Low communication protocols for fair allocation of indivisible goods

TL;DR

This paper investigates the communication complexity of fair division of indivisible goods among multiple agents, providing bounds and protocols for various fairness notions and valuation types.

Contribution

It introduces new bounds and protocols for communication-efficient fair allocation, including approximate proportionality and MMS allocations under different valuation models.

Findings

Expected communication for MMS allocations is constant for unit demand valuations.

Lower bounds established for binary and 2-valued additive valuations.

Protocols achieve $O(rac{m}{n})$ bits for approximate proportional allocations.

Abstract

We study the multi-party randomized communication complexity of computing a fair allocation of $m$ indivisible goods to $n < m$ equally entitled agents. We first consider MMS allocations, allocations that give every agent at least her maximin share. Such allocations are guaranteed to exist for simple classes of valuation functions. We consider the expected number of bits that each agent needs to transmit, on average over all agents. For unit demand valuations, we show that this number is only $O(1)$ (but $\Theta(\log n)$, if one seeks EF1 allocations instead of MMS allocations), for binary additive valuations we show that it is $\Theta(\log \frac{m}{n})$, and for 2-valued additive valuations we show a lower bound of $\Omega(\frac{m}{n})$. For general additive valuations, MMS allocations need not exist. We consider a notion of {\em approximately proportional} (Aprop) allocations, that approximates proportional allocations in two different senses, being both Prop1 (proportional up to one item), and $\frac{n}{2n-1}$-TPS (getting at least a $\frac{n}{2n-1}$ fraction of the {\em truncated proportional share}, and hence also at least a $\frac{n}{2n-1}$ fraction of the MMS). We design randomized protocols that output Aprop allocations, in which the expected average number of bits transmitted per agent is $O(\log m)$. For the stronger notion of MXS ({\em minimum EFX share}) we show a lower bound of $\Omega(\frac{m}{n})$.