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

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 $n$ 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 $O\left( n^{n^{n^{n^{n^{n}}}}} \right)$ 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 $k \in [2, \lceil n/2 \rceil - 1]$, CLB-$k$ allocations are incomparable to envy-free allocations. For $k \geq \lceil n/2 \rceil$, 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.