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.