Communication Complexity of Cake Cutting

We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.Comment: Added efficient communication protocol for the monotone crossing proble

A Cryptographic Moving-Knife Cake-Cutting Protocol

This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the moving-knife protocol can be executed asynchronously by a discrete protocol using a secure auction protocol. The number of cuts is n-1 where n is the number of players, which is the minimum.Comment: In Proceedings IWIGP 2012, arXiv:1202.422

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 a fair 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 for the last fifty years to obtain a bounded envy-free protocol for more than three agents. We propose a discrete and bounded envy-free protocol for four agents

Envy-free cake division without assuming the players prefer nonempty pieces

Consider $n$ players having preferences over the connected pieces of a cake, identified with the interval $[0,1]$. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into $n$ connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when $n$ is a prime number or is equal to $4$. When $n$ is at most $3$, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any $n$. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary

An Algorithmic Framework for Strategic Fair Division

We study the paradigmatic fair division problem of allocating a divisible good among agents with heterogeneous preferences, commonly known as cake cutting. Classical cake cutting protocols are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? To address this question we adopt a novel algorithmic approach, by designing a concrete computational framework for fair division---the class of Generalized Cut and Choose (GCC) protocols}---and reasoning about the game-theoretic properties of algorithms that operate in this model. The class of GCC protocols includes the most important discrete cake cutting protocols, and turns out to be compatible with the study of fair division among strategic agents. In particular, GCC protocols are guaranteed to have approximate subgame perfect Nash equilibria, or even exact equilibria if the protocol's tie-breaking rule is flexible. We further observe that the (approximate) equilibria of proportional GCC protocols---which guarantee each of the $n$ agents a $1/n$-fraction of the cake---must be (approximately) proportional. Finally, we design a protocol in this framework with the property that its Nash equilibrium allocations coincide with the set of (contiguous) envy-free allocations

Efficient Algorithms for Envy-Free Stick Division With Fewest Cuts

Given a set of n sticks of various (not necessarily different) lengths, what is the largest length so that we can cut k equally long pieces of this length from the given set of sticks? We analyze the structure of this problem and show that it essentially reduces to a single call of a selection algorithm; we thus obtain an optimal linear-time algorithm. This algorithm also solves the related envy-free stick-division problem, which Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) recently used as their central primitive operation for the first discrete and bounded envy-free cake cutting protocol with a proportionality guarantee when pieces can be put to waste.Comment: v3 adds more context about the proble