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Card guessing with partial feedback

Part of: Combinatorial probability

Published online by Cambridge University Press:  15 June 2021

Persi Diaconis ,
Ron Graham ,
Xiaoyu He  and
Sam Spiro
Persi Diaconis
Affiliation:
Department of Mathematics and Statistics, Stanford University, Stanford, CA94305, USA
Ron Graham
Affiliation:
Department of Computer Science and Engineering, UCSD, La Jolla, CA92093, USA
Xiaoyu He
Affiliation:
Deparment of Mathematics, Stanford University, Stanford, CA94305, USA
Sam Spiro*
Affiliation:
Department of Mathematics, UCSD, La Jolla, CA92093, USA
*
*Corresponding author. Email: sspiro@ucsd.edu
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Abstract

Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

Keywords

Card GuessingShufflingGames

MSC classification

Primary: 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 1 , January 2022 , pp. 1 - 20
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Research supported by NSF Grant DMS-1954042.

‡

Research supported by NSF Graduate Research Fellowship Grant No. DGE-1656518.

Research supported by NSF Graduate Research Fellowship Grant No. DGE-1650112.

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