New results for the Coupon Collector's problem with swapping and replacement

TL;DR

This paper investigates the Coupon Collector's Problem with replacement and swapping, revealing that key ratios depend mainly on the percentage of replacement stickers and the number of collectors, enabling cost estimation without full simulation.

Contribution

It provides new simulation-based results and a proof for a specific case, showing how replacement and swapping influence collection costs and variances.

Findings

Ratios depend mainly on replacement percentage and number of collectors

Cost and variance estimations are possible via simple calculations and tables

Asymptotically, replacement has a stronger effect than swapping

Abstract

This paper focuses on the Coupon Collector's Problem with replacement (limited purchasing of missing stickers) and swapping. We have simulated combined strategies and found new results, which we were able to prove for a particular case. The ratio of the average number of stickers needed to fill the album by the album size (number of different stickers needed) and the number of collectors as well as the ratio of the variance and the album size depend approximately only on the percentage of replacement stickers (the limited number of stickers that can be bought directly from the vendor) and the number of collectors, but not on the total album size. Thus collectors can estimate the average cost of completion of an album and its standard deviation just based on basic calculations and a table lookup. Additionally we could show that asymptotocally the effect of replacement is stronger than swapping. ----- In diesem Artikel wird das Sammelbilderproblem mit Nachkaufen und Tauschen untersucht. Wir haben mit Hilfe von Simulationen neue Ergebnisse gefunden und sie auch in einem Spezialfall bewiesen. Das Verh\"altnis der mittleren Anzahl zu kaufender Karten pro Sammler als auch das Verh\"altnis der Varianz zur Albumgr\"o{\ss}e h\"angt in sehr guter N\"aherung nur vom Anteil der nachkaufbaren Karten ab, aber nicht von der Albumgr\"o{\ss}e. Damit k\"onnen Sammler anhand des Prozentsatzes der Nachkaufkarten die mittleren Kosten eines Albums sowie deren Standardabweichung nur mit einer Tabelle und Grundrechenarten bestimmen. Ausserdem konnten wir belegen, dass asymptotisch der Effekt des Nachkaufens st\"arker ist als der des Tauschens.