## 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.

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## 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