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


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.


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

JEL Classification

D61 D63 



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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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

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