Advertisement

Economic Theory

, Volume 68, Issue 2, pp 363–401 | Cite as

Monotonicity and competitive equilibrium in cake-cutting

  • Erel Segal-HaleviEmail author
  • Balázs R. Sziklai
Research Article

Abstract

We study monotonicity properties of solutions to the classic problem of fair cake-cutting—dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cutting setting and show that they are violated by common division rules. In contrast, we prove that the Nash-optimal rule—maximizing the product of utilities—is resource-monotonic and population-monotonic, in addition to being Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfare-maximizing rules that is both proportional and resource-monotonic.

Keywords

Fair division Cake-cutting Resource-monotonicity Population-monotonicity Additive utilities Leximin-optimal rule Competitive equilibrium 

JEL Classification

D61 D63 

Notes

Acknowledgements

This paper was born in the COST Summer School on Fair Division in Grenoble, 7/2015 (FairDiv-15). We are grateful to COST and the conference organizers for the wonderful opportunity to meet with fellow researchers from around the globe. In particular, we are grateful to Ioannis Caragiannis, Ulle Endriss and Christian Klamler for sharing their insights on cake-cutting with us. We are also thankful to Marcus Berliant, Shiri Alon-Eron, Herve Moulin, Fedor Sandomirskiy, Christian Blatter, Ilan Nehama, Peter Kristel, Alex Ravsky and Kavi Rama Murthy for their very helpful comments. The reviews by several anonymous referees greatly improved the content and presentation of the paper.

References

  1. Arzi, O.: Cake cutting: achieving efficiency while maintaining fairness. Master’s thesis, Bar-Ilan University, under the supervision of Prof. Yonatan Aumann (2012)Google Scholar
  2. Arzi, O., Aumann, Y., Dombb, Y.: Toss one’s cake, and eat it too: partial divisions can improve social welfare in cake cutting. Soc. Choice Welf. 46(4), 933–954 (2016)Google Scholar
  3. Aumann, R.J., Peleg, B.: A note on Gale’s example. J. Math. Econ. 1(2), 209–211 (1974)Google Scholar
  4. Aziz, H., Ye, C.: Cake cutting algorithms for piecewise constant and piecewise uniform valuations. In: Liu TY, Qi Q, Ye Y (eds) Web and Internet Economics, Lecture Notes in Computer Science, vol. 8877, pp. 1–14. Springer, Berlin (2014). arXiv:1307.2908 (preprint)
  5. Balinski, M., Young, P.H.: Fair Representation: Meeting the Ideal of One Man, One Vote. Yale University Press, New Haven (1982)Google Scholar
  6. Barbanel, J.B.: The Geometry of Efficient Fair Division. Cambridge University Press, Cambridge (2005)Google Scholar
  7. Berliant, M., Thomson, W., Dunz, K.: On the fair division of a heterogeneous commodity. J. Math. Econ. 21(3), 201–216 (1992)Google Scholar
  8. Bogomolnaia, A., Moulin, H.: Competitive fair division under additive utilities. Working Paper (2016)Google Scholar
  9. Bogomolnaia, A., Moulin, H., Sandomirskiy, F., Yanovskaya, E.: Competitive division of a mixed manna. Econometrica 85(6), 1847–1871 (2017)Google Scholar
  10. Braess, D.: Über ein paradoxen der verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)Google Scholar
  11. Brânzei, S.: Computational fair division. Ph.D. thesis, Faculty of Science and Technology in Aarhus university (2015)Google Scholar
  12. Brânzei, S., Miltersen, P.B.: A dictatorship theorem for cake cutting. In: Proceedings of IJCAI’15, pp. 482–488 (2015)Google Scholar
  13. Brânzei, S., Caragiannis, I., Kurokawa, D., Procaccia, A.D.: An algorithmic framework for strategic fair division. In: Proceedings of AAAI’16, pp. 411–417 (2016)Google Scholar
  14. Brânzei, S., Gkatzelis, V., Mehta, R.: Nash social welfare approximation for strategic agents. In: Proceedings of ACM EC’17 (2017). arXiv:1607.01569 (preprint)
  15. Calleja, P., Rafels, C., Tijs, S.: Aggregate monotonic stable single-valued solutions for cooperative games. Int. J. Game Theory 41(4), 899–913 (2012)Google Scholar
  16. Caragiannis, I., Lai, J.K., Procaccia, A.D.: Towards more expressive cake cutting. In: Proceedings of IJCAI’11, pp. 127–132 (2011)Google Scholar
  17. Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A.D., Shah, N., Wang, J.: The unreasonable fairness of maximum Nash welfare. In: Proceedings on ACM EC’16 (2016)Google Scholar
  18. Chambers, C.P.: Allocation rules for land division. J. Econ. Theory 121(2), 236–258 (2005)Google Scholar
  19. Chichilnisky, G., Thomson, W.: The walrasian mechanism from equal division is not monotonic with respect to variations in the number of consumers. J. Public Econ. 32(1), 119–124 (1987)Google Scholar
  20. Cole, R., Gkatzelis, V., Goel, G.: Mechanism design for fair division: allocating divisible items without payments. In: Proceedings ACM EC’13, pp. 251–268. ACM, New York (2013). arXiv:1212.1522 (preprint)
  21. Conitzer, V., Freeman, R., Shah, N.: Fair public decision making. In: Proceedings ACM EC’17, pp. 629–646. ACM, New York (2017)Google Scholar
  22. Dall’Aglio, M.: The Dubins–Spanier optimization problem in fair division theory. J. Comput. Appl. Math. 130(1–2), 17–40 (2001)Google Scholar
  23. Dall’Aglio, M., Di Luca, C.: Finding maxmin allocations in cooperative and competitive fair division. Ann. Oper. Res. 223(1), 121–136 (2014)Google Scholar
  24. Dall’Aglio, M., Hill, T.P.: Maximin share and minimax envy in fair-division problems. J. Math. Anal. Appl. 281(1), 346–361 (2003)Google Scholar
  25. Devanur, N.R., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market equilibrium via a primal–dual algorithm for a convex program. J. ACM 55(5), 22:1–22:18 (2008)Google Scholar
  26. Dubins, L.E., Spanier, E.H.: How to cut a cake fairly. Am. Math. Mon. 68(1), 1–17 (1961)Google Scholar
  27. Dvoretzky, A., Wald, A., Wolfowitz, J.: Relations among certain ranges of vector measures. Pac. J. Math. 1(1), 59–74 (1951)Google Scholar
  28. Eisenberg, E., Gale, D.: Consensus of subjective probabilities: the pari-mutuel method. Ann. Math. Stat. 30(1), 165–168 (1959)Google Scholar
  29. Gale, D.: The linear exchange model. J. Math. Econ. 3(2), 205–209 (1976)Google Scholar
  30. Ghodsi, A., Zaharia, M., Hindman, B., Konwinski, A., Shenker, S., Stoica, I.: Dominant resource fairness: fair allocation of multiple resource types. In: Proceedings of the 8th USENIX Conference on Networked Systems Design and Implementation, USENIX Association, Berkeley, NSDI’11, pp. 323–336 (2011)Google Scholar
  31. Herreiner, D., Puppe, C.: Envy freeness in experimental fair division problems. Theor. Decis. 67(1), 65–100 (2009)Google Scholar
  32. Hill, T.P., Morrison, K.E.: Cutting cakes carefully. Coll. Math. J. 41(4), 281–288 (2010)Google Scholar
  33. Jain, K., Vazirani, V.V.: Eisenberg–Gale markets: algorithms and game-theoretic properties. Games Econ. Behav. 70(1), 84–106 (2010)Google Scholar
  34. Kalai, E.: Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45(7), 1623–1630 (1977)Google Scholar
  35. Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica 43(3), 513–518 (1975)Google Scholar
  36. Kash, I., Procaccia, A.D., Shah, N.: No agent left behind: dynamic fair division of multiple resources. J. Artif. Intell. Res. 51, 579–603 (2014)Google Scholar
  37. Kóczy, L.Á., Biró, P., Sziklai, B.: US vs. European apportionment practices: the conflict between monotonicity and proportionality. In: Endriss, U. (ed.) Trends in Computational Social Choice, Chap 16, pp. 309–325. AI Access, New York (2017)Google Scholar
  38. LiCalzi, M., Nicolò, A.: Efficient egalitarian equivalent allocations over a single good. Econ. Theor. 40(1), 27–45 (2009)Google Scholar
  39. Mas-Colell, A.: Equilibrium theory with possibly satiated preferences. In: Majumdar, M. (ed.) Equilibrium and Dynamics, Chap. 9. Springer, pp 201–213 (1992)Google Scholar
  40. Moulin, H.: Fair Division and Collective Welfare. The MIT Press, Cambridge (2004)Google Scholar
  41. Moulin, H., Thomson, W.: Can everyone benefit from growth? J. Math. Econ. 17(4), 339–345 (1988)Google Scholar
  42. Nash, J.F.: The bargaining problem. Econometrica 18(2), 155–162 (1950)Google Scholar
  43. Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Springer, Berlin (2007)Google Scholar
  44. Procaccia, A.D.: Cake cutting algorithms. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.) Handbook of Computational Social Choice. Cambridge University Press, Cambridge (2016). chap 13Google Scholar
  45. Rawls, J.: A Theory of Justice (1971)Google Scholar
  46. Reijnierse, J.H., Potters, J.A.M.: On finding an envy-free Pareto-optimal division. Math. Program. 83(1–3), 291–311 (1998)Google Scholar
  47. Schilling, R.L., Stoyan, D.: Continuity assumptions in cake-cutting (2016). ArXiv:1611.04988 (preprint)
  48. Segal-Halevi, E.: Redividing the Cake. In: Proceedings of IJCAI’18 (2018). arXiv:1603.00286 (preprint)
  49. Sönmez, T.: Consistency, monotonicity, and the uniform rule. Econ. Lett. 46(3), 229–235 (1994)Google Scholar
  50. Steinhaus, H.: The problem of fair division. Econometrica 16(1), 101–104 (1948)Google Scholar
  51. Stromquist, W., Woodall, D.R.: Sets on which several measures agree. J. Math. Anal. Appl. 108(1), 241–248 (1985)Google Scholar
  52. Sziklai, B., Segal-Halevi, E.: Resource Monotonicity and Population Monotonicity in Connected Cake Cutting (2015). arXiv:1510.05229 (preprint)
  53. Tasnádi, A.: A new proportional procedure for the n-person cake-cutting problem. Econ. Bull. 4(33), 1–3 (2003)Google Scholar
  54. Thomson, W.: The replacement principle in economies with single-peaked preferences. J. Econ. Theory 76(1), 145–168 (1997)Google Scholar
  55. Thomson, W.: Children crying at birthday parties. Why? Econ. Theor. 31(3), 501–521 (2007)Google Scholar
  56. Thomson, W.: Fair allocation rules. In: Handbook of Social Choice and Welfare, pp. 393–506, Elsevier (2011)Google Scholar
  57. Thomson, W.: On the axiomatics of resource allocation: interpreting the consistency principle. Econ. Philos. 28(03), 385–421 (2012)Google Scholar
  58. Vazirani, V.V.: Combinatorial algorithms for market equilibria. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.) Algorithmic Game Theory, pp. 103–134. Cambridge University Press, Cambridge (2007). chap 5Google Scholar
  59. Walsh, T.: Online cake cutting. Algorithmic Decision Theory 6992, 292–305 (2011)Google Scholar
  60. Weller, D.: Fair division of a measurable space. J. Math. Econ. 14(1), 5–17 (1985)Google Scholar
  61. Young, P.H.: On dividing an amount according to individual claims or liabilities. Math. Oper. Res. 12(3), 65–72 (1987)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ariel UniversityArielIsrael
  2. 2.Bar-Ilan UniversityRamat-GanIsrael
  3. 3.Centre for Economic and Regional StudiesHungarian Academy of SciencesBudapestHungary
  4. 4.Department of Operations Research and Actuarial SciencesCorvinus University of BudapestBudapestHungary

Personalised recommendations