Card guessing and the birthday problem for sampling without replacement

He, Jimmy; Ottolini, Andrea

doi:10.48550/arxiv.2108.07355

Cited by 5 publications

(16 citation statements)

References 24 publications

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“…Remark 1.2. In particular, the Theorem applies to the case m i (n) ≡ m for a fixed m. The result about the mean was already shown in [DG81,HO21]. 1.3.…”

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confidence: 65%

Central limit theorem in complete feedback games

Ottolini¹,

Tripathi²

2023

Preprint

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Consider a well-shuffled deck of cards of n different types where each type occurs m times. In a complete feedback game, a player is asked to guess the top card from the deck. After each guess, the top card is revealed to the player and is removed from the deck. The total number of correct guesses in a complete feedback game has attracted significant interest in last few decades. Under different regimes of m, n, the expected number of correct guesses, under the greedy (optimal) strategy, has been obtained by various authors, while there are not many results available about the fluctuations. In this paper, we establish a central limit theorem with Berry-Esseen bounds when m is fixed and n is large. Our results extend to the case of decks where different types may have different multiplicity, under suitable assumptions.

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“…Remark 1.2. In particular, the Theorem applies to the case m i (n) ≡ m for a fixed m. The result about the mean was already shown in [DG81,HO21]. 1.3.…”

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confidence: 65%

Central limit theorem in complete feedback games

Ottolini¹,

Tripathi²

2023

Preprint

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“…The leading order term is H m H n " ln n ln m which, for m fixed, is logarithmic growth in n. The second term in the expansion only depends on m and is thus a constant. Empirically, the result is accurate even for small values of m, n (see [10]).…”

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confidence: 80%

“…A natural remaining question is what happens when both m, n Ñ 8 with the original Zener setup m " n being perhaps particularly interesting. Our main result covers a wide range of these parameters and is applicable as long as the number of different cards n is slightly smaller than exponential in the number of different types m. Such a restriction is necessary: when n becomes disproportionately large compared to m, the result of He & Ottolini [10] shows the behavior to be different.…”

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confidence: 94%

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Guessing cards with complete feedback

Ottolini¹,

Steinerberger²

2022

Preprint

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We consider the following game that has been used as a way of testing claims of extrasensory perception (ESP). One is given a deck of mn cards comprised of n distinct types each of which appears exactly m times: this deck is shuffled and then cards are discarded from the deck one at a time from top to bottom. At each step, a player (whose psychic powers are being tested) tries to guess the type of the card currently on top, which is then revealed to the player before being discarded. We study the expected number Sn,m of correct predictions a player can make: one could always guess the exact same type of card which shows that one can achieve Sn,m ě m. We prove that the optimal (non-psychic) strategy is just slightly better than that and Sn,m " m `π ? 2 ? m ln n `op ? m ln nq whenever pln nq 3`ε ! m. This is very different from the case where m is fixed and n Ñ 8 (He & Ottolini) and similar to the case of fixed n and m Ñ 8 (Graham & Diaconis). The case m " n answers a question of Diaconis.

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“…In [12], a semirestricted version of another classical zero-sum game, Matching Pennies, was introduced. This semi-restricted game was motivated by a certain card guessing game studied by Diaconis and Graham [2] which has recently received a fair amount of attention, see for example [1,3,6,8,10].…”

Section: A More General Settingmentioning

confidence: 99%

Semi-restricted Rock, Paper, Scissors

Spiro¹,

Surya²,

Zeng³

2022

Preprint

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Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of 3n rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly n times during the 3n rounds, while Norman is allowed to play normally without any restrictions. Answering a question of Spiro, we show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is Θ( √ n). Moreover, we study semi-restricted versions of general zero sum games and prove a number of results concerning their optimal strategies and expected scores, which in particular implies our results for semi-restricted RPS.

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Card guessing and the birthday problem for sampling without replacement

Cited by 5 publications

References 24 publications

Central limit theorem in complete feedback games

Central limit theorem in complete feedback games

Guessing cards with complete feedback

Semi-restricted Rock, Paper, Scissors

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