On Hierarchies of Fairness Notions in Cake Cutting: From Proportionality to Super Envy-Freeness
Arnav Mehra, Alexandros Psomas

TL;DR
This paper introduces two new hierarchies of fairness notions in cake cutting, called CHB and CLB, analyzing their relationships with existing fairness concepts and establishing bounds on the query complexity for computing these allocations.
Contribution
The paper defines CHB and CLB hierarchies, explores their relationships with envy-freeness and proportionality, and provides bounds on the query complexity for computing these fairness notions.
Findings
CHB-$n$ allocations are computable with $O(n^4)$ queries.
CHB-$2$ allocations require $ heta(n^2)$ queries to compute.
CLB-$2$ allocations cannot be computed with a bounded number of queries.
Abstract
We consider the classic cake-cutting problem of producing fair allocations for agents, in the Robertson-Webb query model. In this model, it is known that: (i) proportional allocations can be computed using queries, and this is optimal for deterministic protocols; (ii) envy-free allocations (a subset of proportional allocations) can be computed using queries, and the best known lower bound is ; (iii) perfect allocations (a subset of envy-free allocations) cannot be computed using a bounded (in ) number of queries. In this work, we introduce two hierarchies of new fairness notions: Complement Harmonically Bounded (CHB) and Complement Linearly Bounded (CLB). Intuitively, these notions of fairness ask that, for every agent , the collective value that a group of agents has (from the perspective of agent ) is…
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Taxonomy
TopicsOptimization and Packing Problems · Public Spaces through Art · Sustainable Supply Chain Management
