Thou Shalt Covet The Average Of Thy Neighbors' Cakes

We prove an $\Omega(n^2)$ lower bound on the query complexity of local proportionality in the Robertson-Webb cake-cutting model. Local proportionality requires that each agent prefer their allocation to the average of their neighbors' allocations in some undirected social network. It is a weaker fairness notion than envy-freeness, which also has query complexity $\Omega(n^2)$, and generally incomparable to proportionality, which has query complexity $\Theta(n \log n)$. This result separates the complexity of local proportionality from that of ordinary proportionality, confirming the intuition that finding a locally proportional allocation is a more difficult computational problem

Chore Cutting: Envy and Truth

We study the fair division of divisible bad resources with strategic agents who can manipulate their private information to get a better allocation. Within certain constraints, we are particularly interested in whether truthful envy-free mechanisms exist over piecewise-constant valuations. We demonstrate that no deterministic truthful envy-free mechanism can exist in the connected-piece scenario, and the same impossibility result occurs for hungry agents. We also show that no deterministic, truthful dictatorship mechanism can satisfy the envy-free criterion, and the same result remains true for non-wasteful constraints rather than dictatorship. We further address several related problems and directions.Comment: arXiv admin note: text overlap with arXiv:2104.07387 by other author

Cutting a Cake Is Not Always a 'Piece of Cake': A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents' valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory