# A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents

- Published in 2016
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We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from $n$ agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we do not run our protocol to completion, it can find in at most $n^{n+1}$ queries a partial allocation of the cake that achieves proportionality (each agent gets at least $1/n$ of the value of the whole cake) and envy-freeness. Finally we show that an envy-free partial allocation can be computed in $n^{n+1}$ queries such that each agent gets a connected piece that gives the agent at least $1/(3n)$ of the value of the whole cake.

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### BibTeX entry

@article{ADiscreteandBoundedEnvyFreeCakeCuttingProtocolforAnyNumberofAgents, title = {A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents}, abstract = {We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from {\$}n{\$} agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is {\$}n^{\{}n^{\{}n^{\{}n^{\{}n^n{\}}{\}}{\}}{\}}{\$}. We additionally show that even if we do not run our protocol to completion, it can find in at most {\$}n^{\{}n+1{\}}{\$} queries a partial allocation of the cake that achieves proportionality (each agent gets at least {\$}1/n{\$} of the value of the whole cake) and envy-freeness. Finally we show that an envy-free partial allocation can be computed in {\$}n^{\{}n+1{\}}{\$} queries such that each agent gets a connected piece that gives the agent at least {\$}1/(3n){\$} of the value of the whole cake.}, url = {http://arxiv.org/abs/1604.03655v10 http://arxiv.org/pdf/1604.03655v10}, author = {Haris Aziz and Simon Mackenzie}, comment = {}, urldate = {2016-10-13}, archivePrefix = {arXiv}, eprint = {1604.03655}, primaryClass = {cs.DS}, collections = {Attention-grabbing titles,Easily explained,Protocols and strategies,Food,Fun maths facts}, year = 2016 }