The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic

TL;DR

This paper analyzes the probability that in a two-player coupon collection game, the eventual winner was never behind, revealing asymptotic behavior as the number of coupon types grows large.

Contribution

It provides a novel asymptotic analysis of the ballot-type problem in a two-player coupon collection game using renewal decomposition techniques.

Findings

The probability that the eventual winner was never behind is asymptotically 2/d as d→∞.

The tie boundary analysis involves a renewal decomposition with explicit entrance distribution.

The scaled level of the first tie converges to a Rayleigh distribution, and the leader's survival probability relates to Catalan numbers.

Abstract

We study the two-player coupon-collector competition in which two independent collectors draw one coupon each per round from a set of $d$ equally likely coupon types. Myers and Wilf gave finite formulae for several two-player events and explicitly left open the ballot-type problem of finding the probability that the ultimate winner was never behind. We prove that this probability satisfies $$ b_d \sim \frac{2}{d}, \qquad d\to\infty .$$ The proof uses a renewal decomposition at the tie boundary. The first one-sided tie-break has an explicit entrance distribution; its level, scaled by $d^{1/2}$, converges to a Rayleigh law; and, after the break, the leader's survival probability is governed by a Catalan, or gambler's-ruin, harmonic. The main estimate shows that the accumulated defect of this comparison harmonic in the exact simultaneous-round chain is negligible.