Collecting Coupons is Faster with Friends

TL;DR

This paper introduces a theoretical analysis of a distributed coupon collector problem where multiple collectors exchange coupons, showing that such interactions can significantly reduce the sampling effort needed to complete sets.

Contribution

First theoretical analysis of distributed coupon collection with exchanges, demonstrating efficiency gains and connecting to traditional variants.

Findings

Exchanges reduce the number of coupons sampled per collector.

Theoretical bounds are asymptotically tight in most cases.

Raises open questions for finer-grained analysis.

Abstract

In this note, we introduce a distributed twist on the classic coupon collector problem: a set of $m$ collectors wish to each obtain a set of $n$ coupons; for this, they can each sample coupons uniformly at random, but can also meet in pairwise interactions, during which they can exchange coupons. By doing so, they hope to reduce the number of coupons that must be sampled by each collector in order to obtain a full set. This extension is natural when considering real-world manifestations of the coupon collector phenomenon, and has been remarked upon and studied empirically [Hayes and Hannigan 2006, Ahmad et al. 2014, Delmarcelle 2019]. We provide the first theoretical analysis for such a scenario. We find that "coupon collecting with friends" can indeed significantly reduce the number of coupons each collector must sample, and raises interesting connections to the more traditional variants of the problem. While our analysis is in most cases asymptotically tight, there are several open questions raised, regarding finer-grained analysis of both "coupon collecting with friends", and of a long-studied variant of the original problem in which a collector requires multiple full sets of coupons.