Fair Division via the Cake-Cutting Share

TL;DR

Novel notions of fair shares from the perspective of individual agents via the cake-cutting process are defined, which generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy.

Abstract

In this paper, we consider the classic fair division problem of allocating $m$ divisible items to $n$ agents with linear valuations over the items. We define novel notions of fair shares from the perspective of individual agents via the cake-cutting process. These shares generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy. We study what fraction (approximation) of these shares are achievable in the worst case, and present tight and non-trivial approximation bounds as a function of $n$ and $m$. In particular, we show a tight approximation bound of $Θ(\sqrt{n})$ for various notions of such shares. We show this bound via a novel application of dual fitting, which may be of independent interest. We also present a bound of $O(m^{2/3})$ for a strict notion of share, with an almost matching lower bound. We further develop weaker notions of shares whose approximation bounds interpolate smoothly between proportionality and the shares described above. We finally present empirical results showing that our definitions lead to more reasonable shares than the standard fair share notion of proportionality.

Figures (2)

• Figure 1: Figures (a), (b): Box plot of approximation Ratios given by $\mathrm{PROP}$, $\mathrm{CCS}$, and $\mathrm{EFS}$. Figures (c), (d): Plot of mean and inter-quartile range of approximation ratio as a function of $\Delta$ for $\mathrm{EFS}^{\Delta}$.
• Figure 2: Approximation Ratios given by $\mathrm{PROP}$, $\mathrm{CCS}$, and $\mathrm{EFS}$ for Bernoulli distribution and Intrinsic Value.

Theorems & Definitions (19)

• Theorem 1: Proved in Section \ref{['sec:main1']}
• Theorem 2: Proved in Section \ref{['sec:lb1']}
• Theorem 3: Proved in Section \ref{['sec:main2']}
• Theorem 4: Proved in Section \ref{['app:partial']}
• Definition 5: Cake-cutting Share
• Definition 6: Envy-free Share
• Lemma 7
• proof
• Lemma 8
• proof