research-article

The Complexity of Fairness Through Equilibrium

Authors:

Abraham Othman,

Christos Papadimitriou,

Aviad RubinsteinAuthors Info & Claims

ACM Transactions on Economics and Computation (TEAC), Volume 4, Issue 4

Article No.: 20, Pages 1 - 19

https://doi.org/10.1145/2956583

Published: 16 August 2016 Publication History

Abstract

Competitive equilibrium from equal incomes (CEEI) is a well-known fair allocation mechanism with desirable fairness and efficiency properties; however, with indivisible resources, a CEEI may not exist [Foley 1967; Varian 1974; Thomson and Varian 1985]. It was shown in Budish [2011] that in the case of indivisible resources, there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient for some favorable approximation parameters. A heuristic search that attempts to find this approximation is used in practice to assign business school students to courses. In this article, we show that finding the A-CEEI allocation guaranteed to exist by Budish’s theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete.

References

[1]

T. Abbott, D. Kane, and P. Valiant. 2005. On the complexity of two-player win-lose games. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05). 113--122.

Digital Library

[2]

Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. 1998. Proof verification and the hardness of approximation problems. Journal of the ACM 45, 3, 501--555.

Digital Library

[3]

Sanjeev Arora and Shmuel Safra. 1998. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45, 1, 70--122.

Digital Library

[4]

Eric Budish. 2011. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119, 6, 1061--1103. http://EconPapers.repec.org/RePEc:ucp:jpolec:

[5]

Eric Budish, Gerard P. Cachon, Judd Kessler, and Abraham Othman. 2016. Course match: A large-scale implementation of approximate competitive equilibrium from equal incomes for combinatorial allocation. Operations Research. To appear.

[6]

Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 2009. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM 56, 3, Article No. 14.

Digital Library

[7]

Xi Chen, Dimitris Paparas, and Mihalis Yannakakis. 2013. The complexity of non-monotone markets. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC’13). 181--190.

Digital Library

[8]

Xi Chen and Shang-Hua Teng. 2009. Spending is not easier than trading: On the computational equivalence of Fisher and Arrow-Debreu equilibria. In Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC’09). 647--656.

Digital Library

[9]

Xi Chen and Shang-Hua Teng. 2011. A complexity view of markets with social influence. In Proceedings of the 2011 International Conference on Supercomputing (ICS’11). 141--154.

[10]

Bruno Codenotti, Amin Saberi, Kasturi Varadarajan, and Yinyu Ye. 2006. Leontief economies encode nonzero sum two-player games. In Proceedings of the Electronic Colloquium in Computational Complexity (TR-05-055).

[11]

Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. 2009. The complexity of computing a Nash equilibrium. Communications of the ACM 52, 2, 89--97.

Digital Library

[12]

Nikhil R. Devanur, Christos H. Papadimitriou, Amin Saberi, and Vijay V. Vazirani. 2008. Market equilibrium via a primal--Dual algorithm for a convex program. Journal of the ACM 55, 5, Article No. 22.

Digital Library

[13]

Uriel Feige. 1998. A threshold of ln n for approximating set cover. Journal of the ACM 45, 4, 634--652.

Digital Library

[14]

D. K. Foley. 1967. Resource allocation and the public sector. Yale Economic Essays 7, 1, 45--98.

[15]

Ali Ghodsi, Matei Zaharia, Benjamin Hindman, Andy Konwinski, Scott Shenker, and Ion Stoica. 2011. Dominant resource fairness: Fair allocation of multiple resource types. In Proceedings of the 8th USENIX Conference on Networked Systems Design and Implementation (NSDI’11). 323--336. http://dl.acm.org/ citation.cfm?id=1972457.1972490

Digital Library

[16]

Li-Sha Huang and Shang-Hua Teng. 2007. On the approximation and smoothed complexity of Leontief market equilibria. In Proceedings of the 1st Annual International Workshop on Frontiers in Algorithmics (FAW’07). 96--107.

Digital Library

[17]

Shiva Kintali, Laura J. Poplawski, Rajmohan Rajaraman, Ravi Sundaram, and Shang-Hua Teng. 2009. Reducibility among fractional stability problems. In Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). IEEE, Los Alamitos, CA, 283--292.

Digital Library

[18]

Noam Nisan. 2006. Bidding languages for combinatorial auctions. In Combinatorial Auctions, P. Cramton, Y. Shoham, and R. Steinberg (Eds.). MIT Press, Cambridge, MA.

[19]

Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani. 2007. Algorithmic Game Theory. Cambridge University Press, New York, NY.

Digital Library

[20]

Abraham Othman, Eric Budish, and Tuomas Sandholm. 2010. Finding approximate competitive equilibria: Efficient and fair course allocation. In Proceedings of the International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS’10).

Digital Library

[21]

Domotor Palvolgyi. 2009. 2D-TUCKER is PPAD-complete. In Internet and Network Economics. Lecture Notes in Computer Science, Vol. 5929. Springer, 569--574.

Digital Library

[22]

Christos H. Papadimitriou. 1994. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences 48, 3, 498--532.

Digital Library

[23]

Christos H. Papadimitriou and Mihalis Yannakakis. 1991. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 3, 425--440. 10.1016/0022-0000(91)90023-X

[24]

Aviad Rubinstein. 2015. Inapproximability of Nash equilibrium. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15). 409--418.

Digital Library

[25]

Tuomas Sandholm and Craig Boutilier. 2006. Preference elicitation in combinatorial auctions. In Combinatorial Auctions, P. Cramton, Y. Shoham, and R. Steinberg (Eds.). MIT Press, Cambridge, MA.

[26]

William Thomson and Hal R. Varian. 1985. Theories of justice based on symmetry. In Social Goals and Social Organizations: Essays in Memory of Elisha Pazner, L. Hurwicz, D. Schmeidler, and H. Sonnenschein (Eds.). Cambridge University Press, New York, NY.

[27]

Hal R. Varian. 1974. Equity, envy, and efficiency. Journal of Economic Theory 9, 1, 63--91.

[28]

Vijay V. Vazirani and Mihalis Yannakakis. 2011. Market equilibrium under separable, piecewise-linear, concave utilities. Journal of the ACM 58, 3, 10.

Digital Library

Cited By

Brânzei SSandomirskiy F(2024)Algorithms for Competitive Division of ChoresMathematics of Operations Research10.1287/moor.2023.136149:1(398-429)Online publication date: Feb-2024
https://doi.org/10.1287/moor.2023.1361
Chen XKroer CKumar R(2024)The Complexity of Pacing for Second-Price AuctionsMathematics of Operations Research10.1287/moor.2022.000949:4(2109-2135)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2022.0009
Nan TGao YKroer CWilliams BChen YNeville J(2023)Fast and interpretable dynamics for fisher markets via block-coordinate updatesProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i5.25723(5832-5840)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i5.25723
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Index Terms

The Complexity of Fairness Through Equilibrium
1. Applied computing
  1. Law, social and behavioral sciences

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Published In

cover image ACM Transactions on Economics and Computation

ACM Transactions on Economics and Computation Volume 4, Issue 4

Special Issue on EC'14

August 2016

147 pages

ISSN:2167-8375

EISSN:2167-8383

DOI:10.1145/2983294

Editors:
Vincent Conitzer
Duke University
,
Preston McAfee
Microsoft

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Copyright © 2016 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 August 2016

Accepted: 01 November 2015

Revised: 01 June 2015

Received: 01 December 2014

Published in TEAC Volume 4, Issue 4

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Cited By

Brânzei SSandomirskiy F(2024)Algorithms for Competitive Division of ChoresMathematics of Operations Research10.1287/moor.2023.136149:1(398-429)Online publication date: Feb-2024
https://doi.org/10.1287/moor.2023.1361
Chen XKroer CKumar R(2024)The Complexity of Pacing for Second-Price AuctionsMathematics of Operations Research10.1287/moor.2022.000949:4(2109-2135)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2022.0009
Nan TGao YKroer CWilliams BChen YNeville J(2023)Fast and interpretable dynamics for fisher markets via block-coordinate updatesProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i5.25723(5832-5840)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i5.25723
Budish EGao ROthman ARubinstein AZhang QLeyton-Brown KSamuelson LHartline J(2023)Practical algorithms and experimentally validated incentives for equilibrium-based fair division (A-CEEI)Proceedings of the 24th ACM Conference on Economics and Computation10.1145/3580507.3597809(337-368)Online publication date: 9-Jul-2023
https://dl.acm.org/doi/10.1145/3580507.3597809
Chen XPeng BSaha BServedio R(2023)Complexity of Equilibria in First-Price Auctions under General Tie-Breaking RulesProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585195(698-709)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585195
Papadimitriou CPeng B(2023)Public goods games in directed networksGames and Economic Behavior10.1016/j.geb.2023.02.002139(161-179)Online publication date: May-2023
https://doi.org/10.1016/j.geb.2023.02.002
Brânzei SNisan NKoyejo SMohamed SAgarwal ABelgrave DCho KOh A(2022)The query complexity of cake cuttingProceedings of the 36th International Conference on Neural Information Processing Systems10.5555/3600270.3603017(37905-37919)Online publication date: 28-Nov-2022
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Präntare FOsipov GEriksson LPelachaud CTaylor MFaliszewski PMascardi V(2022)Concise Representations and Complexity of Combinatorial Assignment ProblemsProceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems10.5555/3535850.3536086(1714-1716)Online publication date: 9-May-2022
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Grishutin AMusatov D(2022)Discrete Versions of the KKM Lemma and Their PPAD-CompletenessComputer Science – Theory and Applications10.1007/978-3-031-09574-0_11(170-189)Online publication date: 29-Jun-2022
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Babaioff MNisan NTalgam-Cohen I(2021)Competitive Equilibrium with Indivisible Goods and Generic BudgetsMathematics of Operations Research10.1287/moor.2020.106246:1(382-403)Online publication date: 1-Feb-2021
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