Low communication protocols for fair allocation of indivisible goods
Uriel Feige
TL;DR
This work assesses multi-party randomized communication complexity for fair allocations of $m$ indivisible goods to $n$ equal-entitlement agents, focusing on MMS andLovas-relaxations such as Prop1, TPS, and Aprop across unit-demand, binary-additive, two-valued, and additive valuations. It establishes tight or near-tight randomized communication bounds: constant per-agent communication for MMS with unit-demand valuations, $\Theta(\log \frac{m}{n})$ for binary additive, and $\Omega(\frac{m}{n})$ for 2-valued additive valuations; Aprop achieves $O(\log m)$ per agent, while MXS incurs $\Omega(\frac{m}{n})$. For EF1, unit-demand cases admit $\Theta(\log n)$ per-agent complexity under randomness, and additive valuations yield $O(m\log m)$ deterministic communication with per-agent improvements in special cases. The paper also links these results to description complexity, shows NP-hardness for MMS under additive valuations, and highlights substantial gaps between different fairness notions and valuation classes, offering a roadmap of open questions on leveraging randomness and approximate MMS in fair allocation. Overall, the results illustrate a nuanced trade-off between achievable fairness guarantees and communication efficiency in decentralized settings with indivisible goods.
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 $Θ(\log n)$, if one seeks EF1 allocations instead of MMS allocations), for binary additive valuations we show that it is $Θ(\log \frac{m}{n})$, and for 2-valued additive valuations we show a lower bound of $Ω(\frac{m}{n})$. 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 $Ω(\frac{m}{n})$.