The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910207288803328 |
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| author | Long, Christopher D. |
| author_facet | Long, Christopher D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09641 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic Long, Christopher D. Probability Combinatorics 60C05 (Primary) 60J10, 60F05, 05A16 (Secondary) 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. |
| title | The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic |
| topic | Probability Combinatorics 60C05 (Primary) 60J10, 60F05, 05A16 (Secondary) |
| url | https://arxiv.org/abs/2605.09641 |