License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2022.20
URN: urn:nbn:de:0030-drops-159900
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Magen, Roey ; Naor, Moni

Mirror Games Against an Open Book Player

LIPIcs-FUN-2022-20.pdf (0.7 MB)


Mirror games were invented by Garg and Schnieder (ITCS 2019). Alice and Bob take turns (with Alice playing first) in declaring numbers from the set {1,2, …, 2n}. If a player picks a number that was previously played, that player loses and the other player wins. If all numbers are declared without repetition, the result is a draw. Bob has a simple mirror strategy that assures he won't lose and requires no memory. On the other hand, Garg and Schenier showed that every deterministic Alice needs memory of size linear in n in order to secure a draw.
Regarding probabilistic strategies, previous work showed that a model where Alice has access to a secret random perfect matching over {1,2, …, 2n} allows her to achieve a draw in the game w.p. a least 1-1/n and using only polylog bits of memory.
We show that the requirement for secret bits is crucial: for an "open book" Alice with no secrets (Bob knows her memory but not future coin flips) and memory of at most n/4c bits for any c ≥ 2, there is a Bob that wins w.p. close to 1-{2^{-c/2}}.

BibTeX - Entry

  author =	{Magen, Roey and Naor, Moni},
  title =	{{Mirror Games Against an Open Book Player}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{20:1--20:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-159900},
  doi =		{10.4230/LIPIcs.FUN.2022.20},
  annote =	{Keywords: Mirror Games, Space Complexity, Eventown-Oddtown}

Keywords: Mirror Games, Space Complexity, Eventown-Oddtown
Collection: 11th International Conference on Fun with Algorithms (FUN 2022)
Issue Date: 2022
Date of publication: 23.05.2022

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