## 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.

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## Notes

- 1.
- 2.
In an accompanying technical report (Sziklai and Segal-Halevi 2015) we provide similar examples showing that other classic cake-cutting procedures, like Banach–Knaster, Dubins–Spanier and many others, violate both RM and PM.

- 3.
“...there are a number of important issues that should be tackled next pertaining, in particular, to the existence of selections from the no-envy solution satisfying additional properties, examples are monotonicity with respect to the amount to be divided (all agents should benefit from such an increase), and with respect to changes in the number of claimants (all agents initially present should lose in such circumstances).” (Berliant et al. 1992).

- 4.
It is sometimes assumed that

*C*is a pie, i.e., that its endpoints are identified (Thomson 2007). The results in this paper are valid for this model too. - 5.
It is sometimes assumed that the value measures are absolutely continuous with respect to the Lebesgue measure. This means that any slice with zero length (not only a point) is worth 0 for all agents. This is equivalent to the assumption that each value measure is the integral of some “value-density” function, describing the value per unit of length. The absolute-continuity assumption is strictly stronger than our non-atomicity assumption; see Hill and Morrison (2010) and Schilling andStoyan (2016).

- 6.
A more recent protocol, the recursive Cut and Choose, proposed by Tasnádi (2003), violates resource-monotonicity for the same reason.

- 7.
this variant is called the

*Rightmost Mark*rule (Sziklai and Segal-Halevi 2015). - 8.
Note that Weller’s definition of Pareto-optimality (before his Theorem 1) actually defines weak-Pareto-optimality. We are grateful to an anonymous reviewer for this comment.

- 9.
- 10.
Weller (1985) uses a similar price measure in his proof, but does not consider Nash-optimal allocations.

- 11.
- 12.
We are grateful to Fedor Sandomirskiy and an anonymous referee for their help in clarifying this issue.

- 13.
Lemma 13 is not unique to cake-cutting—it is a corollary of a well-known general fact: for a convex set \({\mathbb {M}}\subseteq {\mathbb {R}}^n\), the maximizer of leximin ordering is unique.

- 14.
Note that this argument does not work for the relative-leximin division rule. It is possible that the relative-leximin-optimal division of the enlarged cake is lexicographically smaller than the relative-leximin-optimal division of the smaller cake, since the change in the total cake values changes the order between the agents’ relative values.

- 15.
We are grateful to an anonymous reviewer for this comment.

## References

Arzi, O.: Cake cutting: achieving efficiency while maintaining fairness. Master’s thesis, Bar-Ilan University, under the supervision of Prof. Yonatan Aumann (2012)

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)Aumann, R.J., Peleg, B.: A note on Gale’s example. J. Math. Econ.

**1**(2), 209–211 (1974)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**)Balinski, M., Young, P.H.: Fair Representation: Meeting the Ideal of One Man, One Vote. Yale University Press, New Haven (1982)

Barbanel, J.B.: The Geometry of Efficient Fair Division. Cambridge University Press, Cambridge (2005)

Berliant, M., Thomson, W., Dunz, K.: On the fair division of a heterogeneous commodity. J. Math. Econ.

**21**(3), 201–216 (1992)Bogomolnaia, A., Moulin, H.: Competitive fair division under additive utilities. Working Paper (2016)

Bogomolnaia, A., Moulin, H., Sandomirskiy, F., Yanovskaya, E.: Competitive division of a mixed manna. Econometrica

**85**(6), 1847–1871 (2017)Braess, D.: Über ein paradoxen der verkehrsplanung. Unternehmensforschung

**12**, 258–268 (1968)Brânzei, S.: Computational fair division. Ph.D. thesis, Faculty of Science and Technology in Aarhus university (2015)

Brânzei, S., Miltersen, P.B.: A dictatorship theorem for cake cutting. In: Proceedings of IJCAI’15, pp. 482–488 (2015)

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)

Brânzei, S., Gkatzelis, V., Mehta, R.: Nash social welfare approximation for strategic agents. In: Proceedings of ACM EC’17 (2017). arXiv:1607.01569 (

**preprint**)Calleja, P., Rafels, C., Tijs, S.: Aggregate monotonic stable single-valued solutions for cooperative games. Int. J. Game Theory

**41**(4), 899–913 (2012)Caragiannis, I., Lai, J.K., Procaccia, A.D.: Towards more expressive cake cutting. In: Proceedings of IJCAI’11, pp. 127–132 (2011)

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)

Chambers, C.P.: Allocation rules for land division. J. Econ. Theory

**121**(2), 236–258 (2005)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)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**)Conitzer, V., Freeman, R., Shah, N.: Fair public decision making. In: Proceedings ACM EC’17, pp. 629–646. ACM, New York (2017)

Dall’Aglio, M.: The Dubins–Spanier optimization problem in fair division theory. J. Comput. Appl. Math.

**130**(1–2), 17–40 (2001)Dall’Aglio, M., Di Luca, C.: Finding maxmin allocations in cooperative and competitive fair division. Ann. Oper. Res.

**223**(1), 121–136 (2014)Dall’Aglio, M., Hill, T.P.: Maximin share and minimax envy in fair-division problems. J. Math. Anal. Appl.

**281**(1), 346–361 (2003)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)Dubins, L.E., Spanier, E.H.: How to cut a cake fairly. Am. Math. Mon.

**68**(1), 1–17 (1961)Dvoretzky, A., Wald, A., Wolfowitz, J.: Relations among certain ranges of vector measures. Pac. J. Math.

**1**(1), 59–74 (1951)Eisenberg, E., Gale, D.: Consensus of subjective probabilities: the pari-mutuel method. Ann. Math. Stat.

**30**(1), 165–168 (1959)Gale, D.: The linear exchange model. J. Math. Econ.

**3**(2), 205–209 (1976)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)

Herreiner, D., Puppe, C.: Envy freeness in experimental fair division problems. Theor. Decis.

**67**(1), 65–100 (2009)Hill, T.P., Morrison, K.E.: Cutting cakes carefully. Coll. Math. J.

**41**(4), 281–288 (2010)Jain, K., Vazirani, V.V.: Eisenberg–Gale markets: algorithms and game-theoretic properties. Games Econ. Behav.

**70**(1), 84–106 (2010)Kalai, E.: Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica

**45**(7), 1623–1630 (1977)Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica

**43**(3), 513–518 (1975)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)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)

LiCalzi, M., Nicolò, A.: Efficient egalitarian equivalent allocations over a single good. Econ. Theor.

**40**(1), 27–45 (2009)Mas-Colell, A.: Equilibrium theory with possibly satiated preferences. In: Majumdar, M. (ed.) Equilibrium and Dynamics, Chap. 9. Springer, pp 201–213 (1992)

Moulin, H.: Fair Division and Collective Welfare. The MIT Press, Cambridge (2004)

Moulin, H., Thomson, W.: Can everyone benefit from growth? J. Math. Econ.

**17**(4), 339–345 (1988)Nash, J.F.: The bargaining problem. Econometrica

**18**(2), 155–162 (1950)Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Springer, Berlin (2007)

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 13

Rawls, J.: A Theory of Justice (1971)

Reijnierse, J.H., Potters, J.A.M.: On finding an envy-free Pareto-optimal division. Math. Program.

**83**(1–3), 291–311 (1998)Schilling, R.L., Stoyan, D.: Continuity assumptions in cake-cutting (2016). ArXiv:1611.04988 (

**preprint**)Segal-Halevi, E.: Redividing the Cake. In: Proceedings of IJCAI’18 (2018). arXiv:1603.00286 (

**preprint**)Sönmez, T.: Consistency, monotonicity, and the uniform rule. Econ. Lett.

**46**(3), 229–235 (1994)Steinhaus, H.: The problem of fair division. Econometrica

**16**(1), 101–104 (1948)Stromquist, W., Woodall, D.R.: Sets on which several measures agree. J. Math. Anal. Appl.

**108**(1), 241–248 (1985)Sziklai, B., Segal-Halevi, E.: Resource Monotonicity and Population Monotonicity in Connected Cake Cutting (2015). arXiv:1510.05229 (

**preprint**)Tasnádi, A.: A new proportional procedure for the n-person cake-cutting problem. Econ. Bull.

**4**(33), 1–3 (2003)Thomson, W.: The replacement principle in economies with single-peaked preferences. J. Econ. Theory

**76**(1), 145–168 (1997)Thomson, W.: Children crying at birthday parties. Why? Econ. Theor.

**31**(3), 501–521 (2007)Thomson, W.: Fair allocation rules. In: Handbook of Social Choice and Welfare, pp. 393–506, Elsevier (2011)

Thomson, W.: On the axiomatics of resource allocation: interpreting the consistency principle. Econ. Philos.

**28**(03), 385–421 (2012)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 5

Walsh, T.: Online cake cutting. Algorithmic Decision Theory

**6992**, 292–305 (2011)Weller, D.: Fair division of a measurable space. J. Math. Econ.

**14**(1), 5–17 (1985)Young, P.H.: On dividing an amount according to individual claims or liabilities. Math. Oper. Res.

**12**(3), 65–72 (1987)

## 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.

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## Additional information

This research was supported by the Higher Education Institutional Excellence Program of the Ministry of Human Capacities in the framework of the “Financial and Retail Services” research Project (1783-3/2018/FEKUTSTRAT) at the Corvinus University of Budapest. The authors acknowledge the support of Hungarian National Research, Development and Innovation Office, Grant No. K124550, the ISF Grants 1083/13 and 1394/16, the Doctoral Fellowships of Excellence Program, the Wolfson Chair and the Mordecai and Monique Katz Graduate Fellowship Program at Bar-Ilan University. This research was partially supported by Pallas Athene Domus Educationis Foundation. The views expressed are those of the authors’ and do not necessarily reflect the official opinion of Pallas Athene Domus Educationis Foundation.

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Segal-Halevi, E., Sziklai, B.R. Monotonicity and competitive equilibrium in cake-cutting.
*Econ Theory* **68, **363–401 (2019). https://doi.org/10.1007/s00199-018-1128-6

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### Keywords

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

### JEL Classification

- D61
- D63