Mirror Games Against an Open Book Player

Authors Roey Magen, Moni Naor

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Author Details

Roey Magen
  • Weizmann Institute of Science, Rehovot, Israel
Moni Naor
  • Weizmann Institute of Science, Rehovot, Israel


We thank Boaz Menuhin for very useful discussions and comments and thank Ehud Friedgut for discussions concerning an extension of Berlekamp’s Theorem. We thank the anonymous referees for numerous helpful comments.

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Roey Magen and Moni Naor. Mirror Games Against an Open Book Player. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 20:1-20:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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}}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming models
  • Mirror Games
  • Space Complexity
  • Eventown-Oddtown


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