A discrete and bounded envy-free cake cutting protocol for four agents

We consider the well-studied cake cutting problem in which the goal is to identify an envy-free allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem to obtain a bounded envy-free protocol for more than three agents. The problem has been termed the central open problem in cake cutting. We solve this problem by proposing a discrete and bounded envy-free protocol for four agents.

[1]  Ariel D. Procaccia,et al.  How to Cut a Cake Before the Party Ends , 2013, AAAI.

[2]  Noam Nisan,et al.  Incentive Compatible Two Player Cake Cutting , 2012, WINE.

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

[4]  Haris Aziz,et al.  Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations , 2013, WINE.

[5]  A. Kirman,et al.  Fairness and Envy , 1974 .

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

[7]  A. Razborov.  Communication Complexity , 2011 .

[8]  Simina Brânzei,et al.  A Dictatorship Theorem for Cake Cutting , 2015, IJCAI.

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

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

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

[12]  Branko Milanovic,et al.  Dividing the Spoils: Pensions, Privatization and Reform in Russia's Transition , 2000 .

[13]  Costas Busch,et al.  Hardness Results for Cake Cutting , 2005, Bull. EATCS.

[14]  R. Rosenfeld.  Truth , 2012, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[15]  Oleg Pikhurko.  On Envy-Free Cake Division , 2000, Am. Math. Mon..

[16]  Ariel D. Procaccia,et al.  Cake cutting: not just child's play , 2013, CACM.

[17]  F. Su.  Rental Harmony: Sperner's Lemma in Fair Division , 1999 .

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

[19]  Xiaotie Deng,et al.  Algorithmic Solutions for Envy-Free Cake Cutting , 2012, Oper. Res..

[20]  Richard Cole,et al.  Approximating the Nash Social Welfare with Indivisible Items , 2018, SIAM J. Comput..

[21]  Pascal James Imperato.  Dividing the Spoils , 2018 .

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

[23]  Ian Stewart.  Division without Envy , 1999 .

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

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

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

[27]  Julius B. Barbanel,et al.  Preference relations and measures in the context of fair division , 1995 .

[28]  Erel Segal-Halevi,et al.  Waste Makes Haste: Bounded Time Protocols for Envy-Free Cake Cutting with Free Disposal , 2015, AAMAS.

[29]  Steven J. Brams,et al.  Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond , 2004, Math. Soc. Sci..

[30]  W. Thomson.  Children Crying at Birthday Parties. Why? , 2007 .

[31]  Simina Brânzei,et al.  A note on envy-free cake cutting with polynomial valuations , 2015, Inf. Process. Lett..

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

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