Mind the Gap: Cake Cutting With Separation

Cited by: 0|Bibtex|Views17
Other Links: arxiv.org
Weibo:
We have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting

Abstract:

We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to...More

Code:

Data:

0
Introduction
  • The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street.
  • Each vendor naturally has a preference over potential locations, possibly depending on the proximity to certain attractions or the estimated number of customers visiting that space.
  • This year is different from previous years due to the social distancing guidelines issued by the government—vendors are required to be placed at least two meters apart.
  • Better still, such an allocation can be found by a simple and efficient algorithm (Dubins and Spanier
Highlights
  • The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street
  • We demonstrate that maximin share fairness is an appropriate substitute for proportionality in cake cutting with separation, and analyze it from a computational perspective
  • As is commonly done in cake cutting, we assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake
  • We have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting
  • Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations
  • While the canonical maximin share is a reasonable fairness requirement when agents have equal entitlements to the resource, in certain situations the agents may be endowed with different entitlements (Chakraborty et al 2020; Cseh and Fleiner 2020)
Results
  • As is commonly done in cake cutting, the authors assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake.
  • In Section 3, the authors begin by proving that an allocation that gives every agent at least her maximin share always exists, meaning that maximin share fairness can be guaranteed.
  • Such an allocation can be found by a simple algorithm provided that the algorithm knows the maximin share of each agent.
  • The authors design an algorithm that approximates the maximin share up to an arbitrarily small error, which allows them to compute an allocation wherein each
Conclusion
  • The authors have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting.
  • Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations.
  • This is the motivation behind the model of pie cutting that the authors have addressed in Section 4.
  • A connected cake allocation may not exist even without separation (Segal-Halevi 2019; Crew, Narayanan, and Spirkl 2020)
Summary
  • Introduction:

    The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street.
  • Each vendor naturally has a preference over potential locations, possibly depending on the proximity to certain attractions or the estimated number of customers visiting that space.
  • This year is different from previous years due to the social distancing guidelines issued by the government—vendors are required to be placed at least two meters apart.
  • Better still, such an allocation can be found by a simple and efficient algorithm (Dubins and Spanier
  • Objectives:

    The authors' goal is to compute Il(k), Ir(k) so that the list Ik = (Il(q), Ir(q))q∈[k] is consistent with some s-separated maximin partition of [0, 1]
  • Results:

    As is commonly done in cake cutting, the authors assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake.
  • In Section 3, the authors begin by proving that an allocation that gives every agent at least her maximin share always exists, meaning that maximin share fairness can be guaranteed.
  • Such an allocation can be found by a simple algorithm provided that the algorithm knows the maximin share of each agent.
  • The authors design an algorithm that approximates the maximin share up to an arbitrarily small error, which allows them to compute an allocation wherein each
  • Conclusion:

    The authors have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting.
  • Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations.
  • This is the motivation behind the model of pie cutting that the authors have addressed in Section 4.
  • A connected cake allocation may not exist even without separation (Segal-Halevi 2019; Crew, Narayanan, and Spirkl 2020)
Tables
  • Table1: Summary of the tasks that can and cannot be accomplished by finite algorithms in the Robertson-Webb model for cake cutting and pie cutting. All negative results hold even if the valuations of the agents are piecewise constant (but not given explicitly). The result with an asterisk holds when we do not allow the number of queries that the algorithm makes to depend on the separation parameter s
Download tables as Excel
Related work
Funding
  • This work was partially supported by the European Research Council (ERC) under grant number 639945 (ACCORD), by the Israel Science Foundation under grant number 712/20, and by an NUS Start-up Grant
Reference
  • Alijani, R.; Farhadi, M.; Ghodsi, M.; Seddighin, M.; and Tajik, A. S. 2017. Envy-free mechanisms with minimum number of cuts. In Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI), 312–318.
    Google ScholarLocate open access versionFindings
  • Arunachaleswaran, E. R.; Barman, S.; Kumar, R.; and Rathi, N. 2019. Fair and efficient cake division with connected pieces. In Proceedings of the 15th Conference on Web and Internet Economics (WINE), 57–70.
    Google ScholarLocate open access versionFindings
  • Arunachaleswaran, E. R.; Barman, S.; and Rathi, N. 2019. Fair division with a secretive agent. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence (AAAI), 1732–1739.
    Google ScholarLocate open access versionFindings
  • Aumann, Y.; and Dombb, Y. 2015. The efficiency of fair division with connected pieces. ACM Transactions on Economics and Computation 3(4): 23:1–23:16.
    Google ScholarLocate open access versionFindings
  • Aziz, H.; Chan, H.; and Li, B. 2019. Weighted maxmin fair share allocation of indivisible chores. In Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), 46–52.
    Google ScholarLocate open access versionFindings
  • Babaioff, M.; Nisan, N.; and Talgam-Cohen, I. 2019. Fair allocation through competitive equilibrium from generic incomes. In Proceedings of the 2nd ACM Conference on Fairness, Accountability, and Transparency (ACM FAT*), 180.
    Google ScholarLocate open access versionFindings
  • Barbanel, J. B.; Brams, S. J.; and Stromquist, W. 2009. Cutting a pie is not a piece of cake. American Mathematical Monthly 116(6): 496–514.
    Google ScholarLocate open access versionFindings
  • Bei, X.; Chen, N.; Hua, X.; Tao, B.; and Yang, E. 2012. Optimal proportional cake cutting with connected pieces. In Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI), 1263–1269.
    Google ScholarLocate open access versionFindings
  • Bei, X.; Chen, N.; Huzhang, G.; Tao, B.; and Wu, J. 2017. Cake cutting: Envy and truth. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), 3625–3631.
    Google ScholarLocate open access versionFindings
  • Bei, X.; Huzhang, G.; and Suksompong, W. 2018. Truthful fair division without free disposal. In Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI), 63–69.
    Google ScholarLocate open access versionFindings
  • Bei, X.; Igarashi, A.; Lu, X.; and Suksompong, W. 2021. The price of connectivity in fair division. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI). Forthcoming.
    Google ScholarLocate open access versionFindings
  • Bei, X.; and Suksompong, W. 2021. Dividing a graphical cake. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI). Forthcoming.
    Google ScholarLocate open access versionFindings
  • Bouveret, S.; Cechlarova, K.; Elkind, E.; Igarashi, A.; and Peters, D. 2017. Fair division of a graph. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), 135–141.
    Google ScholarLocate open access versionFindings
  • Brams, S. J.; Jones, M. A.; and Klamler, C. 2008. Proportional pie-cutting. International Journal of Game Theory 36(3–4): 353–367.
    Google ScholarLocate open access versionFindings
  • Brams, S. J.; and Taylor, A. D. 1996. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press.
    Google ScholarFindings
  • Branzei, S.; Caragiannis, I.; Kurokawa, D.; and Procaccia, A. D. 20An algorithmic framework for strategic fair division. In Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), 418–424.
    Google ScholarLocate open access versionFindings
  • Budish, E. 2011. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119(6): 1061–1103.
    Google ScholarLocate open access versionFindings
  • Cechlarova, K.; Dobos, J.; and Pillarova, E. 2013. On the existence of equitable cake divisions. Information Sciences 228: 239–245.
    Google ScholarLocate open access versionFindings
  • Cechlarova, K.; and Pillarova, E. 2012. On the computability of equitable divisions. Discrete Optimization 9(4): 249– 257.
    Google ScholarLocate open access versionFindings
  • Chakraborty, M.; Igarashi, A.; Suksompong, W.; and Zick, Y. 20Weighted envy-freeness in indivisible item allocation. In Proceedings of the 19th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), 231–239.
    Google ScholarLocate open access versionFindings
  • Crew, L.; Narayanan, B.; and Spirkl, S. 2020. Disproportionate division. Bulletin of the London Mathematical Society 52(5): 885–890.
    Google ScholarLocate open access versionFindings
  • Cseh, A.; and Fleiner, T. 2020. The complexity of cake cutting with unequal shares. ACM Transactions on Algorithms 16(3): 29:1–29:21.
    Google ScholarLocate open access versionFindings
  • Dubins, L. E.; and Spanier, E. H. 1961. How to cut a cake fairly. American Mathematical Monthly 68(1): 1–17.
    Google ScholarLocate open access versionFindings
  • Even, S.; and Paz, A. 1984. A note on cake cutting. Discrete Applied Mathematics 7(3): 285–296.
    Google ScholarLocate open access versionFindings
  • Farhadi, A.; Ghodsi, M.; Hajiaghayi, M. T.; Lahaie, S.; Pennock, D.; Seddighin, M.; Seddighin, S.; and Yami, H. 2019. Fair allocation of indivisible goods to asymmetric agents. Journal of Artificial Intelligence Research 64: 1–20.
    Google ScholarLocate open access versionFindings
  • Goldberg, P. W.; Hollender, A.; and Suksompong, W. 2020. Contiguous cake cutting: Hardness results and approximation algorithms. Journal of Artificial Intelligence Research 69: 109–141.
    Google ScholarLocate open access versionFindings
  • Hosseini, H.; Igarashi, A.; and Searns, A. 2020. Fair division of time: Multi-layered cake cutting. In Proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI), 182–188.
    Google ScholarLocate open access versionFindings
  • Kurokawa, D.; Procaccia, A. D.; and Wang, J. 2018. Fair enough: Guaranteeing approximate maximin shares. Journal of the ACM 64(2): 8:1–8:27.
    Google ScholarLocate open access versionFindings
  • Lonc, Z.; and Truszczynski, M. 2020. Maximin share allocations on cycles. Journal of Artificial Intelligence Research 69: 613–655.
    Google ScholarLocate open access versionFindings
  • Menon, V.; and Larson, K. 2017.
    Google ScholarFindings
  • Deterministic, strategyproof, and fair cake cutting. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), 352–358.
    Google ScholarLocate open access versionFindings
  • Procaccia, A. D. 2016. Cake cutting algorithms. In Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; and Procaccia, A. D., eds., Handbook of Computational Social Choice, chapter 13, 311–329. Cambridge University Press.
    Google ScholarLocate open access versionFindings
  • Procaccia, A. D.; and Wang, J. 2017. A lower bound for equitable cake cutting. In Proceedings of the 18th ACM Conference on Economics and Computation (EC), 479–495.
    Google ScholarLocate open access versionFindings
  • Robertson, J.; and Webb, W. 1998. Cake-Cutting Algorithms: Be Fair if You Can. Peters/CRC Press.
    Google ScholarFindings
  • Segal-Halevi, E. 2018. Redividing the cake. In Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI), 498–504.
    Google ScholarLocate open access versionFindings
  • Segal-Halevi, E. 2019. Cake-cutting with different entitlements: How many cuts are needed? Journal of Mathematical Analysis and Applications 480(1): 123382.
    Google ScholarLocate open access versionFindings
  • Segal-Halevi, E. 2020a. Competitive equilibrium for almost all incomes: Existence and fairness. Autonomous Agents and Multi-Agent Systems 34(1): 26:1–26:50.
    Google ScholarLocate open access versionFindings
  • Segal-Halevi, E. 2020b. Fair multi-cake cutting. Discrete Applied Mathematics. Forthcoming.
    Google ScholarFindings
  • Segal-Halevi, E.; Hassidim, A.; and Aumann, Y. 2020. Envy-free division of land. Mathematics of Operations Research. 45(3): 896–922.
    Google ScholarLocate open access versionFindings
  • Segal-Halevi, E.; and Nitzan, S. 2019. Fair cake-cutting among families. Social Choice and Welfare 53(4): 709–740.
    Google ScholarLocate open access versionFindings
  • Segal-Halevi, E.; Nitzan, S.; Hassidim, A.; and Aumann, Y. 2017. Fair and square: Cake-cutting in two dimensions. Journal of Mathematical Economics 70(8): 1–28.
    Google ScholarLocate open access versionFindings
  • Steinhaus, H. 1948. The problem of fair division. Econometrica 16(1): 101–104.
    Google ScholarLocate open access versionFindings
  • Stromquist, W. 1980. How to cut a cake fairly. American Mathematical Monthly 87(8): 640–644.
    Google ScholarLocate open access versionFindings
  • Stromquist, W. 2008. Envy-free cake divisions cannot be found by finite protocols. Electronic Journal of Combinatorics 15: #R11.
    Google ScholarLocate open access versionFindings
  • Su, F. E. 1999. Rental harmony: Sperner’s lemma in fair division. American Mathematical Monthly 106(10): 930– 942.
    Google ScholarLocate open access versionFindings
  • Thomson, W. 2007. Children crying at birthday parties. Why? Economic Theory 31(3): 501–521.
    Google ScholarLocate open access versionFindings
Full Text
Your rating :
0

 

Tags
Comments