 1–19. ACM (2020)
Chaudhury, B.R., Garg, J., Mehlhorn, K., Mehta, R., Misra, P.: Improving EFX guarantees through rainbow cycle number. In: Proceedings of the 22nd ACM Conference on Economics and Computation (EC). pp. 310–311. ACM (2021)
Chaudhury, B.R., Garg, J., Mehta, R.: Fair and efficient allocations under subadditive valuations. In: Proc. of the 35th AAAI Conference on Artificial Intelligence (AAAI) (2021)
Chaudhury, B.R., Kavitha, T., Mehlhorn, K., Sgouritsa, A.: A little charity guarantees almost envy-freeness. SIAM J. Comput. 50(4), 1336–1358 (2021)
Cole, R., Gkatzelis, V.: Approximating the nash social welfare with indivisible items. SIAM J. Comput. 47(3), 1211–1236 (2018)
Etkin, R., Parekh, A., Tse, D.: Spectrum sharing for unlicensed bands. In: In Proceedings of the first IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks (2005)
Garg, J., Kulkarni, P., Kulkarni, R.: Approximating Nash social welfare under submodular valuations through (un)matchings. In: Proc. of the 31st Symposium on Discrete Algorithms (SODA 2020) (2020), to appear
Moulin, H.: Fair division in the internet age. Annual Review of Economics 11 (2019)
Plaut, B., Roughgarden, T.: Almost envy-freeness with general valuations. SIAM J. Discret. Math. 34(2), 1039–1068 (2020)
Pratt, J.W., Zeckhauser, R.J.: The fair and efficient division of the winsor family silver. Management Science 36(11), 1293–1301 (1990)
Procaccia, A.D.: Technical perspective: An answer to fair division’s most enigmatic question. Commun. ACM 63(4), 118 (Mar 2020). https://doi.org/10.1145/3382131, https://doi.org/10.1145/3382131
Steinhaus, H.: The problem of fair division. Econometrica 16, 101–104 (1948)
Vossen, T.W.: Fair allocation concepts in air traffic management. Ph.D. thesis, University of Maryland, College Park (2002)
Walras, L.: Éléments d’économie politique pure, ou théorie de la richesse sociale (Elements of Pure Economics, or the theory of social wealth). Lausanne, Paris (1874), (1899, 4th ed.; 1926, rev ed., 1954, Engl. transl.)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Chaudhury, B.R. (2022). On the Existence of EFX Allocations. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-06678-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-06677-1
Online ISBN: 978-3-031-06678-8
eBook Packages: Computer ScienceComputer Science (R0)