Simon Mackenzie

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

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 nnnnnn. Even if we do not run our protocol to completion, it can find in at most nn+1 queries an envy-free partial allocation of the cake in which each agent gets at least 1/n of the value of the whole cake.

Fair assignment of indivisible objects under ordinal preferences

We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of proportionality and envy-freeness for discrete assignments. The computational complexity of checking whether a fair assignment exists is studied for these fairness notions. We also characterize the conditions under which a fair assignment is guaranteed to exist. For a number of fairness concepts, polynomial-time algorithms are presented to check whether a fair assignment exists. Our algorithmic results also extend to the case of unequal entitlements of agents. Our NP-hardness result, which holds for several variants of envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang (ECAI 2010). We also propose fairness concepts that always suggest a non-empty set of assignments with meaningful fairness properties. Among these concepts, optimal proportionality and optimal weak proportionality appear to be desirable fairness concepts.

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.

On the number of minimal separators in graphs

Weconsider the largest number of minimal separators a graph on n vertices can have. –We give a new proof that this number is in O1+52n·n. –We prove that this number is in ω(1.4457n), improving on the previous best lower bound of Ω(3n/3)⊆ω(1.4422n). This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.

Pillage games with multiple stable sets

We prove that pillage games (Jordan in J Econ Theory 131.1:26–44, 2006, “Pillage and property”, JET) can have multiple stable sets, constructing pillage games with up to

Bounded and envy-free cake cutting

2^{\tfrac{n-1}{3}}

On the Number of Minimal Separators in Graphs

2n-13 stable sets, when the number of agents,