A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents
This solves a long-standing theoretical problem in fair division for computer science, mathematics, and economics, though it is incremental in protocol design.
The paper resolves a major open problem in cake cutting by proposing a discrete and bounded envy-free protocol for any number of agents, requiring at most n^(n^(n^(n^(n^n)))) queries, and also shows that partial allocations with proportionality and envy-freeness can be found in at most n^3(n^2)^n queries.
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 $n^3{(n^2)}^n$ 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 at most $n^3{(n^2)}^n$ queries such that each agent gets a connected piece that gives the agent at least $1/(3n)$ of the value of the whole cake.