Cutting a Cake Is Not Always a "Piece of Cake": A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents’ valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory.

[1]  Massimo Marinacci,et al.  APPLIED MATHEMATICS WORKING PAPER SERIESHOW TO CUT A PIZZA FAIRLY: , 2002 .

[2]  Tatsuaki Okamoto,et al.  Meta-Envy-Free Cake-Cutting Protocols , 2010, MFCS.

[3]  Jack M. Robertson,et al.  Near Exact and Envy Free Cake Division , 1997, Ars Comb..

[4]  William S. Zwicker,et al.  A Moving-Knife Solution to the Four-Person Envy-Free Cake-Division Problem , 1995 .

[5]  John von Neumann,et al.  Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen , 1929 .

[6]  S. Brams,et al.  Better Ways to Cut a Cake , 2006 .

[7]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[8]  Ariel D. Procaccia.  Thou Shalt Covet Thy Neighbor's Cake , 2009, IJCAI.

[9]  R. Schilling,et al.  Counterexamples in Measure and Integration , 2022 .

[10]  Xiaotie Deng,et al.  On the Complexity of Envy-Free Cake Cutting , 2009, ArXiv.

[11]  Jörg Rothe,et al.  Cake-Cutting: Fair Division of Divisible Goods , 2016, Economics and Computation.

[12]  Walter Stromquist,et al.  Envy-Free Cake Divisions Cannot be Found by Finite Protocols , 2008, Electron. J. Comb..

[13]  Simina Brânzei,et al.  An Algorithmic Framework for Strategic Fair Division , 2013, AAAI.

[14]  Lawrence G. Sager.  Handbook of Computational Social Choice , 2015 .

[15]  Simina Brânzei,et al.  Equilibrium analysis in cake cutting , 2013, AAMAS.

[16]  Steven J. Brams,et al.  Fair division - from cake-cutting to dispute resolution , 1998 .

[17]  Avinatan Hassidim,et al.  Auctioning a cake: truthful auctions of heterogeneous divisible goods , 2014, AAMAS.

[18]  Haris Aziz,et al.  A discrete and bounded envy-free cake cutting protocol for four agents , 2015, STOC.

[19]  Leçons sur l'integration , 1898 .

[20]  Ying Wang,et al.  Cutting a Cake for Five People , 2009, AAIM.

[21]  Jörg Rothe,et al.  Degrees of Guaranteed Envy-Freeness in Finite Bounded Cake-Cutting Protocols , 2009, WINE.

[22]  Jrg Rothe,et al.  Economics and Computation, An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division , 2015, Economics and Computation.

[23]  Julius B. Barbanel.  Super Envy-Free Cake Division and Independence of Measures , 1996 .

[24]  Simina Brânzei,et al.  Externalities in Cake Cutting , 2013, IJCAI.

[25]  Ariel D. Procaccia,et al.  Truth, justice, and cake cutting , 2010, Games Econ. Behav..

[26]  Steven J. Brams,et al.  N-Person Cake-Cutting: There May Be No Perfect Division , 2011, Am. Math. Mon..

[27]  Toby Walsh,et al.  Online Cake Cutting , 2010, ADT.

[28]  Gerhard J. Woeginger.  An Approximation Scheme for Cake Division with a Linear Number of Cuts , 2002, ESA.

[29]  Haris Aziz,et al.  A bounded and envy-free cake cutting algorithm , 2020, Commun. ACM.

[30]  Haris Aziz,et al.  A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[31]  William A. Webb,et al.  How to Cut a Cake Fairly Using a Minimal Number of Cuts , 1997, Discret. Appl. Math..

[32]  S. Banach.  Sur le problème de la mesure , 2022 .

[33]  Ariel D. Procaccia,et al.  Towards More Expressive Cake Cutting , 2011, IJCAI.

[34]  Avinatan Hassidim,et al.  Computing socially-efficient cake divisions , 2012, AAMAS.

[35]  Douglas R. Woodall,et al.  Sets on Which Several Measures Agree , 1985 .

[36]  Hans Reijnierse,et al.  On finding an envy-free Pareto-optimal division , 1998, Math. Program..

[37]  Steven J. Brams,et al.  Perfect Cake-Cutting Procedures with Money , 2003 .

[38]  Ariel D. Procaccia.  Cake Cutting Algorithms , 2016, Handbook of Computational Social Choice.

[39]  Steven J. Brams,et al.  Proportional pie-cutting , 2008, Int. J. Game Theory.

[40]  Eva Pillárová,et al.  A near equitable 2-person cake cutting algorithm , 2012 .

[41]  K. Ciesielski.  How good is lebesgue measure? , 1989 .

[42]  Gerhard J. Woeginger,et al.  On the complexity of cake cutting , 2007, Discret. Optim..

[43]  Ariel D. Procaccia,et al.  On Maxsum Fair Cake Divisions , 2012, AAAI.

[44]  Ning Chen,et al.  Optimal Proportional Cake Cutting with Connected Pieces , 2012, AAAI.

[45]  Ariel D. Procaccia,et al.  Optimal Envy-Free Cake Cutting , 2011, AAAI.

[46]  Julius B. Barbanel.  Game-theoretic algorithms for fair and strongly fair cake division with entitlements , 1996 .

[47]  F. Su.  The Banach-Tarski Paradox , 1990 .

[48]  Yonatan Aumann,et al.  Throw One's Cake - and Eat It Too , 2011, SAGT.

[49]  S. Banach,et al.  Théorie des opérations linéaires , 1932 .

[50]  Erel Segal-Halevi,et al.  Fair and Square: Cake-Cutting in Two Dimensions , 2014, ArXiv.

[51]  Jack M. Robertson,et al.  Cake-cutting algorithms - be fair if you can , 1998 .

[52]  Dao-Zhi Zeng,et al.  Approximate Envy-Free Procedures , 2000 .

[53]  Katarína Cechlárová,et al.  On the existence of equitable cake divisions , 2013, Inf. Sci..

[54]  Yonatan Aumann,et al.  The Efficiency of Fair Division with Connected Pieces , 2010, WINE.

[55]  Dietrich Stoyan,et al.  Continuity Assumptions in Cake-Cutting , 2016, ArXiv.

[56]  Katarína Cechlárová,et al.  On the computability of equitable divisions , 2012, Discret. Optim..