A Generalized Coupon Collector Problem

TL;DR

This paper analyzes a generalized coupon collector problem where multiple coupons are drawn each time, showing asymptotic behavior and providing an exact algorithm for finite cases, supported by numerical verification.

Contribution

It introduces a generalized model with multiple coupons per draw, derives asymptotic formulas, and develops an exact algorithm for finite cases.

Findings

Asymptotic average runs needed: (n log n)/d + (n/d)(m-1) log log n + O(mn)

Exact algorithm for finite cases to compute average runs

Numerical results confirm theoretical predictions

Abstract

This paper provides analysis to a generalized version of the coupon collector problem, in which the collector gets $d$ distinct coupons each run and she chooses the one that she has the least so far. On the asymptotic case when the number of coupons $n$ goes to infinity, we show that on average $\frac{n\log n}{d} + \frac{n}{d}(m-1)\log\log{n}+O(mn)$ runs are needed to collect $m$ sets of coupons. An efficient exact algorithm is also developed for any finite case to compute the average needed runs exactly. Numerical examples are provided to verify our theoretical predictions.