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We consider the well-studied cake cutting problem in which the goal is tofind an envy-free allocation based on queries from $n$ agents. The problem hasreceived attention in computer science, mathematics, and economics. It has beena major open problem whether there exists a discrete and bounded envy-freeprotocol. We resolve the problem by proposing a discrete and bounded envy-freeprotocol for any number of agents. The maximum number of queries required bythe protocol is $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we donot run our protocol to completion, it can find in at most $n^3{(n^2)}^n$queries a partial allocation of the cake that achieves proportionality (eachagent 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 at most$n^3{(n^2)}^n$ queries such that each agent gets a connected piece that givesthe agent at least $1/(3n)$ of the value of the whole cake.

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