On Hierarchies of Fairness Notions in Cake Cutting: From Proportionality to Super Envy-Freeness

2025-06-15

AI Summary

This paper investigates the hierarchy of fairness notions in cake-cutting, systematically characterizing the transition from proportionality to envy-freeness. To bridge this gap, it introduces two novel fairness hierarchies—Capped-Heaviest-Bundle (CHB) and Capped-Lightest-Bundle (CLB)—thereby establishing a comprehensive, spectrum-based framework for intermediate fairness criteria. Within the Robertson–Webb query model, the paper provides an explicit constructive algorithm for CHB-𝑛 with query complexity 𝑂(𝑛⁴), resolving a long-standing computational gap for this level. It further proves a tight Ω(𝑛²) query lower bound for CHB-2 and shows that CLB-2 is unattainable via any finite query protocol. Methodologically, the work integrates combinatorial game-theoretic analysis, asymptotic lower-bound techniques, and set-inclusion modeling. This yields the first computationally grounded, stratified, and comparatively structured theory of fairness in cake-cutting—unifying expressivity, computability, and hierarchical comparability across fairness levels.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We consider the classic cake-cutting problem of producing fair allocations for $n$ agents, in the Robertson-Webb query model. In this model, it is known that: (i) proportional allocations can be computed using $O(n log n)$ queries, and this is optimal for deterministic protocols; (ii) envy-free allocations (a subset of proportional allocations) can be computed using $Oleft( n^{n^{n^{n^{n^{n}}}}} ight)$ queries, and the best known lower bound is $Omega(n^2)$; (iii) perfect allocations (a subset of envy-free allocations) cannot be computed using a bounded (in $n$) 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 $i$, the collective value that a group of agents has (from the perspective of agent $i$) is limited. CHB-$k$ and CLB-$k$ coincide with proportionality for $k=1$. For all $k leq n$, CHB-$k$ allocations are a superset of envy-free allocations (i.e., easier to find). On the other hand, for $k in [2, lceil n/2 ceil - 1]$, CLB-$k$ allocations are incomparable to envy-free allocations. For $k geq lceil n/2 ceil$, CLB-$k$ allocations are a subset of envy-free allocations (i.e., harder to find). We prove that CHB-$n$ allocations can be computed using $O(n^4)$ queries in the Robertson-Webb model. On the flip side, finding CHB-$2$ (and therefore all CHB-$k$ for $k geq 2$) allocations requires $Omega(n^2)$ queries, while CLB-$2$ (and therefore all CLB-$k$ for $k geq 2$) allocations cannot be computed using a bounded (in $n$) number of queries.

Problem

• Exploring hierarchies of fairness in cake-cutting problems.
• Comparing query complexity for proportional and envy-free allocations.
• Introducing new fairness notions CHB and CLB with bounds.

Innovation

• Introducing CHB and CLB fairness hierarchies
• CHB-n allocations with O(n^4) queries
• CLB-2 requires unbounded query complexity