Thou Shalt Covet The Average Of Thy Neighbors' Cakes
Jamie Tucker-Foltz

TL;DR
This paper establishes a quadratic lower bound on the query complexity for achieving local proportionality in cake-cutting, showing it is computationally harder than proportionality but easier than envy-freeness.
Contribution
It proves an $oldsymbol{ ext{Omega}(n^2)}$ lower bound on the query complexity for local proportionality, distinguishing it from other fairness notions.
Findings
Local proportionality requires $oldsymbol{ ext{Omega}(n^2)}$ queries.
It is weaker than envy-freeness but harder than proportionality.
The result confirms local proportionality's higher computational complexity.
Abstract
We prove an 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 , and generally incomparable to proportionality, which has query complexity . 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.
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Taxonomy
TopicsGame Theory and Voting Systems · Game Theory and Applications · Auction Theory and Applications
