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Optimal no-feedback card guessing after a fixed number of riffle shuffles — arXiv:2611.04339

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.04339
License (as archived): CC0-1.0
Archived: 2026-11-04T12:47:15.429Z
SHA-256 of the archived text: e7b13f4f42f94c5dcff570852ae3236b9a472a55e13e78f141c847c710cbc841
Stable URL of this page: https://metraipolis.com/documento/dc_cc13799fe4e9

How to cite

Optimal no-feedback card guessing after a fixed number of riffle shuffles — arXiv:2611.04339 — arXiv (en), https://arxiv.org/abs/2611.04339 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cc13799fe4e9 (Metraipolis archive dc_cc13799fe4e9, sha256 e7b13f4f42f94c5dcff570852ae3236b9a472a55e13e78f141c847c710cbc841)

BibTeX

@misc{dccc13799fe4e9,
  title = {Optimal no-feedback card guessing after a fixed number of riffle shuffles — arXiv:2611.04339},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.04339},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cc13799fe4e9; sha256 e7b13f4f42f94c5dcff570852ae3236b9a472a55e13e78f141c847c710cbc841},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.04339. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

An ordered deck of $n$ cards is subjected to $k$ independent Gilbert--Shannon--Reeds riffle shuffles. A player then guesses each card without feedback, with repeated guesses allowed. For every fixed $k\ge1$, we prove that, once $n$ is sufficiently large, an optimal guess at position $i$ in the top half of the deck is $\lfloor i/2^k\rfloor+1$; the bottom-half strategy is obtained by reflection. We also show that the optimal expected number of correct guesses is $2\sqrt{n/((2^k-1)\pi)}+O_k(1)$. Our starting point is the result of our earlier work that the label at each position is distributed as an equally weighted mixture of $2^k$ components, each a shifted sum of two independent binomial random variables (a Poisson-binomial distribution). The proof of the optimal strategy combines three ingredients: a uniform concentration estimate that separates the mixture components, exact binomial inequalities that control the gaps between competing peaks, and a coefficient comparison that resolves modal ties. The expected-score asymptotic is proved independently of the strategy, using only the peak heights of the mixture components.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Probability (math.PR) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.04339 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-04T12:32:02.007Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).