Guessing about Guessing: Practical Strategies for Card Guessing with Feedback

Diaconis, Persi; Graham, Ron; Spiro, Sam

doi:10.48550/arxiv.2012.04019

Cited by 6 publications

(10 citation statements)

References 12 publications

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“…Thus in the adversarial models one can not hope for Guesser to get asymptotically more than m correct guesses, which they can always guarantee by guessing the same card type each round. When the deck is shuffled uniformly at random, it is known [6] that Guesser can play so that they get m + Ω(m 1/2 ) correct guesses when n is sufficiently large, but we do not know of any such strategy that works in the adversarial setting. Question 5.1.…”

Section: Discussionmentioning

confidence: 99%

“…This game is called the complete feedback model, and it was motivated by several real world problems related to clinical trials [2,7] and to extrasensory perception experiments [4]. We refer the interested reader to [6] for more information on the history of this model, as well as to variants of the model which involve different levels of feedback.…”

Section: Introductionmentioning

confidence: 99%

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Online Card Games

Spiro

2021

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Consider the following one player game. A deck containing m copies of n different card types is shuffled uniformly at random. Each round the player tries to guess the next card in the deck, and then the card is revealed and discarded. It was shown by Diaconis, Graham, He, and Spiro that if m is fixed, then the maximum expected number of correct guesses that the player can achieve is asymptotic to H m log n, where H m is the mth harmonic number.In this paper we consider an adversarial version of this game where a second player shuffles the deck according to some (possibly non-uniform) distribution. We prove that a certain greedy strategy for the shuffler is the unique optimal strategy in this game, and that the guesser can achieve at most log n expected correct guesses asymptotically for fixed m against this greedy strategy.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Online Card Games

Spiro

2021

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“…As for other variants of Card Guessing, the literature considers Card Guessing with partial feedback (was the guess correct or not) and no feedback at all. The optimal [10] and near-optimal [8] guessing strategies for partial feedback requires little to no memory, and so goes for the optimal strategy for no feedback [9]. We suggest the study of a general theory of when we can convert a feedback type into a low memory guessing.…”

Section: Low Memory Dealer: a Conjecturementioning

confidence: 93%

Keep That Card in Mind: Card Guessing with Limited Memory

Menuhin¹,

Naor²

2021

Preprint

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A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of n cards (labeled 1, ..., n). For n turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then the card is discarded from the deck. The Guesser receives a point for each correctly guessed card.With perfect memory, a Guesser can keep track of all cards that were played so far and pick at random a card that has not appeared so far, yielding in expectation ln n correct guesses. With no memory, the best a Guesser can do will result in a single guess in expectation.We consider the case of a memory bounded Guesser that has m < n memory bits. We show that the performance of such a memory bounded Guesser depends much on the behavior of the Dealer. In more detail, we show that there is a gap between the static case, where the Dealer draws cards from a properly shuffled deck or a prearranged one, and the adaptive case, where the Dealer draws cards thoughtfully, in an adversarial manner. Specifically:1. We show a Guesser with O(log 2 n) memory bits that scores a near optimal result against any static Dealer.2. We show that no Guesser with m bits of memory can score better than O( √ m) correct guesses, thus, no Guesser can score better than min{ √ m, ln n}, i.e., the above Guesser is optimal.3. We show an efficient adaptive Dealer against which no Guesser with m memory bits can make more than ln m + 2 ln log n + O(1) correct guesses in expectation.These results are (almost) tight, and we prove them using compression arguments that harness the guessing strategy for encoding.

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“…Therefore, we can apply SIS to find the number of correct card guesses using greedy for large decks of cards. Note that in the literature, the exact expectation of correct guesses using greedy was only known for small deck sizes (see e.g., [18,20]).…”

Section: Applicationsmentioning

confidence: 99%

Sequential importance sampling for estimating expectations over the space of perfect matchings

Alimohammadi¹,

Diaconis²,

Roghani³

et al. 2021

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This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1 − )-approximation for the number of perfect matchings of a λ-dense bipartite graph, using) samples. With size n on each side and for 1 2 > λ > 0, a λ-dense bipartite graph has all degrees greater than (λ + 1 2 )n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements.Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

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Guessing about Guessing: Practical Strategies for Card Guessing with Feedback

Cited by 6 publications

References 12 publications

Online Card Games

Online Card Games

Keep That Card in Mind: Card Guessing with Limited Memory

Sequential importance sampling for estimating expectations over the space of perfect matchings

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